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+parse/train/SklTQCNtvS/SklTQCNtvS_layout.pdf filter=lfs diff=lfs merge=lfs -text +parse/train/rJeW1yHYwH/rJeW1yHYwH_span.pdf filter=lfs diff=lfs merge=lfs -text +parse/train/rJeW1yHYwH/rJeW1yHYwH_layout.pdf filter=lfs diff=lfs merge=lfs -text +parse/train/rJeW1yHYwH/rJeW1yHYwH_origin.pdf filter=lfs diff=lfs merge=lfs -text +parse/train/BJlBSkHtDS/BJlBSkHtDS_layout.pdf filter=lfs diff=lfs merge=lfs -text +parse/train/BJlBSkHtDS/BJlBSkHtDS_span.pdf filter=lfs diff=lfs merge=lfs -text +parse/train/BJlBSkHtDS/BJlBSkHtDS_origin.pdf filter=lfs diff=lfs merge=lfs -text +parse/train/IOqr2ZyXHz1/IOqr2ZyXHz1_span.pdf filter=lfs diff=lfs merge=lfs -text diff --git a/parse/dev/E0zOKxQsZhN/E0zOKxQsZhN.md b/parse/dev/E0zOKxQsZhN/E0zOKxQsZhN.md new file mode 100644 index 0000000000000000000000000000000000000000..931255f79718f1c81fcedb33182a47e3d158e2a9 --- /dev/null +++ b/parse/dev/E0zOKxQsZhN/E0zOKxQsZhN.md @@ -0,0 +1,505 @@ +# RECURRENT MODEL-FREE RL IS A STRONGBASELINE FOR MANY POMDPS + +Anonymous authors Paper under double-blind review + +# ABSTRACT + +Many problems in RL, such as meta RL, robust RL, and generalization in RL, can be cast as POMDPs. In theory, simply augmenting model-free RL with memory, such as recurrent neural networks, provides a general approach to solving all types of POMDPs. However, prior work has found that such recurrent model-free $R L$ methods tend to perform worse than more specialized algorithms that are designed for specific types of POMDPs. This paper revisits this claim. We find that a careful architecture and hyperparameter decisions yield a recurrent model-free implementation that performs on par with (and occasionally substantially better than) more sophisticated recent techniques in their respective domains. We also release a simple and efficient implementation of recurrent model-free RL for future work to use as a baseline for POMDPs.1 + +# 1 INTRODUCTION + +While reinforcement learning (RL) is often cast as the problem of learning a single fully observable task, also known as MDP, training and testing on that same task, most real-world applications of RL demand some degree of transfer and partial observability. For example, visual navigation (Zhu et al., 2017) requires adaptation to unseen scenes with occlusion in observations, and human-robot collaboration requires that robots infer the intentions of human collaborators. (Chen et al., 2018). + +Many subareas in RL study problems that are special cases of POMDPs, and we summarize them in Table 1. For example, meta RL (Duan et al., 2016; Schmidhuber, 1987; Thrun & Pratt, 2012; Wang et al., 2017) is a POMDP where certain aspects of the reward function or (less commonly) dynamics function are unobserved but held constant through one episode. The robust RL problem (Bagnell et al., 2001; Pattanaik et al., 2018; Pinto et al., 2017; Rajeswaran et al., 2017a) assumes that certain aspects of the dynamics or reward function are + +![](images/0263b0377870a9e706cac9a587cc7239ea6cfcc9daae0f0b1f499a4ca8db104b.jpg) +Figure 1: Implementation Matters for Recurrent Model-Free RL. This paper identifies critical design decisions for recurrent model-free RL that outperforms not only prior implementations (e.g. PPO-GRU and A2C-GRU from Kostrikov (2018)), but also purposedesigned methods (e.g. VRM from Han et al. (2020)). We also show Markovian policies as lower bounds for reference. The y-axis is normalized return given the return of oracle policy (Raffin et al., 2021). + +unknown, aiming at finding optimal policies that perform against adversarially-chosen perturbations. Generalization in RL (Cobbe et al., 2019; Packer et al., 2018; Whiteson et al., 2011; Zhang et al., 2018a) focuses on unobserved aspects of the dynamics or reward function that are novel during testing, using an average-case objective instead of a worst-case objective like robust RL. Recent work has proposed efficient and performant algorithms for solving these specialized problem settings. However, these algorithms often make assumptions that preclude their application to other classes of POMDPs. For example, methods for robust RL are rarely used for the meta RL setting due to objective mismatch; methods for meta RL are rarely used for general POMDPs due to the stationarity assumption in meta RL. + +Nonetheless, many prior works have used a simple baseline that is applicable to all POMDPs: model-free RL equipped with a recurrent policy and (sometimes) value function (Duan et al., 2016; Fakoor et al., 2020; Igl et al., 2018; Packer et al., 2018; Rakelly et al., 2019; Wang et al., 2017; Yu et al., 2019). We will refer to this approach as recurrent model-free RL. This baseline is simultaneously simple (requiring changing only a few lines of code from a model-free RL algorithm) and general. However, prior work has consistently found that recurrent model-free RL performs poorly across a wide range of problem settings, including meta RL (Rakelly et al., 2019; Zintgraf et al., 2020), general POMDPs (Han et al., 2020; Igl et al., 2018), robust RL (Zhang et al., 2021), and generalization in RL (Packer et al., 2018). One common explanation is that specialized algorithms that are tailored to specific types of POMDPs are very likely to outperform recurrent model-free RL because they (implicitly) encode inductive biases for solving these specific tasks. For example, algorithms for meta RL may leverage the assumption that the underlying dynamics (while unknown) are fixed, and the underlying goals are fixed within one episode (Rakelly et al., 2019; Zintgraf et al., 2020); algorithms for robust RL may assume that the dynamics parameters are known (Rajeswaran et al., 2017a) and dynamics is Lipschitz continuous (Jiang et al., 2021). + +This paper challenges this explanation. We argue that, contrary to popular belief, recurrent modelfree RL is competitive with recent state-of-the-art algorithms across a range of different POMDP settings. Similar to prior work in Markovian on-policy RL methods (Andrychowicz et al., 2021; Engstrom et al., 2020), our experiments show that implementation in recurrent model-free RL matters. Fig. 1 shows a typical scenario in PyBullet occlusion environments (Coumans & Bai, 2016) to support this argument. Through extensive experiments, we show that the careful design and implementation of recurrent model-free RL is critical to its performance. Design decisions, such as the actor-critic architecture, conditioning on previous actions and/or rewards, the underlying model-free RL algorithms, and context length in RNNs, are especially crucial. + +The main contribution of this paper is a performant implementation of recurrent-model free RL. We demonstrate that simple yet important design decisions, such as the underlying RL algorithm and the context length, yield a recurrent model-free RL algorithm that performs on par with prior specialized POMDP algorithms on the environments those algorithms were designed to solve. Ablation experiments identify the importance of these design decisions. We also open-sourced our code that is easy to use and memory-efficient. + +# 2 BACKGROUND + +MDP. A Markov decision process (MDP) (Bellman, 1957) is a tuple $( S , A , T , T _ { 0 } , R , H , \gamma )$ , where $s$ is the set of states, $\mathcal { A }$ is the set of actions, $T : \mathcal { S } \times \mathcal { A } \times \mathcal { S } [ 0 , 1 ]$ is the transition function (dynamics), $T _ { 0 } : { \mathcal { S } } [ 0 , 1 ]$ is the initial state distribution, $R : \mathcal { S } \times \mathcal { A } \times \mathcal { S } \to \mathbb { R }$ is the reward function, $H \in \mathbb { N }$ is the time horizon, and $\gamma \in \ [ 0 , 1 )$ is the discount factor. Solving an MDP requires learning a memoryless policy $\pi : \mathcal { S \times A } \to [ 0 , 1 ]$ that maximizes the expected discounted return: $\begin{array} { r } { \pi ^ { * } = \arg \operatorname* { m a x } _ { \pi } \mathbb { E } _ { s _ { t } , a _ { t } , r _ { t } \sim T , \pi } \left[ \sum _ { t = 0 } ^ { H - 1 } \gamma ^ { t } r _ { t + 1 } \ | \ s _ { 0 } \right] } \end{array}$ . For any MDP, there exists an optimal policy that is both memoryless and deterministic (Puterman, 2014). MaxEnt RL algorithms (Ziebart, 2010), such as SAC (Haarnoja et al., 2018a), add an entropy bonus to the RL objective. + +POMDP. A partially observable Markov decision process (POMDP) (Astr ˚ om¨ , 1965) is a tuple $( S , A , \mathcal { O } , T , T _ { 0 } , O , O _ { 0 } , R , H , \gamma )$ , where the underlying process is an MDP $( S , A , T , T _ { 0 } , R , H , \gamma )$ . Let $\mathcal { O }$ be the set of observations and let $O : S \times A \times \mathcal { O } [ 0 , 1 ]$ be the emission function. Let the observable trajectory up to time-step $t$ be $\tau _ { 0 : t } = ( o _ { 0 } , a _ { 0 } , o _ { 1 } , r _ { 1 } , \dots , a _ { t - 1 } , o _ { t } , r _ { t } )$ , the memory-based policy in the most general form is defined as $\pi ( \boldsymbol { a } _ { t } \mid \tau _ { 0 : t } )$ , conditioning on the whole history. At the first time step $t = 0$ , an initial state $s _ { 0 } \sim T _ { 0 } ( \cdot )$ and initial observation $\phantom { } O _ { 0 } \sim O _ { 0 } ( \cdot \mid s _ { 0 } )$ are sampled. At any time-step $t \in \{ 0 , \ldots , H - 1 \}$ , the policy emits the action $a _ { t } \in \mathcal A$ to the system, the system updates the state following the dynamics, $s _ { t + 1 } \sim T ( \cdot \mid s _ { t } , a _ { t } )$ , then the next observation is sampled $\tilde { o _ { t + 1 } } \sim O ( \cdot \mid s _ { t + 1 } , a _ { t } )$ and the reward is computed as $r _ { t + 1 } = R ( s _ { t } , a _ { t } , s _ { t + 1 } )$ . + +We refer to the part of the state $s _ { t }$ at current time-step $t$ that can be directly unveiled from current observation $o _ { t }$ as the observable state sot , and the rest part of the state as the hidden state s ht . We call the hidden state $s _ { t } ^ { h }$ stationary if it does not change within an episode. In this scenario, the policy objective can be rewritten as $\pi ^ { * } =$ arg $\begin{array} { r } { \operatorname* { m a x } _ { \boldsymbol { \pi } } \mathbb { E } _ { s ^ { h } \sim T _ { 0 } } \left[ \mathbb { E } _ { s _ { t } , a _ { t } , r _ { t } \sim T , O , O _ { 0 } , \boldsymbol { \pi } } \left[ \sum _ { t = 0 } ^ { H - 1 } \gamma ^ { t } r _ { t + 1 } \ | \ s ^ { h } \right] \right] } \end{array}$ for the average-case POMDP objec + +Table 1: The summary of selected POMDP subareas. For each subarea, we list the information of the hidden state $s ^ { h }$ including its appearance in dynamics and reward function and its stationarity during one trajectory. We also list the policy input space that are connected with the hidden states, where $\bigcirc$ , a, r, and d refer to the sequence of observations, actions, rewards, and done signals, respectively. Finally, we list the RL objective in terms of average-case or worst-case, and whether there is a domain shift between training and testing environments. We append the check $( \checkmark )$ or cross mark $( { \pmb x } )$ with $^ *$ if it applies to some but not all the work in that subarea. The notation in this table will be covered in Sec. 2. + +
Subareashin dynam-sics?sin re-ward?Isshsta-tionary?Policy in-put spaceRL objec-tiveDomainshift?
“Standard”POMDP√√XoarAvgX
Meta RLX*√√oardAvgX
Robust RL*XoaWorst×
Generalizationin RL*X**oaAvg
+ +tive, or POMD $\begin{array} { r } { \pi ^ { * } = \arg \operatorname* { m a x } _ { \pi } \operatorname* { m i n } _ { s ^ { h } \in \mathrm { s u p p } ( T _ { 0 } ) } \mathbb { E } _ { s _ { t } , a _ { t } , r _ { t } \sim T , O , O _ { 0 } , \pi } \left[ \sum _ { t = 0 } ^ { H - 1 } \gamma ^ { t } r _ { t + 1 } \mid s ^ { h } \right] } \end{array}$ for the worst-case + +# 3 RELATED WORK + +In this section, we discuss several subareas of RL that both explicitly and implicitly solve POMDPs, as well as algorithms proposed for these specialized settings. Table 1 summarizes these subareas. + +RL for “Standard” POMDPs. We use the term “standard” to refer to prior work that explicitly labels the problems studied as POMDPs. Common tasks include scenarios where the states are partially occluded (Heess et al., 2015), different states correspond to the same observation (perceptual aliasing (Whitehead & Ballard, 1990)), random frames are dropped (Hausknecht & Stone, 2015), observations use egocentric images (Zhu et al., 2017), or the observations are perturbed with random noise (Meng et al., 2021). These POMDPs often have hidden states that are non-stationary and affect both the rewards and the dynamics. POMDPs are hard to solve (Littman, 1996; Papadimitriou & Tsitsiklis, 1987) because of the curse of dimensionality: the size of the history grows linearly with the horizon length. Many prior POMDP algorithms (Cassandra et al., 1994; Kaelbling et al., 1998) attempt to infer the state from the past sequence of observations, and then apply standard RL techniques to that inferred state. However, the exact inference requires the knowledge of the dynamics, emission, and reward functions, and is intractable in all except the most simple settings. A common strategy for solving these general POMDPs is to use recurrent policies, which take the entire history of past observations as inputs (Bakker, 2001; Schmidhuber, 1991; Wierstra et al., 2007). This strategy is very simple and general, and can be applied to arbitrary tasks without knowledge of the task structure (e.g., whether the hidden states change within an episode) across long time horizons (Duan et al., 2016). These recurrent strategies can be further subdivided into model-free methods (Hausknecht & Stone, 2015; Heess et al., 2015; Meng et al., 2021) , where the single objective is to maximize the return, and model-based methods (Freeman et al., 2019; Han et al., 2020; Igl et al., 2018; Watter et al., 2015) that have explicit objectives on modeling the belief states and use them as the inputs of memoryless policies. The recurrent model-free RL that we focus on belongs to the class of model-free off-policy memory-based algorithms. + +Meta RL. Meta RL, also called “learning to learn” (Schmidhuber, 1987; Thrun & Pratt, 2012), focuses on POMDPs where some parameters in the rewards or (less commonly) dynamics are varied from episode to episode, but remain fixed within a single episode, which represent different tasks with different values (Humplik et al., 2019). The meta RL setting is almost the same as multitask RL (Wilson et al., 2007; Yu et al., 2019), but differs in that multi-task RL can observe the task parameters, making it an MDP instead of a POMDP. Algorithms for meta RL can be roughly categorized based on how the adaptation step is performed. Gradient-based algorithms (Fakoor et al., 2020; Finn et al., 2017; Hochreiter et al., 2001) run a few gradient steps on the pre-trained models to adapt. Memory or context-based algorithms use RNNs to implicitly adapt, which can be further subdivided into implicit and explicit task inference methods. Implicit task inference methods (Duan et al., 2016; Wang et al., 2017) use RL objective only to learn recurrent policies. Explicit task inference methods (Rakelly et al., 2019; Zintgraf et al., 2020) train an extra inference model to explicitly estimate task embeddings (i.e., a representation of the unobserved parameters) by variational inference. Task embeddings are then used as additional inputs to memoryless policies. + +Robust RL. The goal of robust RL is to find a policy that maximizes returns in the worst-case environments. Early work in the control and operations research community (Khalil et al., 1996; Nilim & Ghaoui, 2005) and RL community (Bagnell et al., 2001; Morimoto & Doya, 2005) focused on linear or finite systems. Prior work designs deep RL algorithms that are robust against a variety of adversarial attacks, including attacks on the dynamics (Jiang et al., 2021; Rajeswaran et al., 2017a), observations (Huang et al., 2017; Pattanaik et al., 2018; Zhang et al., 2021), and actions (Gleave et al., 2020; Pinto et al., 2017; Tessler et al., 2019). Treating the robust RL problem as a POMDP, rather than an MDP (as done in most prior work), unlocks a key capability for RL agents, because agents can use their memory to identify the hidden states of the current adversarial environment, although previous work (Jiang et al., 2021; Rajeswaran et al., 2017a) only train Markovian policies on POMDPs. While some work find memory-based policies are more robust to the adversarial attacks than Markovian policies (Russo & Proutiere \` , 2021; Zhang et al., 2021), they train these baselines in a single MDP without adversaries, which differs from our training setting where the recurrent model-free RL can have access to a set of MDPs. + +Generalization in RL. The goal of generalization in RL is to make RL algorithms perform well in test domains that are unseen during training, which emphasizes the average case on the novel test domains instead of the worse case in the possibly seen test domains as in robust RL. Prior work have studied generalization to initial states in the same MDP (Rajeswaran et al., 2017b; Whiteson et al., 2011; Zhang et al., 2018b), random disturbance in dynamics (Rajeswaran et al., 2017b), states (Stulp et al., 2011), observations (Song et al., 2020; Zhang et al., 2018a), and actions (Srouji et al., 2018), and different modes in procedurally generated games (Cobbe et al., 2019; Farebrother et al., 2018; Justesen et al., 2018). Among them, Packer et al. (2018) provides a benchmark on both in-distribution (ID) and out-of-distribution (OOD) generalization to different dynamics parameters, and Zhao et al. (2019) extends the benchmark by introducing random noise in states, observations, and actions. Algorithms for improving generalization in RL can be roughly divided into classic regularization methods such as weight decay, dropout, batch normalization, and entropy regularization (Cobbe et al., 2020; Farebrother et al., 2018; Igl et al., 2019), model architectures (Raileanu & Fergus, 2021; Srouji et al., 2018), data augmentation through randomization (Lee et al., 2020; Tobin et al., 2017), Although introducing observational noise and the change in dynamics parameters will transform MDPs to POMDPs, few work study memory-based policies such as model-free recurrent RL with mixed results. Same algorithm RL2 (Duan et al., 2016) was found to perform badly in Packer et al. (2018) but relatively well in Yu et al. (2019). + +# 4 DESIGN CONSIDERATIONS FOR RECURRENT MODEL-FREE RL + +Implementing a recurrent model-free RL algorithm requires making a number of design decisions. This section describes the decisions that we found most important to make recurrent model-free RL competitive with more complex, recent algorithms. We will focus on continuous control problems with state-based inputs (i.e., not image-based inputs). Importantly, we assume that the policy can observe the reward and done signals (the end of one episode during one trial (Duan et al., 2016)) from the environment during evaluation. This assumption is common in prior work (Han et al., 2020; Zintgraf et al., 2020), but many recurrent model-free implementations do not provide the agent with information. In the following paragraphs, we will describe the important decision factors in recurrent model-free RL. Table 2 summarizes how prior work and our method makes these design decisions when implementing recurrent model-free RL. + +Recurrent Off-Policy Actor-Critic Architecture. The first important design decision is whether the recurrent policy (actor) and the recurrent Q-value function (critic) use shared RNN encoder (and embedders) or use separate ones. In the experiment section (Sec. 5.2) we will show that a shared encoder would cause large gradient norm in the (off-policy) recurrent actor-critic and thus hinder learning, while separate encoders can greatly mitigate this issue and learn efficiently. This echoes prior work (Fakoor et al., 2020; Meng et al., 2021; Sun et al., 2021; Wang et al., 2020) that also use separate encoders in their (off-policy) recurrent actor-critic. To avoid running an inordinate number of experiments, we will use the separate architecture in the rest of the paper. + +Policy Input Space. The next consideration is the input space of the model-free policy. The maximal input space of policy to emit an action $a _ { t }$ at time $t$ , should be the history of all quantities that the policy has observed, namely the past observations $o _ { 0 : t }$ , the past actions $a _ { 0 : t - 1 }$ , the past rewards $r _ { 0 : t }$ , and the past done signals $d _ { 0 : t }$ , which was already employed in the early work (Duan et al., 2016). Generally, the input space of optimal policy should only depend on the quantities that have connections with hidden states (defined in Sec. 2) (Izadi & Precup, 2005; Poupart & Boutilier, 2002). We show the policy input spaces that are connected with the hidden states for the discussed subareas in the “Inputs” column of Table 1. While prior work often only conditions the recurrent RL baseline on previous observations (and actions) (Han et al., 2020; Igl et al., 2018; Kostrikov, 2018; Meng et al., 2021; Wang et al., 2020; Yang & Nguyen, 2021), our experiments in Sec. 5.2 find that additionally conditioning on other previous information, such as previous rewards, can increase reward by up to $3 0 \%$ . + +Table 2: How the prior work and our method implement the recurrent model-free RL as their own method or baseline. We can see that none of the prior work share the same set of decision variables, some of which have bad choices that may lead to the poor performance reported in the prior work. Our method covers a range of choices in these decision factors and finds the combinations in the last rows that lead to the best performance in terms of the average performance across the experimented environments in each subarea. + +
AlgorithmDomainArchEncoderInputsLenRL
Duan et al. (2016)Meta RLseparateGRUoard1000TRPO,PPO
Wang et al. (2017)Meta RLsharedLSTMoart5-150A2C
Baseline from Rakelly et al. (2019)Meta RLseparateGRUoard100PPO
Baseline from Zintgraf et al. (2020)Meta RLseparateGRUoardMaxA2C,PPO
Baseline from Fakoor et al. (2020)MetaRLseparateGRUoar10-25TD3
Baseline from Yu et al. (2019)Meta RLseparateGRUoard500PPO
Kostrikov (2018)POMDPsharedGRU05-2048PPO,A2C
Wang et al. (2020)POMDPseparateLSTMoa150TD3, SAC
Meng et al. (2021)POMDPseparateLSTMoa1-5TD3
Yang & Nguyen (2021)POMDPseparatebothoaMaxTD3, SAC
Baseline from Igl et al. (2018)POMDPsharedGRUoa25A2C
Baseline from Han et al. (2020)POMDPsharedLSTM064SAC
Baseline from Zhang et al. (2021)Robust RLseparateLSTM0100PPO
Baseline1 from Packer et al. (2018)GeneralizationsharedLSTM0128-512PPO,A2C
Baseline2 from Packer et al. (2018)GeneralizationseparateLSTMoard128-512PPO,A2C
Our methodMeta RLseparateLSTMoard64TD3
Our methodPOMDPseparateGRUoa64TD3
Our methodRobust RLseparateLSTM064TD3
Our methodGeneralizationseparateLSTM064TD3
+ +Model-free RL Algorithms. Recurrent model-free RL can be understood as applying an off-theshelf model-free RL algorithm with an actor and a Q function parametrized to take sequences of inputs. As such, the choice of the underlying model-free RL algorithm is paramount. Most prior work on continuous control POMDP problems used on-policy algorithms, such as A2C (Mnih et al., 2016), TRPO (Schulman et al., 2015) or PPO (Schulman et al., 2017). While off-policy algorithms such as TD3 (Fujimoto et al., 2018) and SAC (Haarnoja et al., 2018a;b) greatly improve the performance in continuous control MDP problems in terms of sample efficiency and asymptotic performance, these methods are rarely used in recurrent model-free RL baselines (Rakelly et al., 2019; Zhang et al., 2020; Zintgraf et al., 2020). In the experiment section (Sec. 5.1), we will show that using these off-policy algorithms for recurrent model-free RL provides results that are better than using on-policy algorithms and are comparable to their specialized methods in POMDP. This echoes the finding that model-free off-policy TD3-Context (Fakoor et al., 2020) can be better than the specialized method PEARL (Rakelly et al., 2019) in meta RL. + +RNN Variants and Context Length. RNN training is known to be unstable, especially with long sequences input (Bengio et al., 1994). The RNN variants like LSTM (Hochreiter & Schmidhuber, 1997) and GRU (Chung et al., 2014) mitigate the training issues, but still may fail to learn long-term dependencies (Trinh et al., 2018). In POMDP problems, these dependencies reflect the memory that an agent must have to solve a task. For example, a POMDP that hides velocities from observations theoretically requires a short memory length to infer velocities through consecutive positions (Meng et al., 2021). Prior work in POMDPs choose a variety set of context lengths for RNNs from 1 to 2048 (see the “Len” column of Table 2), and we select three representatives of short (5), medium (64), and long length (larger than 100) in the experiments (Sec. 5) for comparison. We also try both LSTM and GRU as RNN variants to compare their performance. We find that the optimal context length and RNN variant are task-specific (see Sec. 5.2). + +# 5 EXPERIMENTS + +Our experiments aim to answer two questions. First, how does a well-tuned implementation of recurrent model-free RL compare to specialized POMDP methods, such as purpose-designed meta RL and robust RL algorithms? To give these prior methods the strongest possible footing, we will compare prior methods on the specific problem types for which they were developed (i.e., meta RL algorithms were tested on meta RL tasks). Our second question studies which design decisions are essential for recurrent model-free RL. We put the environment details in Appendix D. + +Code Implementation. We release a modular and highly-configurable implementation of recurrent (off-policy) model-free RL: url. Our implementation is efficient in terms of computer memory compared to previous off-policy RL methods for POMDPs $2 0 0 \mathrm { x }$ less RAM than Han et al. (2020) and $9 \mathbf { x }$ less GPU memory than Dorfman et al. (2020)). Please see the appendix A for details, including an explanation of why our implementation is more memory-efficient than prior work. + +# 5.1 RECURRENT MODEL-FREE RL IS COMPARABLE WITH PRIOR SPECIALIZED METHODS + +While prior work has studied a range of different POMDP settings (e.g., meta RL, occluded observations), recurrent model-free RL is a ubiquitous baseline (Han et al., 2020; Humplik et al., 2019; Igl et al., 2018; Rakelly et al., 2019; Zintgraf et al., 2020). However, prior work consistently report that this baseline is reported to be unperformed to more specialized methods. This section casts doubt on that claim, showing that a well-tuned implementation of recurrent model-free RL can perform at least as well as more specialized methods. + +We study four subareas of POMDPs: the “standard” POMDP, meta RL, robust RL, and generalization in RL. We tune a wide range of decision factors shown in Sec. 4 in our implemented recurrent model-free RL. Appendix A.3 shows the details of the tuning options. For each subarea, we show the performance of a single variant that works best across the environments in that subarea, compared with the prior specialized methods in this subsection. In other words, the following plots of each subarea report the same model-free recurrent RL algorithm with the same hyperparameters. The exact configurations of each subarea can be found in the last four rows of Table 2. Under this restricted setting, we find that our implementation can actually outperform prior (specialized) methods by a wide margin across the four subareas. + +For each plot of learning curves, we show three approaches as reference. First, an Oracle policy has access to the POMDP hidden states, turning the POMDP into an MDP; this policy should therefore be treated as an upper bound on the performance that any POMDP method should receive. Second, as a lower bound, we use a Markovian policy to solve the POMDP. Both Oracle policy and Markovian policy are trained with the same hyperparameters as our recurrent model-free RL implementation. Third, we add a Random policy, which represents a trivial lower bound. We show the full learning curves in Appendix E.1 due to the space limit. See Appendix B for details about the implementation of these comparisons. + +“Standard” POMDP. Our first experiments look at the “standard” POMDPs that typically occlude some part of states in the environment. We will compare against VRM (Han et al., 2020), a recent state-of-the-art model-based POMDP algorithm. We directly apply the environment design of VRM paper that occludes either positions&angles or velocities of the simulated robot in PyBullet (Coumans & Bai, 2016). There are 8 environments {Hopper, Ant, Walker, Cheetah}- $\{ \mathrm { P } , \mathrm { V } \}$ , where “-P” stands for observing positions&angles only, and “-V” stands for observing velocities only. Fig. 1 and Fig. 18 in appendix show that the best single variant of our model-free recurrent RL implementation outperform VRM in 6 out of 8 environments, especially in {Cheetah,Hopper}- $\{ \mathrm { P } \}$ (over $8 0 \%$ of the Oracles). Our results suggest that, while the variational dynamics model used by VRM may be useful for some tasks, a simple recurrent model-free RL baseline can outperform VRM if properly tuned. While we are primarily interested in sample complexity, but not compute, it is worth noting that our recurrent model-free RL implementation is substantially more efficient than the open-source VRM implementation, training $5 \times$ faster and requiring at most $2 0 0 \times$ less RAM usage (see Appendix A). + +Meta RL. We next compare the recurrent model-free RL to the meta RL setting, where some indicator of the task is unobserved. We compare our implementation of recurrent model-free RL to a specialized state-of-the-art method, VariBAD (Zintgraf et al., 2020) that explicitly learns the task embeddings by variational model-based objectives. As suggested by Dorfman et al. (2020), we modified VariBAD to use SAC instead of PPO. This change also allows us for fair comparison with our implementation of recurrent model-free RL, which uses SAC and TD3. We adopt the three environments used in Dorfman et al. (2020) for experiments, including Semi-Circle and CheetahVel, and we also adapt Wind to make it harder to solve. Figure 2 shows that our best single variant outperforms VariBAD and even reaches Oracles in the two meta RL environments, Cheetah-Vel and Wind, leaving Semi-Circle in the appendix. Prior work (Rakelly et al., 2019; Zintgraf et al., 2020) show that disentangling task inference and control can stabilize training. However, our experiments suggest that joint training of task inference and control could also have comparable performance if well implemented. Additionally, because recurrent model-free RL is trained end-to-end, without using pre-trained task representations saved in the replay buffer like the off-policy VariBAD (Dorfman et al., 2020), our implementation does not have non-stationarity issue in task representations. + +![](images/30edb2beedd34b03559cfe4a7cb9d2b19f7fdd16870545558111d5f04ec4f72b.jpg) +Figure 2: Learning curves on two meta RL environments. The single best variant of our implementation on recurrent model-free RL can surpass the specialized meta RL method off-policy VariBAD (Dorfman et al., 2020), and match the performance of an “Oracle” policy that gets to observe the hidden state. + +![](images/473825d8305f612f2647b1794d5d4c29303a403a99877872238870f73e2911e4.jpg) +Figure 3: Learning curves on one robust RL environment, Cheetah-Robust. We show the average returns (left figure) and worst returns (right figure) of each method. The single best variant of our implementation on recurrent model-free RL can greatly outperform the specialized robust RL method MRPO (Jiang et al., 2021), and surpass the Oracle that are trained with same simulation and gradient steps. + +Robust RL. Thirdly, we focus on the robust RL that aims to maximize the worst returns over the tasks. We choose the recent specialized algorithm MRPO (Jiang et al., 2021) as the compared method, and adopt their used environments based on SunBlaze benchmark (Packer et al., 2018). These environments have hidden states that are fixed during one episode, including the density and the friction coefficients of the simulated robots, namely {Cheetah, Hopper, Walker $\}$ -Robust. Fig. 3 shows both the average return and worst return of our single best variant and MRPO on the three environments, where the worst return is measured by the average return in the worst $1 0 \%$ testing tasks following the practice in Jiang et al. (2021). The results are quite surprising: although our implementation, using average-case RL objective and without access to the hidden states, is not expected to surpass MRPO and Oracle with access to hidden states in worst return, we found that our best variant vastly outperforms the specialized MRPO and Oracle in both average return and worst return, with over $8 0 \%$ fewer simulation steps. Our implementation benefits from its memory and off-policy algorithms, while MRPO might suffer from its Markovian on-policy algorithm and a bit ideal Lipschitz assumption in dynamics. Nevertheless, our implementation is around $1 7 . 5 \mathrm { x }$ slower than MRPO given the same simulation steps (see Appendix A), so we only run it with 3M steps with a limited time budget. + +Generalization in RL. Finally, we focus on the SunBlaze benchmark from Packer et al. (2018) for investigating generalization in RL, including {Hopper, Cheetah}-Generalize. We pick the best specialized method in the tables of final performance in Packer et al. (2018), Markovian on-policy robust RL method EPOpt-PPO-FF (Rajeswaran et al., 2017a). Fig. 4 show the interpolation and extrapolation success rates in one environment, where in the interpolation the testing tasks have same distribution of hidden states as that of the training tasks, while in the extrapolation the testing distribution is disjoint from that of training. We can see that our model-free method is on par with the EPOpt-PPO-FF in the interpolation benchmark, while EPOpt-PPO-FF requires access to the dynamics parameters but ours does not. In the extrapolation benchmark, our method greatly outperforms the previous method, although our objective does not consider extrapolation. + +![](images/77085ccdb2c33145a870334a711f81035c3a228a1dd349da3475e0735128d0c8.jpg) +Figure 4: Learning curves on generalization in one RL environment, Hopper-Generalize. We show the interpolation success rates (left figures) and extrapolation success rates (right figures) of each method. The single best variant of our implementation on recurrent model-free RL can be par with the specialized method EPOpt-PPO-FF (Rajeswaran et al., 2017a) in interpolation and outperform it in extrapolation. The data of EPOpt-PPO-FF and A2C-RC (a recurrent model-free on-policy RL method) are copied from the Table 7 & 8 in Packer et al. (2018). + +Overall, we can see that with careful tuning on recurrent model-free RL, it can at least perform as well as the specialized or more complicated methods, in various kinds of POMDPs. + +# 5.2 WHAT MATTERS IN RECURRENT MODEL-FREE RL ALGORITHMS? + +In the previous subsection, we showed that recurrent model-free RL can perform on par with the specialized (state-of-the-art) methods, then a natural question comes: Why our implementation of recurrent model-free RL outperforms the implementation used in prior work? + +Our analysis will focus on ablating the five important design decisions introduced in Sec. 4: the actor-critic architecture (Arch), the policy input space (Inputs), the underlying model-free RL algorithm (RL), the RNN encoder (Encoder), and the RNN context length (Len). See Table. 2 for a summary of how prior work made these design decisions. Due to the space limit, we show the ablation results in some but not all the environments to compare the performance between the best single variant and the other variant that only differs in one decision factor. We also provide “single factor analysis” plots for each decision factor by averaging the performance over the other factors in Appendix E.2. + +Recurrent Off-Policy Actor-Critic Architecture. First, we ran some experiments with both shared and separate architectures on two toy POMDP environments. Fig. 5 show the results in one of them (see Appendix E.3 for the other). We can see that the shared architecture failed to learn, compared to the separate architecture. The large RNN gradient norm in the shared architecture suggests that the actor and critic losses may cause conflicts in gradient update. Our results echo prior work (Fakoor et al., 2020; Meng et al., + +![](images/1a7c066033dd192c76200952bccffa2034a7b549306d97b25307dd5892ba7fd1.jpg) +Figure 5: Comparison between shared and separate recurrent actor-critic architecture with all the other hyperparameters same, on Semi-Circle, a toy meta RL environment. We show the performance metric (left) and also gradient norm of the RNN encoder(s) (right, in log-scale). For the separate architecture, :critic and :actor refer to the separate RNN in critic and actor networks, respectively. + +2021; Sun et al., 2021) that only consider separate RNN encoders that can achieve high asymptotic rewards, and also echo that (Han et al., 2020) shows poor results in the shared architecture of SAC-LSTM. + +Policy Input Space. The 1st row of Table 3 shows the effect of policy input space in a POMDP environment Walker-P. The reward signals could help reveal the missing information of the velocity of the robot base, which is occluded in Walker-P. Similarly, the single factor analysis on policy input in Fig. 12 shows that oar is among the best in “-P” environments. Therefore, it is reasonable that adding previous rewards into policy inputs can increase the performance. + +Table 3: Ablation results in our implementation of recurrent model-free RL. In this table, we show how a single change in one decision factor from the variant that is best on average in that subarea, could significantly increase the performance. The first column shows how we change the single decision factor, and the last column shows the performance comparison between the best variant in that subarea (left) and the ablated one (right). For robust RL and generalization in RL, we show the performance metric in worst returns and extrapolation success rates, respectively. + +
Change in one decision factorSubareaEnvPerformance comparison
Inputs:oa→oar“Standard”POMDPWalker-P981.6 → 1345.0 (1.3×)
RL: TD3 →SAC“Standard”POMDPAnt-P310.7 → 2123.5 (6.8×)
Encoder: LSTM→GRURobust RLWalker-Robust765.9 → 931.3 (1.2×)
Len: 64 →400Meta RLCheetah-Vel-85.2 -→ -74.6 (+14%)
Len: 64→5GeneralizationHopper-Generalize0.292 -→ 0.415 (1.4×)
Len: 64→5“Standard"POMDPWalker-V121.4 -→ 264.3 (2.2×)
+ +Model-free RL Algorithms. Table 2 shows that TD3 dominates all the best variants in each subarea, which may be due to dense reward setting in most environments. However, the 2nd row of Table 3 shows the effect of RL algorithm in a POMDP environment Ant-P. SAC is significantly better than TD3 (increase by $6 . 8 \times$ , surpassing the PPO-GRU (Kostrikov, 2018) in Fig. 1), possibly due to strategic exploration where the action noise conditions on the history instead of being independent. This is prominent mainly in Ant-P as it might be much harder than the others. + +RNN Variants and Context Length. Generally, there is no significant difference between LSTM and GRU (see the single factor analysis in Appendix E.2), which is understandable as both are designed for general purpose. Howover, the 3rd row of Table 3 shows the effect of RNN encoder in a robust RL environment. We can see replacing LSTM with GRU can increase the worst-case metric in Walker-Robust. For the context length in RNNs, a medium length (64) dominates in all the best variants in each subarea (see Table 2), which could be viewed as a trade-off between memory capacity and computation costs. However, the remaining rows of Table 3 show the mixed effects of context length in RNNs. Both increasing and decreasing the context length can boost the performance in different environments. Specifically, decreasing the length from 64 to 5 makes our method surpass VRM in Walker-V (increase by $2 . 2 \times \phantom { }$ . This might explain why the prior methods adopt a wide range of context lengths from 1 to 2048 (see Table 2). Therefore, the choice of context length is problem-specific and can require tuning. + +Summary. We now summarize the main findings of our experiments: + +1. Using separate weights for the recurrent actor and recurrent critic boosts performance, likely because it avoids gradient explosion (Fig. 5 and Fig. 17 in Appendix). +2. Using state-of-the-art off-policy RL algorithms as the backbone in recurrent model-free RL can improve asymptotic performance (Fig. 1 and Figures in Appendix E.1). +3. The context length for the recurrent actor and critic has a large influence on task performance, but the optimal length is task-specific. Reasonable values are 5 to 500, and 64 is a good start (Rows 4–6 in Table 3 and Figures in Appendix E.2). +4. It is important that the inputs to the recurrent actor and critic, such as past observations and past returns, contain enough information to infer the POMDP hidden states (Row 1 in Table 3 and Figures in Appendix E.2). + +These findings may provide a useful initialization for researchers studying recurrent model-free RL. + +# 6 CONCLUSION + +In this paper, we show that a carefully-designed implementation of recurrent model-free RL can perform well across a range of POMDP domains, often on par with (if not significantly better than) prior methods that are specifically designed for specific types of POMDPs. Our ablation experiments demonstrate the importance of key design decisions, such as the underlying RL algorithm and RNN context length. While the best choices for some decisions (such as using separate RNNs for the actor and the critic) are consistent across domains, the best choices for other decisions (such as RNN context length) are problem-dependent. + +# REPRODUCIBILITY STATEMENT + +We release our code at the url for reproducibility. In this code repository, we provide instructions on how to run our method and the compared methods in this paper on all the environments involved in the experiment section. We provide the default configuration files for training and evaluating these algorithms that are adopted in our experiments. We also attach the numeric results of all the bar charts shown in the experiment section. + +ETHICS STATEMENT + +We do not believe that our work has direct ethical or societal implications. + +# REFERENCES + +Marcin Andrychowicz, Anton Raichuk, Piotr Stanczyk, Manu Orsini, Sertan Girgin, Raphael Marinier, L ¨ eonard ´ Hussenot, Matthieu Geist, Olivier Pietquin, Marcin Michalski, Sylvain Gelly, and Olivier Bachem. What matters for on-policy deep actor-critic methods? A large-scale study. In 9th International Conference on Learning Representations, ICLR 2021, Virtual Event, Austria, May 3-7, 2021, 2021. + +Karl Johan Astr ˚ om. Optimal control of markov processes with incomplete state information i. ¨ Journal of Mathematical Analysis and Applications, 10:174–205, 1965. + +J Andrew Bagnell, Andrew Y Ng, and Jeff G Schneider. Solving uncertain markov decision processes. 2001. + +Bram Bakker. Reinforcement learning with long short-term memory. 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In 8th International Conference on Learning Representations, ICLR 2020, Addis Ababa, Ethiopia, April 26-30, 2020, 2020. + +# A CODE-LEVEL DETAILS + +In this section, we first introduce the outline of code design, especially the replay buffer for sequences, and then compare the system usage, including computing speed, RAM, and GPU memory with previous POMDP methods. + +# A.1 CODE DESIGN + +Easy to use. Our code can be either used as an API to call the recurrent model-free RL class or a framework to tune the details in the class. The recurrent model-free RL class takes the hyperparameters of RNN encoder type, shared or separate actor-critic architecture, and whether include previous observations, and/or actions, and/or rewards into the inputs, to generate different instances. The details of the hyperparameter tuning set are shown in Sec. A.3. + +Memory-efficient replay buffer for sequences. Moreover, we design an efficient replay buffer for off-policy RL methods to cope with sequential inputs. Previous methods (Han et al., 2020; Yang & Nguyen, 2021) mainly use three-dimensional replay buffer to store sequential inputs, with the dimensions of (num episodes, max episode length, observation dimension), taking observation storage as example. This kind of implementation becomes memory-inefficient if the actual episode length is far smaller than the max episode length (then there will be many padded zeros (Dorfman et al., 2020)). Instead, we manage to implement a two-dimensional replay buffer of shape (num transitions, observation dimension) for observation storage, which also records the locations where each stored episode ends. In case of actual episodes that are shorter than provided sampled sequence length, the buffer also generates on-the-fly masks to indicate if the corresponding transitions are valid, so that we do not need to save zero-padded observations in the buffer. This enables the policy to receive a batch of previous experiences in a tensor-like data structure when sampling from the replay buffer. To sum up, our replay buffer can support varying-length sequence inputs and subsequence sampling without zero padding. + +Flexible training speed. Finally, our code supports flexible training speed by controlling the ratio of the numbers of gradient updates in RL w.r.t. the environment rollout steps. The training speed is approximately proportional to the ratio if the simulator speed is much faster than the policy gradient update. Typically, the ratio is less than or equal to 1.0 to enjoy higher training speed. + +# A.2 SYSTEM USAGE + +Table 4 shows the typical system usage of our method and the compared specialized methods on different environments. The time cost for our method and VariBAD depends on how many processes in parallel are run on a single GPU – our method is run with 8 processes on a single GPU while VariBAD is run with one process due to large GPU memory usage. From the results we can see that our method is memory-efficient in both RAM and GPU, and has an acceptable training speed with default hyperparameters. The computer system we used during the experiments includes a GeForce RTX 2080 Ti Graphic Card (with 11GB memory) and Intel(R) Xeon(R) Gold 6148 CPU $@$ 2.40GHz (with 250GB RAM and 80 cores). + +Table 4: Comparison between our method and specialized methods in system usage. The time costs are evaluated within 1M environment steps. Both VRM and MRPO are run on CPUs and MRPO does not have a replay buffer (shown in N/A). VariBAD requires the assumption of fixed episode length for the RAM cost. + +
MethodEnvironmentTime costRAMGPU memory
Ours (GRU)Hopper-V22.5 h0(1)1.2 GB
VRM (Han et al., 2020)Hopper-V102 h0(200)N/A
OursSemi-Circle12h0(1)1 GB
VariBAD (Dorfman et al., 2020)Semi-Circle2.3 h0(1)*9.5 GB
OursCheetah-Robust7h0(1)1.1 GB
MRPO (Jiang et al., 2021)Cheetah-Robust0.4 hN/AN/A
+ +# A.3 OUR HYPERPARAMETER TUNING SET + +Our proposed method has the following decision factors (introduced in Sec. 4) to tune in the experiments with the following options (the names in brackets are abbreviated ones): + +• Actor-Critic architecture (Arch): share the encoder weights between the recurrent actor and recurrent critic or not, namely shared and separate. +• Model-free RL algorithms (RL): td3 (Fujimoto et al., 2018) or sac (Haarnoja et al., 2018a) +• Encoder architecture (Encoder): lstm (Hochreiter & Schmidhuber, 1997) or gru (Cho et al., 2014) +• Policy input space (Inputs): o, oa, or, oar, oard (the notation is introduced in Sec. 4; depending on the POMDPs, see “Policy input” row in Table 5) +• Context length (Len): short (5), medium (64), long (larger than 100, depending on the POMDPs). + +For each instance, we label it with the names of all the hyperparameters it used in lowercase as notation. For example, td3-lstm-64-or-separate in Fig. 5 refers to the instance that uses the separate actor-critic architecture, TD3 RL algorithm, LSTM encoder, the policy input space of previous observations and reward sequences, and RNN context length of 64. + +# B TRAINING DETAILS + +![](images/bdf52f41abe2592c14a688161716770e854e2501df874d95c4b850a55ebaab8e.jpg) +Figure 6: The network architecture of our implementation on recurrent model-free RL. The left part shows the actor network, and the right shows the critic network. Each block shows a trainable module, with independent weights. We italicize the previous action and reward encoders as they are optional. By default, each encoder has one hidden layer, each RNN is one-layer LSTM or GRU, each MLP has two hidden layers. + +Fig. 6 shows our (separate) recurrent actor-critic architecture. Table 5 shows the main hyperparameters we adopt for each subarea. We did not tune these hyperparameters, except that we adjusted the number of gradient steps so that all the experiments could be completed in 36 hours. + +For Markovian policies (SAC and TD3), we remove the encoders and RNNs from the actor-critic architecture, and train them with same hyperparameters as those of recurrent policies. For each task, we report the results of either SAC or TD3, whichever achieves higher returns. For Oracle policies, we use the well-tuned results from Table 1 (“SAC w/ unstructured row”) in Raffin et al. (2021), which is based on Stable Baseline3 (Raffin, 2020), for “standard” POMDPs. For the other subareas, we have to run the Markovian policies (SAC and TD3) with access to the hidden states, using the same training hyperparameters as those of recurrent policies. But these Oracle policies might be not well-tuned given same environment and gradient steps, especially in robust RL and generalization in RL. + +We also show the settings of the specialized methods we compared in the main paper in Table 6. Note that our recurrent model-free RL share the exactly same settings as (off-policy) VariBAD (Dorfman et al., 2020) and VRM (Han et al., 2020). For MRPO (Jiang et al., 2021) and EPOPT (Rajeswaran et al., 2017a), they adopt totally different settings, i.e. on-policy Markovian approaches to MDPs (with access to the ground-truth state (s) of environment). Thus, in fact, MRPO and EPOPT should be more viewed as oracle policies as upper bounds of recurrent model-free RL. + +Table 5: Hyperparameter summary in our implementation of model-free recurrent RL. For each subarea, we report the hidden layer size of each module, RL and training hyperparameters. For Meta-RL, we take the model on Cheetah-Vel as example, which follows the architecture design of off-policy VariBAD (Dorfman et al., 2020). The hidden size of observation-action encoder is the sum of that of observation encoder, previous action encoder (if exists), and reward encoder (if exists). + +
Meta-RL* POMDP Robust RL Gen. in RL
HiddenlayersizeObserv.encoderPrev. Action encoder*Reward encoder*RNNMLP[32][16][16][128][128,128,128] [256,256]
RLhparamsLearning rateDiscount factor ySmoothing coef T SAC temperatureTD3 noisesReplay buffer sizeBatch size3e-40.990.005 automatically updated by Haarnoja et al. (2018b)default values from Fujimoto et al. (2018)1e632 64
TraininghparamsEnvironment stepsGradient steps5M 1.5M 3M0.1M 1.5M 0.6M
PolicyinputsLargest input spaceBest input spaceoard oar oa oaroard oa 0 0
+ +Table 6: Settings of the specialized methods we compared in the main paper. For Meta-RL, we take the model on Cheetah-Vel as example. + +
Meta-RL*POMDPRobustRLGen. in RL
Approach(off-policy) VariBADVRMMRPOEPOPT
Memory-based?√√XX
Off-policy?√√XX
Input spaceoardoarSS
Access to hidden states?XX√√
+ +# C EVALUATION DETAILS + +The bar charts in Fig. 1 and 18 and Table 3 adopt the final performance of each method. We run each instance/variant in our method and each compared method with 4 random seeds. The final performance is calculated by the average performance of the last $20 \%$ environment steps across the 4 seeds. + +In terms of selecting the best variant in our method for each subarea, we first compute the final performance of each variant, then normalize the final performance into $[ 0 , 1 ]$ , and finally select the best variant in the average of the normalized final performance across all the environments in each subarea. + +For the bar charts in Fig. 1 and 18, we show the normalized returns of each method, calculated by R−RminR −R ∈ [0, 1], where R is the raw average return of that method and Rmax and Rmin are the maximum and minimum of all the methods including Oracle policy and Random policy. + +# D ENVIRONMENT DETAILS + +# D.1 “STANDARD” POMDP + +Except for the classic Pendulum environment, we use PyBullet (Coumans & Bai, 2016) as the simulator for “standard” POMDP environments. As the practice in VRM (Han et al., 2020), we remove all the position/angle-related entries in the observation space for “-V” environments and velocityrelated entries for “-P” environments, to transform the original MDP into POMDP. + +{Pendulum,Ant,Cheetah,Hopper,Walker}-P. The “-P” stands for the environments that keep position-related entries by removal of velocity-related entries. Thus, the observed state $s ^ { o }$ includes positions $p$ , while the hidden state $s ^ { h }$ is the velocities $v$ . + +{Pendulum,Ant,Cheetah,Hopper,Walker}-V. The “-V” stands for the environments that keep velocity-related entries by removal of position-related entries. Thus, the observed state $s ^ { o }$ includes positions $v$ , while the hidden state $s ^ { h }$ is the velocities $p$ . + +# D.2 META RL + +Semi-Circle. We directly follow off-policy version of VariBAD (Dorfman et al., 2020). The observed state $s ^ { o }$ includes the agent’s 2D position $p$ , and the hidden state $s ^ { h }$ is referred to the goal state $p _ { g }$ . The goal state only appears in reward function: $R ( s _ { t } ^ { o } , s _ { t + 1 } ^ { o } , a _ { t } , s ^ { h } ) \equiv R ( p _ { t + 1 } , p _ { g } { \bar { ) } } =$ $\mathbb { 1 } ( \lVert p _ { t + 1 } - p _ { g } \rVert _ { 2 } \leq r )$ . The dynamic function $T$ is independent of the goal state. + +Cheetah-Vel. We directly follow Dorfman et al. (2020) using MuJoCo (Todorov et al., 2012) simulator of HalfCheetah-v2. The hidden state $s ^ { h }$ is the target velocity $v _ { g }$ and observed state $s ^ { o }$ includes the velocity $v$ . Reward function includes both the hidden state and action: $R ( s _ { t } ^ { o } , s _ { t + 1 } ^ { o } , a _ { t } , s ^ { h } ) \equiv R ( v _ { t } , v _ { g } , a _ { t } ) = - \| v _ { t } - v _ { g } \| _ { 1 } - 0 . 0 5 \| a _ { t } \| _ { 2 }$ . The dynamic function $T$ is independent of the goal state. + +Wind. We modified the parameters of Wind environment in Dorfman et al. (2020) to make it harder to solve. The agent must navigate to a fixed (but unknown) goal $p _ { g }$ within a distance of $D = 1$ from its fixed initial state. Similarly to Semi-Circle, the reward function is goal conditioned but without hidden state: $R ( s _ { t } ^ { o } , s _ { t + 1 } ^ { o } , a _ { t } , \bar { s ^ { h } } ) \equiv R ( p _ { t + 1 } , p _ { g } ) = \mathbb { 1 } ( \| p _ { t + 1 } - p _ { g } \| _ { 2 } \bar { \leq ^ { } { r } } )$ . The hidden state $s ^ { h }$ appear in the deterministic dynamics as a noise term, i.e. $s _ { t + 1 } ^ { o } = s _ { t } ^ { o } + a _ { t } + s ^ { h }$ , where $s ^ { h }$ is sampled from $U [ - 0 . 0 8 , 0 . 0 8 ]$ at the initial time-step and then kept fixed. + +# D.3 ROBUST RL + +{Hopper,Walker,Cheetah}-Robust. We directly adopt the environments used in MRPO (Jiang et al., 2021). In each environment, the hidden state is the dynamics parameters including the density and friction coefficients of the simulated robot in roboschool, adapted from the SunBlaze (Packer et al., 2018). The exact ranges of the hidden states in each environment can be found in Table 1 of MRPO (Jiang et al., 2021). We evaluate the algorithms with 100 tasks in each environment, and use the average of them as average returns, and the average of the worst $1 0 \%$ of them as worst returns, following MRPO paper. + +# D.4 GENERALIZATION IN RL + +{Hopper,Cheetah}-Generalize. We directly adopt the environments used in SunBlaze (Packer et al., 2018). In each environment, the hidden state is the dynamics parameters including the density, friction coefficients, and the power of the simulated robot in roboschool. The exact ranges of both interpolation and extrapolation in the hidden state distribution for each environment can be found in Table 1 of SunBlaze (Packer et al., 2018). We follow the practice of SunBlaze to evaluate interpolation and extrapolation success rates. + +# E FULL EXPERIMENTAL RESULTS + +# E.1 LEARNING CURVES OF ALL THE COMPARED METHODS + +In this subsection, we show all the learning curves of all the compared methods (including Oracle policy as upper bound, Markovian and Random policies as lower bounds) in each subarea, namely “Standard” POMDPs (Fig. 7 and Fig. 8), meta RL (Fig. 9), robust RL (Fig. 10), and generalization in RL (Fig. 11). The final performance of these learning curves generate the bar charts in Sec. 5.1. + +![](images/85ccefe946b801cd15f2e89cf6c81a75ab616a8ab9469593b794c32433d306e9.jpg) +Figure 7: Learning curves on “standard” POMDP environments that preserve positions & angles but occlude velocities in the states in PyBullet (Coumans & Bai, 2016) (namely “-P”). We show the results from the single best variant of our implementation on recurrent model-free RL, the popular recurrent model-free on-policy implementation (PPO-GRU, A2C-GRU) (Kostrikov, 2018), and also model-based method VRM (Han et al., 2020). Note that VRM is around ${ 5 } \mathbf { x }$ slower than ours, so we have to run 0.5M environment steps for it. + +# E.2 SINGLE FACTOR ANALYSIS ON OUR METHOD + +Our analysis will focus on ablating the important design decisions: the actor-critic architecture (Arch), the policy input space (Inputs), the underlying model-free RL algorithm (RL), the RNN encoder (Encoder), and the RNN context length (Len). As there are several decision variables in our method, we could only show the results of each variable in the plots by averaging the performance over the other variables, which we call single factor analysis. In this kind of analysis, we can only say one value is better than another in one factor in the average sense (not the maximal sense); therefore it can show the robustness of each factor, but cannot tell the best choices (showed in Sec. 5.1). We show single factor analysis plots in each subarea, namely “Standard” POMDPs (Fig. 12 and Fig. 13), meta RL (Fig. 14), robust RL (Fig. 15), and generalization in RL (Fig. 16). + +From these plots, we can see that each decision factor can make a difference in some environments. For example, the choice of RL algorithm is crucial in Ant-P (Fig. 12), Cheetah-V (Fig. 13), Wind (Fig. 14) and Hopper-Generalize (Fig. 16). The context length is essential in all the “-P” environments (Fig. 12), Cheetah-Vel (Fig. 14), and both the generalization environments (Fig. 16). The policy input space can make a difference in most “-P” environments (Fig. 12) possibly because oar contains the information of missing velocities. + +![](images/9b9ca379534e68d909afc28de5d21154b0d5cd869969737e00fe2763f6376029.jpg) +Figure 8: Learning curves on “standard” POMDP environments that preserve velocities but occlude positions $\pmb { \& }$ angles in the states in PyBullet (Coumans & Bai, 2016) (namely “-V”). We show the results from the single best variant of our implementation on recurrent model-free RL, the popular recurrent model-free on-policy implementation (PPO-GRU, A2C-GRU) (Kostrikov, 2018), and also model-based method VRM (Han et al., 2020). Note that VRM is around ${ 5 } \mathrm { x }$ slower than ours, so we have to run 0.5M environment steps for it. + +# E.3 ADDITIONAL RESULTS ON SEPARATE VS SHARED RECURRENT ACTOR-CRITICARCHITECTURE + +In Fig. 5 of Sec. 5.2, we show the importance of selecting a separate recurrent actor-critic architecture in a meta RL environment, Semi-Circle. Now we show the result in another POMDP environment, Pendulum-V, which occludes the positions and angles, in Fig. 17. We can see that the shared encoder architecture is also worse than the separate one, possibly due to the different gradient scales in actor and critic losses w.r.t. the encoder. + +# E.4 ADDITIONAL RESULTS ON COMPARISON WITH VRM + +Both Fig. 1 and Fig. 18 shows the final performance of the same single variant of our implementation, but the former shows our results with 1.5M simulation steps while the latter shows our results with $0 . 5 \mathbf { M }$ simulation steps to match with those of VRM due to the time budget. + +![](images/3894fdfca46be8394637e0e88be03a60d4ad34c6d20b7fdc0897c0d605dcfe11.jpg) +Figure 9: Learning curves on meta RL environments. We show the results from the single best variant of our implementation on recurrent model-free RL, the specialized meta RL method VariBAD (Dorfman et al., 2020) + +![](images/dfa7dc791a89631fd6f6bc1cdf6f0871db3745292eea57b55d353289f7d243a6.jpg) +Figure 10: Learning curves on robust RL environments. We show the average returns (left figures) and worst returns (right figures) from the single best variant of our implementation on recurrent model-free RL, the specialized robust RL method MRPO (Jiang et al., 2021). Note that our method is much slower than MRPO, so we have to run our method within 3M environment steps. But the results show that our method have much better sample efficiency over MRPO. + +![](images/66064063e5800389fd2dbd583070929708fb7b1e199ff2798820509b4e39ad54.jpg) +Figure 11: Learning curves on generalization in RL environments. We show the interpolation success rates (left figures) and extrapolation success rates (right figures) from the single best variant of our implementation on recurrent model-free RL. We also show the final performance of the specialized method EPOpt-PPO-FF (Rajeswaran et al., 2017a) and another recurrent model-free (on-policy) RL method (A2C-RC) copied from the Table 7 & 8 in Packer et al. (2018). + +![](images/d9a540b849cdaba473400f6d2b726dee14f5c2a4d5c468fa53d01c31cd5000fa.jpg) +Figure 12: Ablation study of our implementation on “standard” POMDP environments that preserve positions $\pmb { \& }$ angles but occlude velocities in the states in pybullet (Coumans & Bai, 2016) (namely $\mathbf { \tilde { \Sigma } ^ { 6 6 } - } \mathbf { P } ^ { 9 } \mathbf { \bar { \Sigma } } ,$ ). We show the single factor analysis on the 4 decision factors including RL, Encoder, Len, and Inputs for each environment. + +![](images/32445265e20f1f545fd08fe0230bf577d359a0980e43099b5bce76317dc9ecf4.jpg) +Figure 13: Ablation study of our implementation on “standard” POMDP environments that preserve velocities but occlude positions & angles in the states in pybullet (Coumans & Bai, 2016) (namely “-V”). We show the single factor analysis on the 4 decision factors including RL, Encoder, Len, and Inputs for each environment. + +![](images/26813d40fd770010f14b184f9791b978c08ed29426a0c77c554d40a4d575ad91.jpg) +Figure 14: Ablation study of our implementation on meta RL environments. We show the single factor analysis on covering all the decision factors + +![](images/af848a7b1bd8022f0a5a01bf2cddbe2366af14d2091951fb72fcae8b645a9a7b.jpg) +Figure 15: Ablation study of our implementation on robust RL environments. We show the single factor analysis on the 4 decision factors including RL, Encoder, Len, and Inputs for each environment in both average returns and worst returns. + +![](images/fc734dbfd8668706406e93d90ff0fd4650c5d5adff372b1c88b6ff670ae61a5a.jpg) +Figure 16: Ablation study of our implementation on generalization in RL environments. We show the single factor analysis on the 3 decision factors including RL, Len, and Inputs for each environment in both interpolation and extraploation success rates. + +![](images/5ff5f0fcb3a9f61051dd00d39ed12967134c54b38301ce44e33adf4838a9fa2f.jpg) +Figure 17: Comparison between shared and separate recurrent actor-critic architecture with all the other hyperparameters same, on Pendulum-V, a toy “standard” POMDP environment. We show the performance metric (left) and also the gradient norm of the RNN encoder(s) (right, in log-scale). For the separate architecture, :critic and :actor refer to the separate RNN in critic and actor networks, respectively. + +![](images/87a809e96598bc16313fea09489dbdb6bb785477299c66ee97a72274e50173bc.jpg) +Figure 18: Final normalized returns of our implemented recurrent model-free RL algorithm with the same hyperparameters, and the prior method VRM (Han et al., 2020) across the eight environments in “standard” POMDPs, each of which trained in 0.5M simulation steps. + +# F ABLATION STUDY ON RNN ARCHITECTURES + +Our implementation of recurrent model-free RL uses the popular 1-layer LSTM or GRU. To investigate the effect of RNN architecture, we ablate two RNN variants: one is 2-layer stacked LSTM/GRU, the other is Particle-Filter RNN (PF-RNN (Ma et al., 2020a)). PF-RNN maintains a stochastic belief (posterior distribution) through a set of weighted particles to better capture the uncertainty and multi-modality, compared to classic RNN’s deterministic belief. DPFRL (Ma et al., 2020b) applies PF-GRU to POMDP tasks, using the mean of particles and MGF features of particles as the belief state. We follow the implementation of PF-GRU in DPFRL (Ma et al., 2020b) to replace regular GRU, keeping the other model components the same. + +We try 2-layer LSTM/GRU, on our best variant across all the environments. We run PF-GRU on “standard” POMDP environments where the best variant also uses GRU for fair comparison, and do not try PF-LSTM as it is not adopted in DPFRL (Ma et al., 2020b). + +Since 2-layer LSTM/GRU doubles the training time of 1-layer LSTM/GRU given the same gradient updating frequency, we have to decrease the frequency from 1.0 to 0.6 in “standard” POMDPs. Similarly, PF-GRU costs $1 5 \times$ than 1-layer GRU, so we also have to decrease the frequency from 1.0 to 0.6. All the other hyperparameters remain the same. + +![](images/75896b02d02693733e2773334d33463d9735c1ded84e90b3d11b64954c25f30b.jpg) +Figs. 19 to 23 show all the learning curves of the ablation study on RNN architectures. Except for rare cases (e.g. Ant-P, Walker-V), the RNN variants perform worse than or are par with 1-layer LSTM/GRU, although in most cases the RNN variants can still outperform the Markovian policies. Possible reasons are that the RNN variants need hyperparameter tuning and more training samples to converge. +Figure 19: Ablation study on RNN architecture in “standard” POMDPs (“-P”). We show two models that are only different from our best single variant in the RNN architectures, namely using 2-layer stacked GRU (td3-2gru) and PF-GRU (Ma et al., 2020a;b) (td3-pfgru), instead of 1-layer GRU. + +![](images/058c7ea47e1c159b6fe098fd66aa4e39e8383fe8a07e5ec40a45895b54c4ebb4.jpg) +Figure 20: Ablation study on RNN architecture in “standard” POMDPs (“-V”). We show two models that are only different from our best single variant in the RNN architectures, namely using 2-layer stacked GRU (td3-2gru) and PF-GRU (Ma et al., 2020a;b) (td3-pfgru), instead of 1-layer GRU. + +![](images/1d04cd47f46e4193da7f651143562f09f9344d1245c759d8448902ccfb028f35.jpg) +Figure 21: Ablation study on RNN architecture in Meta RL. We show one model that are only different from our best single variant in the RNN architecture, namely using 2-layer stacked LSTM (td3-2lstm), instead of 1-layer LSTM. + +![](images/764e5249b62d350489348f58de8742e57e05989fadfe593ce37c71ee8c06e097.jpg) +Figure 22: Ablation study on RNN architecture in robust RL. We show one model that are only different from our best single variant in the RNN architecture, namely using 2-layer stacked LSTM (td3-2lstm), instead of 1-layer LSTM. + +![](images/eb61e0519c56998e2edeb2bb48ea3ac6587ada65032b50129bb78d91c80618ee.jpg) +Figure 23: Ablation study on RNN architecture in generalization in RL. We show one model that are only different from our best single variant in the RNN architecture, namely using 2-layer stacked LSTM (td3-2lstm), instead of 1-layer LSTM. + +# G ADDITIONAL RESULTS ON COMPARISON WITH OFF-POLICY VARIBAD + +In our experiment section (Sec. 5) and Fig. 9, we show the learning curves of our method and our re-implemented off-policy VariBAD (Dorfman et al., 2020). To rule out the possibility of reimplementation bugs in the off-policy VariBAD, we ran the official off-policy VariBAD implementation2, denoted as VariBAD-BOReL. + +In Fig. 24, we show official off-policy VariBAD (VariBAD-BOReL in short), our re-implemented off-policy VariBAD (VariBAD in short), and one variant of our recurrent model-free RL, which uses the same decision factors as VariBAD-BOReL (SAC as RL algorithm, GRU as encoder, oar as input, 400 as context length) for fair comparison. We also try Ant-Dir from Dorfman et al. (2020) paper, a more challenging meta RL environment than Cheetah-Vel. + +In Semi-Circle and Wind, VariBAD-BOReL has similar performance as our re-implemented version (VariBAD). + +In Cheetah-Vel, we find VariBAD-BOReL outperforms our re-implemented version, but still has a gap from what they reported in Fig. 11 in the appendix of Dorfman et al. (2020). Our implementation still outperforms VariBAD-BOReL, supporting our claim in the main paper. + +Finally, in the newly added and more challenging Ant-Dir, we found our recurrent model-free RL can even greatly surpass Oracle (trained with same environment and gradient steps), while VariBADBOReL somehow has extremely low performance (worse than Random) and suffers from numerical issue to terminate early. + +![](images/2bee8e27956779c2058d9477678f8a6064ff9f8f6cd810dfb28714dc2934413c.jpg) +Figure 24: Additional learning curves on meta RL environments. We show the results from the single variant of our implementation on recurrent model-free RL (sac-gru-oar-400), and the specialized meta RL method off-policy VariBAD (Dorfman et al., 2020) (their official implementation VariBAD-BOReL and our re-implementation VariBAD). \ No newline at end of file diff --git a/parse/dev/E0zOKxQsZhN/E0zOKxQsZhN_middle.json b/parse/dev/E0zOKxQsZhN/E0zOKxQsZhN_middle.json new file mode 100644 index 0000000000000000000000000000000000000000..021c9d25d87694aa3392b434fcbfc8227ff7cd56 --- /dev/null +++ b/parse/dev/E0zOKxQsZhN/E0zOKxQsZhN_middle.json @@ -0,0 +1,63473 @@ +{ + "pdf_info": [ + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 78, + 504, + 116 + ], + "lines": [ + { + "bbox": [ + 105, + 78, + 505, + 98 + ], + "spans": [ + { + "bbox": [ + 105, + 78, + 505, + 98 + ], + "score": 1.0, + "content": "RECURRENT MODEL-FREE RL IS A STRONG", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 97, + 347, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 97, + 347, + 118 + ], + "score": 1.0, + "content": "BASELINE FOR MANY POMDPS", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 112, + 136, + 244, + 157 + ], + "lines": [ + { + "bbox": [ + 113, + 136, + 201, + 147 + ], + "spans": [ + { + "bbox": [ + 113, + 136, + 201, + 147 + ], + "score": 1.0, + "content": "Anonymous authors", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 112, + 146, + 245, + 159 + ], + "spans": [ + { + "bbox": [ + 112, + 146, + 245, + 159 + ], + "score": 1.0, + "content": "Paper under double-blind review", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5 + }, + { + "type": "title", + "bbox": [ + 278, + 186, + 333, + 199 + ], + "lines": [ + { + "bbox": [ + 277, + 186, + 335, + 200 + ], + "spans": [ + { + "bbox": [ + 277, + 186, + 335, + 200 + ], + "score": 1.0, + "content": "ABSTRACT", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 143, + 210, + 468, + 330 + ], + "lines": [ + { + "bbox": [ + 142, + 210, + 469, + 222 + ], + "spans": [ + { + "bbox": [ + 142, + 210, + 469, + 222 + ], + "score": 1.0, + "content": "Many problems in RL, such as meta RL, robust RL, and generalization in RL,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 141, + 221, + 470, + 234 + ], + "spans": [ + { + "bbox": [ + 141, + 221, + 470, + 234 + ], + "score": 1.0, + "content": "can be cast as POMDPs. 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(Chen et al., 2018).", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 484, + 296, + 615 + ], + "lines": [ + { + "bbox": [ + 106, + 484, + 297, + 495 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 297, + 495 + ], + "score": 1.0, + "content": "Many subareas in RL study problems that are", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 493, + 297, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 493, + 297, + 507 + ], + "score": 1.0, + "content": "special cases of POMDPs, and we summa-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 505, + 297, + 517 + ], + "spans": [ + { + "bbox": [ + 106, + 505, + 297, + 517 + ], + "score": 1.0, + "content": "rize them in Table 1. 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However, prior work has found that such recurrent model-free", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 142, + 254, + 470, + 267 + ], + "spans": [ + { + "bbox": [ + 142, + 255, + 156, + 264 + ], + "score": 0.49, + "content": "R L", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 254, + 470, + 267 + ], + "score": 1.0, + "content": "methods tend to perform worse than more specialized algorithms that are de-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 265, + 470, + 277 + ], + "spans": [ + { + "bbox": [ + 141, + 265, + 470, + 277 + ], + "score": 1.0, + "content": "signed for specific types of POMDPs. This paper revisits this claim. We find that", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 276, + 470, + 289 + ], + "spans": [ + { + "bbox": [ + 141, + 276, + 470, + 289 + ], + "score": 1.0, + "content": "a careful architecture and hyperparameter decisions yield a recurrent model-free", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 142, + 288, + 470, + 299 + ], + "spans": [ + { + "bbox": [ + 142, + 288, + 470, + 299 + ], + "score": 1.0, + "content": "implementation that performs on par with (and occasionally substantially better", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 298, + 470, + 311 + ], + "spans": [ + { + "bbox": [ + 141, + 298, + 470, + 311 + ], + "score": 1.0, + "content": "than) more sophisticated recent techniques in their respective domains. We also", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 309, + 469, + 321 + ], + "spans": [ + { + "bbox": [ + 141, + 309, + 469, + 321 + ], + "score": 1.0, + "content": "release a simple and efficient implementation of recurrent model-free RL for fu-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 321, + 321, + 331 + ], + "spans": [ + { + "bbox": [ + 141, + 321, + 321, + 331 + ], + "score": 1.0, + "content": "ture work to use as a baseline for POMDPs.1", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 10, + "bbox_fs": [ + 141, + 210, + 470, + 331 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 342, + 206, + 353 + ], + "lines": [ + { + "bbox": [ + 105, + 340, + 208, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 208, + 356 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 357, + 296, + 477 + ], + "lines": [ + { + "bbox": [ + 106, + 356, + 297, + 369 + ], + "spans": [ + { + "bbox": [ + 106, + 356, + 297, + 369 + ], + "score": 1.0, + "content": "While reinforcement learning (RL) is often cast", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 368, + 297, + 380 + ], + "spans": [ + { + "bbox": [ + 106, + 368, + 297, + 380 + ], + "score": 1.0, + "content": "as the problem of learning a single fully observ-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 378, + 297, + 390 + ], + "spans": [ + { + "bbox": [ + 106, + 378, + 297, + 390 + ], + "score": 1.0, + "content": "able task, also known as MDP, training and test-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 390, + 296, + 401 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 296, + 401 + ], + "score": 1.0, + "content": "ing on that same task, most real-world appli-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 401, + 297, + 411 + ], + "spans": [ + { + "bbox": [ + 106, + 401, + 297, + 411 + ], + "score": 1.0, + "content": "cations of RL demand some degree of transfer", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 411, + 297, + 423 + ], + "spans": [ + { + "bbox": [ + 106, + 411, + 297, + 423 + ], + "score": 1.0, + "content": "and partial observability. For example, visual", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 423, + 297, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 423, + 297, + 434 + ], + "score": 1.0, + "content": "navigation (Zhu et al., 2017) requires adapta-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 434, + 297, + 445 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 297, + 445 + ], + "score": 1.0, + "content": "tion to unseen scenes with occlusion in observa-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 444, + 297, + 456 + ], + "spans": [ + { + "bbox": [ + 106, + 444, + 297, + 456 + ], + "score": 1.0, + "content": "tions, and human-robot collaboration requires", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 455, + 297, + 467 + ], + "spans": [ + { + "bbox": [ + 106, + 455, + 297, + 467 + ], + "score": 1.0, + "content": "that robots infer the intentions of human col-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 467, + 231, + 478 + ], + "spans": [ + { + "bbox": [ + 106, + 467, + 231, + 478 + ], + "score": 1.0, + "content": "laborators. (Chen et al., 2018).", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 22, + "bbox_fs": [ + 106, + 356, + 297, + 478 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 484, + 296, + 615 + ], + "lines": [ + { + "bbox": [ + 106, + 484, + 297, + 495 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 297, + 495 + ], + "score": 1.0, + "content": "Many subareas in RL study problems that are", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 493, + 297, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 493, + 297, + 507 + ], + "score": 1.0, + "content": "special cases of POMDPs, and we summa-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 505, + 297, + 517 + ], + "spans": [ + { + "bbox": [ + 106, + 505, + 297, + 517 + ], + "score": 1.0, + "content": "rize them in Table 1. For example, meta RL", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 516, + 297, + 528 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 297, + 528 + ], + "score": 1.0, + "content": "(Duan et al., 2016; Schmidhuber, 1987; Thrun", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 527, + 297, + 538 + ], + "spans": [ + { + "bbox": [ + 106, + 527, + 297, + 538 + ], + "score": 1.0, + "content": "& Pratt, 2012; Wang et al., 2017) is a POMDP", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 107, + 539, + 297, + 549 + ], + "spans": [ + { + "bbox": [ + 107, + 539, + 297, + 549 + ], + "score": 1.0, + "content": "where certain aspects of the reward function or", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 549, + 297, + 560 + ], + "spans": [ + { + "bbox": [ + 106, + 549, + 297, + 560 + ], + "score": 1.0, + "content": "(less commonly) dynamics function are unob-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 561, + 296, + 571 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 296, + 571 + ], + "score": 1.0, + "content": "served but held constant through one episode.", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 570, + 297, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 297, + 583 + ], + "score": 1.0, + "content": "The robust RL problem (Bagnell et al., 2001;", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 581, + 297, + 594 + ], + "spans": [ + { + "bbox": [ + 106, + 581, + 297, + 594 + ], + "score": 1.0, + "content": "Pattanaik et al., 2018; Pinto et al., 2017; Ra-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 593, + 297, + 604 + ], + "spans": [ + { + "bbox": [ + 106, + 593, + 297, + 604 + ], + "score": 1.0, + "content": "jeswaran et al., 2017a) assumes that certain as-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 604, + 297, + 615 + ], + "spans": [ + { + "bbox": [ + 106, + 604, + 297, + 615 + ], + "score": 1.0, + "content": "pects of the dynamics or reward function are", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 484, + 297, + 615 + ] + }, + { + "type": "image", + "bbox": [ + 306, + 360, + 502, + 521 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 306, + 360, + 502, + 521 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 306, + 360, + 502, + 521 + ], + "spans": [ + { + "bbox": [ + 306, + 360, + 502, + 521 + ], + "score": 0.972, + "type": "image", + "image_path": "0263b0377870a9e706cac9a587cc7239ea6cfcc9daae0f0b1f499a4ca8db104b.jpg" + } + ] + } + ], + "index": 46, + "virtual_lines": [ + { + "bbox": [ + 306, + 360, + 502, + 372.38461538461536 + ], + "spans": [], + "index": 40 + }, + { + "bbox": [ + 306, + 372.38461538461536, + 502, + 384.7692307692307 + ], + "spans": [], + "index": 41 + }, + { + "bbox": [ + 306, + 384.7692307692307, + 502, + 397.1538461538461 + ], + "spans": [], + "index": 42 + }, + { + "bbox": [ + 306, + 397.1538461538461, + 502, + 409.53846153846143 + ], + "spans": [], + "index": 43 + }, + { + "bbox": [ + 306, + 409.53846153846143, + 502, + 421.9230769230768 + ], + "spans": [], + "index": 44 + }, + { + "bbox": [ + 306, + 421.9230769230768, + 502, + 434.30769230769215 + ], + "spans": [], + "index": 45 + }, + { + "bbox": [ + 306, + 434.30769230769215, + 502, + 446.6923076923075 + ], + "spans": [], + "index": 46 + }, + { + "bbox": [ + 306, + 446.6923076923075, + 502, + 459.07692307692287 + ], + "spans": [], + "index": 47 + }, + { + "bbox": [ + 306, + 459.07692307692287, + 502, + 471.4615384615382 + ], + "spans": [], + "index": 48 + }, + { + "bbox": [ + 306, + 471.4615384615382, + 502, + 483.8461538461536 + ], + "spans": [], + "index": 49 + }, + { + "bbox": [ + 306, + 483.8461538461536, + 502, + 496.23076923076894 + ], + "spans": [], + "index": 50 + }, + { + "bbox": [ + 306, + 496.23076923076894, + 502, + 508.6153846153843 + ], + "spans": [], + "index": 51 + }, + { + "bbox": [ + 306, + 508.6153846153843, + 502, + 520.9999999999997 + ], + "spans": [], + "index": 52 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 304, + 522, + 504, + 611 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 304, + 521, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 304, + 521, + 505, + 532 + ], + "score": 1.0, + "content": "Figure 1: Implementation Matters for Recurrent", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 303, + 530, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 303, + 530, + 505, + 543 + ], + "score": 1.0, + "content": "Model-Free RL. This paper identifies critical design", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 304, + 541, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 304, + 541, + 505, + 552 + ], + "score": 1.0, + "content": "decisions for recurrent model-free RL that outperforms", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 303, + 551, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 303, + 551, + 505, + 562 + ], + "score": 1.0, + "content": "not only prior implementations (e.g. PPO-GRU and", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 304, + 561, + 505, + 572 + ], + "spans": [ + { + "bbox": [ + 304, + 561, + 505, + 572 + ], + "score": 1.0, + "content": "A2C-GRU from Kostrikov (2018)), but also purpose-", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 303, + 570, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 303, + 570, + 505, + 582 + ], + "score": 1.0, + "content": "designed methods (e.g. VRM from Han et al. (2020)).", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 304, + 580, + 505, + 592 + ], + "spans": [ + { + "bbox": [ + 304, + 580, + 505, + 592 + ], + "score": 1.0, + "content": "We also show Markovian policies as lower bounds for", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 304, + 591, + 505, + 602 + ], + "spans": [ + { + "bbox": [ + 304, + 591, + 505, + 602 + ], + "score": 1.0, + "content": "reference. The y-axis is normalized return given the", + "type": "text" + } + ], + "index": 60 + }, + { + "bbox": [ + 303, + 601, + 461, + 612 + ], + "spans": [ + { + "bbox": [ + 303, + 601, + 461, + 612 + ], + "score": 1.0, + "content": "return of oracle policy (Raffin et al., 2021).", + "type": "text" + } + ], + "index": 61 + } + ], + "index": 57 + } + ], + "index": 51.5 + }, + { + "type": "text", + "bbox": [ + 107, + 615, + 505, + 714 + ], + "lines": [ + { + "bbox": [ + 106, + 615, + 504, + 627 + ], + "spans": [ + { + "bbox": [ + 106, + 615, + 504, + 627 + ], + "score": 1.0, + "content": "unknown, aiming at finding optimal policies that perform against adversarially-chosen perturba-", + "type": "text" + } + ], + "index": 62 + }, + { + "bbox": [ + 105, + 623, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 506, + 639 + ], + "score": 1.0, + "content": "tions. Generalization in RL (Cobbe et al., 2019; Packer et al., 2018; Whiteson et al., 2011; Zhang", + "type": "text" + } + ], + "index": 63 + }, + { + "bbox": [ + 105, + 637, + 504, + 649 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 504, + 649 + ], + "score": 1.0, + "content": "et al., 2018a) focuses on unobserved aspects of the dynamics or reward function that are novel dur-", + "type": "text" + } + ], + "index": 64 + }, + { + "bbox": [ + 105, + 648, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 505, + 660 + ], + "score": 1.0, + "content": "ing testing, using an average-case objective instead of a worst-case objective like robust RL. Recent", + "type": "text" + } + ], + "index": 65 + }, + { + "bbox": [ + 106, + 659, + 504, + 671 + ], + "spans": [ + { + "bbox": [ + 106, + 659, + 504, + 671 + ], + "score": 1.0, + "content": "work has proposed efficient and performant algorithms for solving these specialized problem set-", + "type": "text" + } + ], + "index": 66 + }, + { + "bbox": [ + 106, + 669, + 505, + 682 + ], + "spans": [ + { + "bbox": [ + 106, + 669, + 505, + 682 + ], + "score": 1.0, + "content": "tings. However, these algorithms often make assumptions that preclude their application to other", + "type": "text" + } + ], + "index": 67 + }, + { + "bbox": [ + 105, + 680, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 680, + 505, + 694 + ], + "score": 1.0, + "content": "classes of POMDPs. For example, methods for robust RL are rarely used for the meta RL setting", + "type": "text" + } + ], + "index": 68 + }, + { + "bbox": [ + 105, + 691, + 505, + 704 + ], + "spans": [ + { + "bbox": [ + 105, + 691, + 505, + 704 + ], + "score": 1.0, + "content": "due to objective mismatch; methods for meta RL are rarely used for general POMDPs due to the", + "type": "text" + } + ], + "index": 69 + }, + { + "bbox": [ + 105, + 703, + 251, + 714 + ], + "spans": [ + { + "bbox": [ + 105, + 703, + 251, + 714 + ], + "score": 1.0, + "content": "stationarity assumption in meta RL.", + "type": "text" + } + ], + "index": 70 + } + ], + "index": 66, + "bbox_fs": [ + 105, + 615, + 506, + 714 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 83, + 505, + 247 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "Nonetheless, many prior works have used a simple baseline that is applicable to all POMDPs:", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "model-free RL equipped with a recurrent policy and (sometimes) value function (Duan et al., 2016;", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 104, + 505, + 116 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 505, + 116 + ], + "score": 1.0, + "content": "Fakoor et al., 2020; Igl et al., 2018; Packer et al., 2018; Rakelly et al., 2019; Wang et al., 2017; Yu", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 505, + 127 + ], + "score": 1.0, + "content": "et al., 2019). We will refer to this approach as recurrent model-free RL. This baseline is simulta-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 138 + ], + "score": 1.0, + "content": "neously simple (requiring changing only a few lines of code from a model-free RL algorithm) and", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "score": 1.0, + "content": "general. However, prior work has consistently found that recurrent model-free RL performs poorly", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 148, + 504, + 160 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 504, + 160 + ], + "score": 1.0, + "content": "across a wide range of problem settings, including meta RL (Rakelly et al., 2019; Zintgraf et al.,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 159, + 505, + 171 + ], + "spans": [ + { + "bbox": [ + 106, + 159, + 505, + 171 + ], + "score": 1.0, + "content": "2020), general POMDPs (Han et al., 2020; Igl et al., 2018), robust RL (Zhang et al., 2021), and", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 169, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 505, + 183 + ], + "score": 1.0, + "content": "generalization in RL (Packer et al., 2018). One common explanation is that specialized algorithms", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 182, + 505, + 193 + ], + "spans": [ + { + "bbox": [ + 106, + 182, + 505, + 193 + ], + "score": 1.0, + "content": "that are tailored to specific types of POMDPs are very likely to outperform recurrent model-free", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 191, + 505, + 204 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 505, + 204 + ], + "score": 1.0, + "content": "RL because they (implicitly) encode inductive biases for solving these specific tasks. For example,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 203, + 504, + 215 + ], + "spans": [ + { + "bbox": [ + 106, + 203, + 504, + 215 + ], + "score": 1.0, + "content": "algorithms for meta RL may leverage the assumption that the underlying dynamics (while unknown)", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 213, + 504, + 226 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 504, + 226 + ], + "score": 1.0, + "content": "are fixed, and the underlying goals are fixed within one episode (Rakelly et al., 2019; Zintgraf et al.,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 225, + 505, + 237 + ], + "spans": [ + { + "bbox": [ + 106, + 225, + 505, + 237 + ], + "score": 1.0, + "content": "2020); algorithms for robust RL may assume that the dynamics parameters are known (Rajeswaran", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 236, + 393, + 248 + ], + "spans": [ + { + "bbox": [ + 106, + 236, + 393, + 248 + ], + "score": 1.0, + "content": "et al., 2017a) and dynamics is Lipschitz continuous (Jiang et al., 2021).", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 107, + 253, + 505, + 352 + ], + "lines": [ + { + "bbox": [ + 106, + 253, + 504, + 265 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 504, + 265 + ], + "score": 1.0, + "content": "This paper challenges this explanation. We argue that, contrary to popular belief, recurrent model-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 264, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 505, + 276 + ], + "score": 1.0, + "content": "free RL is competitive with recent state-of-the-art algorithms across a range of different POMDP", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 274, + 505, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 505, + 287 + ], + "score": 1.0, + "content": "settings. Similar to prior work in Markovian on-policy RL methods (Andrychowicz et al., 2021;", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 286, + 504, + 298 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 504, + 298 + ], + "score": 1.0, + "content": "Engstrom et al., 2020), our experiments show that implementation in recurrent model-free RL mat-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 297, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 505, + 308 + ], + "score": 1.0, + "content": "ters. Fig. 1 shows a typical scenario in PyBullet occlusion environments (Coumans & Bai, 2016) to", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 307, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 505, + 320 + ], + "score": 1.0, + "content": "support this argument. Through extensive experiments, we show that the careful design and imple-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 318, + 504, + 330 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 504, + 330 + ], + "score": 1.0, + "content": "mentation of recurrent model-free RL is critical to its performance. Design decisions, such as the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 329, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 505, + 342 + ], + "score": 1.0, + "content": "actor-critic architecture, conditioning on previous actions and/or rewards, the underlying model-free", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 341, + 374, + 353 + ], + "spans": [ + { + "bbox": [ + 106, + 341, + 374, + 353 + ], + "score": 1.0, + "content": "RL algorithms, and context length in RNNs, are especially crucial.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 357, + 505, + 424 + ], + "lines": [ + { + "bbox": [ + 106, + 358, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 505, + 369 + ], + "score": 1.0, + "content": "The main contribution of this paper is a performant implementation of recurrent-model free RL. We", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 369, + 505, + 380 + ], + "spans": [ + { + "bbox": [ + 106, + 369, + 505, + 380 + ], + "score": 1.0, + "content": "demonstrate that simple yet important design decisions, such as the underlying RL algorithm and", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 379, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 505, + 393 + ], + "score": 1.0, + "content": "the context length, yield a recurrent model-free RL algorithm that performs on par with prior spe-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 390, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 505, + 403 + ], + "score": 1.0, + "content": "cialized POMDP algorithms on the environments those algorithms were designed to solve. Ablation", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 402, + 506, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 506, + 414 + ], + "score": 1.0, + "content": "experiments identify the importance of these design decisions. We also open-sourced our code that", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 412, + 253, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 253, + 425 + ], + "score": 1.0, + "content": "is easy to use and memory-efficient.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26.5 + }, + { + "type": "title", + "bbox": [ + 108, + 434, + 200, + 447 + ], + "lines": [ + { + "bbox": [ + 104, + 433, + 201, + 450 + ], + "spans": [ + { + "bbox": [ + 104, + 433, + 201, + 450 + ], + "score": 1.0, + "content": "2 BACKGROUND", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 106, + 459, + 505, + 555 + ], + "lines": [ + { + "bbox": [ + 105, + 459, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 384, + 471 + ], + "score": 1.0, + "content": "MDP. 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Solving an MDP", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 503, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 262, + 515 + ], + "score": 1.0, + "content": "requires learning a memoryless policy", + "type": "text" + }, + { + "bbox": [ + 262, + 504, + 342, + 515 + ], + "score": 0.87, + "content": "\\pi : \\mathcal { S \\times A } \\to [ 0 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 503, + 505, + 515 + ], + "score": 1.0, + "content": "that maximizes the expected discounted", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 513, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 104, + 513, + 137, + 533 + ], + "score": 1.0, + "content": "return:", + "type": "text" + }, + { + "bbox": [ + 137, + 514, + 343, + 534 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\pi ^ { * } = \\arg \\operatorname* { m a x } _ { \\pi } \\mathbb { E } _ { s _ { t } , a _ { t } , r _ { t } \\sim T , \\pi } \\left[ \\sum _ { t = 0 } ^ { H - 1 } \\gamma ^ { t } r _ { t + 1 } \\ | \\ s _ { 0 } \\right] } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 513, + 505, + 533 + ], + "score": 1.0, + "content": ". For any MDP, there exists an optimal", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 531, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 505, + 545 + ], + "score": 1.0, + "content": "policy that is both memoryless and deterministic (Puterman, 2014). MaxEnt RL algorithms (Ziebart,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 542, + 456, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 456, + 556 + ], + "score": 1.0, + "content": "2010), such as SAC (Haarnoja et al., 2018a), add an entropy bonus to the RL objective.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 34.5 + }, + { + "type": "text", + "bbox": [ + 106, + 566, + 505, + 667 + ], + "lines": [ + { + "bbox": [ + 105, + 565, + 506, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 506, + 580 + ], + "score": 1.0, + "content": "POMDP. 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Let the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 599, + 506, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 254, + 614 + ], + "score": 1.0, + "content": "observable trajectory up to time-step", + "type": "text" + }, + { + "bbox": [ + 254, + 601, + 259, + 610 + ], + "score": 0.68, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 599, + 272, + 614 + ], + "score": 1.0, + "content": "be", + "type": "text" + }, + { + "bbox": [ + 272, + 601, + 425, + 612 + ], + "score": 0.87, + "content": "\\tau _ { 0 : t } = ( o _ { 0 } , a _ { 0 } , o _ { 1 } , r _ { 1 } , \\dots , a _ { t - 1 } , o _ { t } , r _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 599, + 506, + 614 + ], + "score": 1.0, + "content": ", the memory-based", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 610, + 506, + 625 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 287, + 625 + ], + "score": 1.0, + "content": "policy in the most general form is defined as", + "type": "text" + }, + { + "bbox": [ + 287, + 612, + 333, + 623 + ], + "score": 0.9, + "content": "\\pi ( \\boldsymbol { a } _ { t } \\mid \\tau _ { 0 : t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 610, + 506, + 625 + ], + "score": 1.0, + "content": ", conditioning on the whole history. 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In this scenario, the policy objective can be rewritten as", + "type": "text" + }, + { + "bbox": [ + 473, + 705, + 504, + 716 + ], + "score": 0.86, + "content": "\\pi ^ { * } =", + "type": "inline_equation" + } + ], + "index": 51 + }, + { + "bbox": [ + 101, + 711, + 504, + 740 + ], + "spans": [ + { + "bbox": [ + 101, + 711, + 121, + 740 + ], + "score": 1.0, + "content": "arg", + "type": "text" + }, + { + "bbox": [ + 122, + 715, + 347, + 736 + ], + "score": 0.88, + "content": "\\begin{array} { r } { \\operatorname* { m a x } _ { \\boldsymbol { \\pi } } \\mathbb { E } _ { s ^ { h } \\sim T _ { 0 } } \\left[ \\mathbb { E } _ { s _ { t } , a _ { t } , r _ { t } \\sim T , O , O _ { 0 } , \\boldsymbol { \\pi } } \\left[ \\sum _ { t = 0 } ^ { H - 1 } \\gamma ^ { t } r _ { t + 1 } \\ | \\ s ^ { h } \\right] \\right] } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 711, + 504, + 740 + ], + "score": 1.0, + "content": "for the average-case POMDP objec-", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 50 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 83, + 505, + 247 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "Nonetheless, many prior works have used a simple baseline that is applicable to all POMDPs:", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "model-free RL equipped with a recurrent policy and (sometimes) value function (Duan et al., 2016;", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 104, + 505, + 116 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 505, + 116 + ], + "score": 1.0, + "content": "Fakoor et al., 2020; Igl et al., 2018; Packer et al., 2018; Rakelly et al., 2019; Wang et al., 2017; Yu", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 505, + 127 + ], + "score": 1.0, + "content": "et al., 2019). We will refer to this approach as recurrent model-free RL. This baseline is simulta-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 138 + ], + "score": 1.0, + "content": "neously simple (requiring changing only a few lines of code from a model-free RL algorithm) and", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "score": 1.0, + "content": "general. However, prior work has consistently found that recurrent model-free RL performs poorly", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 148, + 504, + 160 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 504, + 160 + ], + "score": 1.0, + "content": "across a wide range of problem settings, including meta RL (Rakelly et al., 2019; Zintgraf et al.,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 159, + 505, + 171 + ], + "spans": [ + { + "bbox": [ + 106, + 159, + 505, + 171 + ], + "score": 1.0, + "content": "2020), general POMDPs (Han et al., 2020; Igl et al., 2018), robust RL (Zhang et al., 2021), and", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 169, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 505, + 183 + ], + "score": 1.0, + "content": "generalization in RL (Packer et al., 2018). One common explanation is that specialized algorithms", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 182, + 505, + 193 + ], + "spans": [ + { + "bbox": [ + 106, + 182, + 505, + 193 + ], + "score": 1.0, + "content": "that are tailored to specific types of POMDPs are very likely to outperform recurrent model-free", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 191, + 505, + 204 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 505, + 204 + ], + "score": 1.0, + "content": "RL because they (implicitly) encode inductive biases for solving these specific tasks. For example,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 203, + 504, + 215 + ], + "spans": [ + { + "bbox": [ + 106, + 203, + 504, + 215 + ], + "score": 1.0, + "content": "algorithms for meta RL may leverage the assumption that the underlying dynamics (while unknown)", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 213, + 504, + 226 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 504, + 226 + ], + "score": 1.0, + "content": "are fixed, and the underlying goals are fixed within one episode (Rakelly et al., 2019; Zintgraf et al.,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 225, + 505, + 237 + ], + "spans": [ + { + "bbox": [ + 106, + 225, + 505, + 237 + ], + "score": 1.0, + "content": "2020); algorithms for robust RL may assume that the dynamics parameters are known (Rajeswaran", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 236, + 393, + 248 + ], + "spans": [ + { + "bbox": [ + 106, + 236, + 393, + 248 + ], + "score": 1.0, + "content": "et al., 2017a) and dynamics is Lipschitz continuous (Jiang et al., 2021).", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 82, + 505, + 248 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 253, + 505, + 352 + ], + "lines": [ + { + "bbox": [ + 106, + 253, + 504, + 265 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 504, + 265 + ], + "score": 1.0, + "content": "This paper challenges this explanation. We argue that, contrary to popular belief, recurrent model-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 264, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 505, + 276 + ], + "score": 1.0, + "content": "free RL is competitive with recent state-of-the-art algorithms across a range of different POMDP", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 274, + 505, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 505, + 287 + ], + "score": 1.0, + "content": "settings. Similar to prior work in Markovian on-policy RL methods (Andrychowicz et al., 2021;", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 286, + 504, + 298 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 504, + 298 + ], + "score": 1.0, + "content": "Engstrom et al., 2020), our experiments show that implementation in recurrent model-free RL mat-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 297, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 505, + 308 + ], + "score": 1.0, + "content": "ters. Fig. 1 shows a typical scenario in PyBullet occlusion environments (Coumans & Bai, 2016) to", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 307, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 505, + 320 + ], + "score": 1.0, + "content": "support this argument. Through extensive experiments, we show that the careful design and imple-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 318, + 504, + 330 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 504, + 330 + ], + "score": 1.0, + "content": "mentation of recurrent model-free RL is critical to its performance. Design decisions, such as the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 329, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 505, + 342 + ], + "score": 1.0, + "content": "actor-critic architecture, conditioning on previous actions and/or rewards, the underlying model-free", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 341, + 374, + 353 + ], + "spans": [ + { + "bbox": [ + 106, + 341, + 374, + 353 + ], + "score": 1.0, + "content": "RL algorithms, and context length in RNNs, are especially crucial.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 253, + 505, + 353 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 357, + 505, + 424 + ], + "lines": [ + { + "bbox": [ + 106, + 358, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 505, + 369 + ], + "score": 1.0, + "content": "The main contribution of this paper is a performant implementation of recurrent-model free RL. We", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 369, + 505, + 380 + ], + "spans": [ + { + "bbox": [ + 106, + 369, + 505, + 380 + ], + "score": 1.0, + "content": "demonstrate that simple yet important design decisions, such as the underlying RL algorithm and", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 379, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 505, + 393 + ], + "score": 1.0, + "content": "the context length, yield a recurrent model-free RL algorithm that performs on par with prior spe-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 390, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 505, + 403 + ], + "score": 1.0, + "content": "cialized POMDP algorithms on the environments those algorithms were designed to solve. Ablation", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 402, + 506, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 506, + 414 + ], + "score": 1.0, + "content": "experiments identify the importance of these design decisions. We also open-sourced our code that", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 412, + 253, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 253, + 425 + ], + "score": 1.0, + "content": "is easy to use and memory-efficient.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 358, + 506, + 425 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 434, + 200, + 447 + ], + "lines": [ + { + "bbox": [ + 104, + 433, + 201, + 450 + ], + "spans": [ + { + "bbox": [ + 104, + 433, + 201, + 450 + ], + "score": 1.0, + "content": "2 BACKGROUND", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 106, + 459, + 505, + 555 + ], + "lines": [ + { + "bbox": [ + 105, + 459, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 384, + 471 + ], + "score": 1.0, + "content": "MDP. A Markov decision process (MDP) (Bellman, 1957) is a tuple", + "type": "text" + }, + { + "bbox": [ + 384, + 459, + 474, + 471 + ], + "score": 0.91, + "content": "( S , A , T , T _ { 0 } , R , H , \\gamma )", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 459, + 505, + 471 + ], + "score": 1.0, + "content": ", where", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 470, + 505, + 482 + ], + "spans": [ + { + "bbox": [ + 106, + 471, + 114, + 480 + ], + "score": 0.78, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 470, + 196, + 482 + ], + "score": 1.0, + "content": "is the set of states,", + "type": "text" + }, + { + "bbox": [ + 197, + 471, + 206, + 480 + ], + "score": 0.78, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 470, + 293, + 482 + ], + "score": 1.0, + "content": "is the set of actions,", + "type": "text" + }, + { + "bbox": [ + 294, + 470, + 401, + 481 + ], + "score": 0.89, + "content": "T : \\mathcal { S } \\times \\mathcal { A } \\times \\mathcal { S } [ 0 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 470, + 505, + 482 + ], + "score": 1.0, + "content": "is the transition function", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 481, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 481, + 158, + 492 + ], + "score": 1.0, + "content": "(dynamics),", + "type": "text" + }, + { + "bbox": [ + 158, + 481, + 226, + 493 + ], + "score": 0.91, + "content": "T _ { 0 } : { \\mathcal { S } } [ 0 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 481, + 353, + 492 + ], + "score": 1.0, + "content": "is the initial state distribution,", + "type": "text" + }, + { + "bbox": [ + 353, + 482, + 448, + 492 + ], + "score": 0.9, + "content": "R : \\mathcal { S } \\times \\mathcal { A } \\times \\mathcal { S } \\to \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 481, + 505, + 492 + ], + "score": 1.0, + "content": "is the reward", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 492, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 492, + 146, + 505 + ], + "score": 1.0, + "content": "function,", + "type": "text" + }, + { + "bbox": [ + 147, + 493, + 182, + 503 + ], + "score": 0.9, + "content": "H \\in \\mathbb { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 492, + 287, + 505 + ], + "score": 1.0, + "content": "is the time horizon, and", + "type": "text" + }, + { + "bbox": [ + 288, + 492, + 333, + 504 + ], + "score": 0.91, + "content": "\\gamma \\in \\ [ 0 , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 492, + 505, + 505 + ], + "score": 1.0, + "content": "is the discount factor. Solving an MDP", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 503, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 262, + 515 + ], + "score": 1.0, + "content": "requires learning a memoryless policy", + "type": "text" + }, + { + "bbox": [ + 262, + 504, + 342, + 515 + ], + "score": 0.87, + "content": "\\pi : \\mathcal { S \\times A } \\to [ 0 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 503, + 505, + 515 + ], + "score": 1.0, + "content": "that maximizes the expected discounted", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 513, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 104, + 513, + 137, + 533 + ], + "score": 1.0, + "content": "return:", + "type": "text" + }, + { + "bbox": [ + 137, + 514, + 343, + 534 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\pi ^ { * } = \\arg \\operatorname* { m a x } _ { \\pi } \\mathbb { E } _ { s _ { t } , a _ { t } , r _ { t } \\sim T , \\pi } \\left[ \\sum _ { t = 0 } ^ { H - 1 } \\gamma ^ { t } r _ { t + 1 } \\ | \\ s _ { 0 } \\right] } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 513, + 505, + 533 + ], + "score": 1.0, + "content": ". 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For each subarea, we list the information of the hid-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 60, + 505, + 74 + ], + "spans": [ + { + "bbox": [ + 105, + 60, + 140, + 74 + ], + "score": 1.0, + "content": "den state", + "type": "text" + }, + { + "bbox": [ + 140, + 60, + 150, + 70 + ], + "score": 0.87, + "content": "s ^ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 60, + 505, + 74 + ], + "score": 1.0, + "content": "including its appearance in dynamics and reward function and its stationarity during one trajectory.", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 107, + 72, + 505, + 83 + ], + "spans": [ + { + "bbox": [ + 107, + 72, + 408, + 83 + ], + "score": 1.0, + "content": "We also list the policy input space that are connected with the hidden states, where", + "type": "text" + }, + { + "bbox": [ + 409, + 74, + 415, + 82 + ], + "score": 0.29, + "content": "\\bigcirc", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 72, + 505, + 83 + ], + "score": 1.0, + "content": ", a, r, and d refer to the", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 84, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 84, + 505, + 95 + ], + "score": 1.0, + "content": "sequence of observations, actions, rewards, and done signals, respectively. 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We append the check", + "type": "text" + }, + { + "bbox": [ + 226, + 105, + 239, + 115 + ], + "score": 0.66, + "content": "( \\checkmark )", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 105, + 293, + 116 + ], + "score": 1.0, + "content": "or cross mark", + "type": "text" + }, + { + "bbox": [ + 293, + 105, + 305, + 115 + ], + "score": 0.58, + "content": "( { \\pmb x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 105, + 324, + 116 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 325, + 105, + 331, + 114 + ], + "score": 0.27, + "content": "^ *", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 105, + 505, + 116 + ], + "score": 1.0, + "content": "if it applies to some but not all the work in that", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 116, + 324, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 116, + 324, + 127 + ], + "score": 1.0, + "content": "subarea. The notation in this table will be covered in Sec. 2.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3 + }, + { + "type": "table", + "bbox": [ + 106, + 130, + 509, + 232 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 106, + 130, + 509, + 232 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 130, + 509, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 130, + 509, + 232 + ], + "score": 0.983, + "html": "
Subareashin dynam-sics?sin re-ward?Isshsta-tionary?Policy in-put spaceRL objec-tiveDomainshift?
“Standard”POMDP√√XoarAvgX
Meta RLX*√√oardAvgX
Robust RL*XoaWorst×
Generalizationin RL*X**oaAvg
", + "type": "table", + "image_path": "6575bd0685a9fe3dbec262e65cdf86c5b1403c859d5dfc021b5af4eec2b47ed7.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 106, + 130, + 509, + 164.0 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 106, + 164.0, + 509, + 198.0 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 106, + 198.0, + 509, + 232.0 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "table_footnote", + "bbox": [ + 106, + 234, + 505, + 263 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 101, + 227, + 510, + 265 + ], + "spans": [ + { + "bbox": [ + 101, + 227, + 137, + 265 + ], + "score": 1.0, + "content": "tive, or POMD", + "type": "text" + }, + { + "bbox": [ + 138, + 234, + 428, + 254 + ], + "score": 0.77, + "content": "\\begin{array} { r } { \\pi ^ { * } = \\arg \\operatorname* { m a x } _ { \\pi } \\operatorname* { m i n } _ { s ^ { h } \\in \\mathrm { s u p p } ( T _ { 0 } ) } \\mathbb { E } _ { s _ { t } , a _ { t } , r _ { t } \\sim T , O , O _ { 0 } , \\pi } \\left[ \\sum _ { t = 0 } ^ { H - 1 } \\gamma ^ { t } r _ { t + 1 } \\mid s ^ { h } \\right] } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 227, + 510, + 265 + ], + "score": 1.0, + "content": "for the worst-case", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + } + ], + "index": 9.0 + }, + { + "type": "title", + "bbox": [ + 108, + 273, + 212, + 286 + ], + "lines": [ + { + "bbox": [ + 104, + 273, + 213, + 288 + ], + "spans": [ + { + "bbox": [ + 104, + 273, + 213, + 288 + ], + "score": 1.0, + "content": "3 RELATED WORK", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 293, + 504, + 316 + ], + "lines": [ + { + "bbox": [ + 106, + 293, + 506, + 306 + ], + "spans": [ + { + "bbox": [ + 106, + 293, + 506, + 306 + ], + "score": 1.0, + "content": "In this section, we discuss several subareas of RL that both explicitly and implicitly solve POMDPs,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 305, + 496, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 305, + 496, + 318 + ], + "score": 1.0, + "content": "as well as algorithms proposed for these specialized settings. Table 1 summarizes these subareas.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5 + }, + { + "type": "text", + "bbox": [ + 107, + 325, + 505, + 568 + ], + "lines": [ + { + "bbox": [ + 105, + 324, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 505, + 339 + ], + "score": 1.0, + "content": "RL for “Standard” POMDPs. We use the term “standard” to refer to prior work that explicitly", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 336, + 505, + 350 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 505, + 350 + ], + "score": 1.0, + "content": "labels the problems studied as POMDPs. Common tasks include scenarios where the states are par-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 347, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 505, + 360 + ], + "score": 1.0, + "content": "tially occluded (Heess et al., 2015), different states correspond to the same observation (perceptual", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 359, + 504, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 359, + 504, + 371 + ], + "score": 1.0, + "content": "aliasing (Whitehead & Ballard, 1990)), random frames are dropped (Hausknecht & Stone, 2015),", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 370, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 505, + 383 + ], + "score": 1.0, + "content": "observations use egocentric images (Zhu et al., 2017), or the observations are perturbed with ran-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 381, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 106, + 381, + 505, + 393 + ], + "score": 1.0, + "content": "dom noise (Meng et al., 2021). These POMDPs often have hidden states that are non-stationary and", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 391, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 505, + 404 + ], + "score": 1.0, + "content": "affect both the rewards and the dynamics. POMDPs are hard to solve (Littman, 1996; Papadimitriou", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 402, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 106, + 402, + 505, + 416 + ], + "score": 1.0, + "content": "& Tsitsiklis, 1987) because of the curse of dimensionality: the size of the history grows linearly", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 414, + 504, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 414, + 504, + 426 + ], + "score": 1.0, + "content": "with the horizon length. Many prior POMDP algorithms (Cassandra et al., 1994; Kaelbling et al.,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 424, + 505, + 437 + ], + "spans": [ + { + "bbox": [ + 106, + 424, + 505, + 437 + ], + "score": 1.0, + "content": "1998) attempt to infer the state from the past sequence of observations, and then apply standard RL", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 436, + 504, + 447 + ], + "spans": [ + { + "bbox": [ + 106, + 436, + 504, + 447 + ], + "score": 1.0, + "content": "techniques to that inferred state. However, the exact inference requires the knowledge of the dy-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 446, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 505, + 459 + ], + "score": 1.0, + "content": "namics, emission, and reward functions, and is intractable in all except the most simple settings. A", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 458, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 505, + 470 + ], + "score": 1.0, + "content": "common strategy for solving these general POMDPs is to use recurrent policies, which take the en-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 468, + 504, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 504, + 481 + ], + "score": 1.0, + "content": "tire history of past observations as inputs (Bakker, 2001; Schmidhuber, 1991; Wierstra et al., 2007).", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 479, + 505, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 505, + 492 + ], + "score": 1.0, + "content": "This strategy is very simple and general, and can be applied to arbitrary tasks without knowledge", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 491, + 506, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 506, + 503 + ], + "score": 1.0, + "content": "of the task structure (e.g., whether the hidden states change within an episode) across long time", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 502, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 505, + 513 + ], + "score": 1.0, + "content": "horizons (Duan et al., 2016). These recurrent strategies can be further subdivided into model-free", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 512, + 504, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 504, + 525 + ], + "score": 1.0, + "content": "methods (Hausknecht & Stone, 2015; Heess et al., 2015; Meng et al., 2021) , where the single objec-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 523, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 505, + 536 + ], + "score": 1.0, + "content": "tive is to maximize the return, and model-based methods (Freeman et al., 2019; Han et al., 2020; Igl", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 533, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 505, + 547 + ], + "score": 1.0, + "content": "et al., 2018; Watter et al., 2015) that have explicit objectives on modeling the belief states and use", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 544, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 505, + 558 + ], + "score": 1.0, + "content": "them as the inputs of memoryless policies. The recurrent model-free RL that we focus on belongs", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 556, + 362, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 362, + 569 + ], + "score": 1.0, + "content": "to the class of model-free off-policy memory-based algorithms.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 24.5 + }, + { + "type": "text", + "bbox": [ + 107, + 578, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 577, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 505, + 590 + ], + "score": 1.0, + "content": "Meta RL. Meta RL, also called “learning to learn” (Schmidhuber, 1987; Thrun & Pratt, 2012),", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 588, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 505, + 601 + ], + "score": 1.0, + "content": "focuses on POMDPs where some parameters in the rewards or (less commonly) dynamics are varied", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "score": 1.0, + "content": "from episode to episode, but remain fixed within a single episode, which represent different tasks", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "score": 1.0, + "content": "with different values (Humplik et al., 2019). The meta RL setting is almost the same as multi-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 622, + 505, + 633 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 505, + 633 + ], + "score": 1.0, + "content": "task RL (Wilson et al., 2007; Yu et al., 2019), but differs in that multi-task RL can observe the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "score": 1.0, + "content": "task parameters, making it an MDP instead of a POMDP. Algorithms for meta RL can be roughly", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 643, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 505, + 656 + ], + "score": 1.0, + "content": "categorized based on how the adaptation step is performed. Gradient-based algorithms (Fakoor", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 653, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 505, + 667 + ], + "score": 1.0, + "content": "et al., 2020; Finn et al., 2017; Hochreiter et al., 2001) run a few gradient steps on the pre-trained", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "models to adapt. Memory or context-based algorithms use RNNs to implicitly adapt, which can", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "be further subdivided into implicit and explicit task inference methods. Implicit task inference", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "methods (Duan et al., 2016; Wang et al., 2017) use RL objective only to learn recurrent policies.", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "Explicit task inference methods (Rakelly et al., 2019; Zintgraf et al., 2020) train an extra inference", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "model to explicitly estimate task embeddings (i.e., a representation of the unobserved parameters)", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "by variational inference. Task embeddings are then used as additional inputs to memoryless policies.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 42.5 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "score": 1.0, + "content": "3", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 50, + 505, + 127 + ], + "lines": [ + { + "bbox": [ + 106, + 49, + 504, + 61 + ], + "spans": [ + { + "bbox": [ + 106, + 49, + 504, + 61 + ], + "score": 1.0, + "content": "Table 1: The summary of selected POMDP subareas. For each subarea, we list the information of the hid-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 60, + 505, + 74 + ], + "spans": [ + { + "bbox": [ + 105, + 60, + 140, + 74 + ], + "score": 1.0, + "content": "den state", + "type": "text" + }, + { + "bbox": [ + 140, + 60, + 150, + 70 + ], + "score": 0.87, + "content": "s ^ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 60, + 505, + 74 + ], + "score": 1.0, + "content": "including its appearance in dynamics and reward function and its stationarity during one trajectory.", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 107, + 72, + 505, + 83 + ], + "spans": [ + { + "bbox": [ + 107, + 72, + 408, + 83 + ], + "score": 1.0, + "content": "We also list the policy input space that are connected with the hidden states, where", + "type": "text" + }, + { + "bbox": [ + 409, + 74, + 415, + 82 + ], + "score": 0.29, + "content": "\\bigcirc", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 72, + 505, + 83 + ], + "score": 1.0, + "content": ", a, r, and d refer to the", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 84, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 84, + 505, + 95 + ], + "score": 1.0, + "content": "sequence of observations, actions, rewards, and done signals, respectively. 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Subareashin dynam-sics?sin re-ward?Isshsta-tionary?Policy in-put spaceRL objec-tiveDomainshift?
“Standard”POMDP√√XoarAvgX
Meta RLX*√√oardAvgX
Robust RL*XoaWorst×
Generalizationin RL*X**oaAvg
", + "type": "table", + "image_path": "6575bd0685a9fe3dbec262e65cdf86c5b1403c859d5dfc021b5af4eec2b47ed7.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 106, + 130, + 509, + 164.0 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 106, + 164.0, + 509, + 198.0 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 106, + 198.0, + 509, + 232.0 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "table_footnote", + "bbox": [ + 106, + 234, + 505, + 263 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 101, + 227, + 510, + 265 + ], + "spans": [ + { + "bbox": [ + 101, + 227, + 137, + 265 + ], + "score": 1.0, + "content": "tive, or POMD", + "type": "text" + }, + { + "bbox": [ + 138, + 234, + 428, + 254 + ], + "score": 0.77, + "content": "\\begin{array} { r } { \\pi ^ { * } = \\arg \\operatorname* { m a x } _ { \\pi } \\operatorname* { m i n } _ { s ^ { h } \\in \\mathrm { s u p p } ( T _ { 0 } ) } \\mathbb { E } _ { s _ { t } , a _ { t } , r _ { t } \\sim T , O , O _ { 0 } , \\pi } \\left[ \\sum _ { t = 0 } ^ { H - 1 } \\gamma ^ { t } r _ { t + 1 } \\mid s ^ { h } \\right] } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 227, + 510, + 265 + ], + "score": 1.0, + "content": "for the worst-case", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + } + ], + "index": 9.0 + }, + { + "type": "title", + "bbox": [ + 108, + 273, + 212, + 286 + ], + "lines": [ + { + "bbox": [ + 104, + 273, + 213, + 288 + ], + "spans": [ + { + "bbox": [ + 104, + 273, + 213, + 288 + ], + "score": 1.0, + "content": "3 RELATED WORK", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 293, + 504, + 316 + ], + "lines": [ + { + "bbox": [ + 106, + 293, + 506, + 306 + ], + "spans": [ + { + "bbox": [ + 106, + 293, + 506, + 306 + ], + "score": 1.0, + "content": "In this section, we discuss several subareas of RL that both explicitly and implicitly solve POMDPs,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 305, + 496, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 305, + 496, + 318 + ], + "score": 1.0, + "content": "as well as algorithms proposed for these specialized settings. Table 1 summarizes these subareas.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 293, + 506, + 318 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 325, + 505, + 568 + ], + "lines": [ + { + "bbox": [ + 105, + 324, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 505, + 339 + ], + "score": 1.0, + "content": "RL for “Standard” POMDPs. We use the term “standard” to refer to prior work that explicitly", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 336, + 505, + 350 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 505, + 350 + ], + "score": 1.0, + "content": "labels the problems studied as POMDPs. Common tasks include scenarios where the states are par-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 347, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 505, + 360 + ], + "score": 1.0, + "content": "tially occluded (Heess et al., 2015), different states correspond to the same observation (perceptual", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 359, + 504, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 359, + 504, + 371 + ], + "score": 1.0, + "content": "aliasing (Whitehead & Ballard, 1990)), random frames are dropped (Hausknecht & Stone, 2015),", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 370, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 505, + 383 + ], + "score": 1.0, + "content": "observations use egocentric images (Zhu et al., 2017), or the observations are perturbed with ran-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 381, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 106, + 381, + 505, + 393 + ], + "score": 1.0, + "content": "dom noise (Meng et al., 2021). These POMDPs often have hidden states that are non-stationary and", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 391, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 505, + 404 + ], + "score": 1.0, + "content": "affect both the rewards and the dynamics. POMDPs are hard to solve (Littman, 1996; Papadimitriou", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 402, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 106, + 402, + 505, + 416 + ], + "score": 1.0, + "content": "& Tsitsiklis, 1987) because of the curse of dimensionality: the size of the history grows linearly", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 414, + 504, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 414, + 504, + 426 + ], + "score": 1.0, + "content": "with the horizon length. Many prior POMDP algorithms (Cassandra et al., 1994; Kaelbling et al.,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 424, + 505, + 437 + ], + "spans": [ + { + "bbox": [ + 106, + 424, + 505, + 437 + ], + "score": 1.0, + "content": "1998) attempt to infer the state from the past sequence of observations, and then apply standard RL", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 436, + 504, + 447 + ], + "spans": [ + { + "bbox": [ + 106, + 436, + 504, + 447 + ], + "score": 1.0, + "content": "techniques to that inferred state. However, the exact inference requires the knowledge of the dy-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 446, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 505, + 459 + ], + "score": 1.0, + "content": "namics, emission, and reward functions, and is intractable in all except the most simple settings. A", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 458, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 505, + 470 + ], + "score": 1.0, + "content": "common strategy for solving these general POMDPs is to use recurrent policies, which take the en-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 468, + 504, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 504, + 481 + ], + "score": 1.0, + "content": "tire history of past observations as inputs (Bakker, 2001; Schmidhuber, 1991; Wierstra et al., 2007).", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 479, + 505, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 505, + 492 + ], + "score": 1.0, + "content": "This strategy is very simple and general, and can be applied to arbitrary tasks without knowledge", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 491, + 506, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 506, + 503 + ], + "score": 1.0, + "content": "of the task structure (e.g., whether the hidden states change within an episode) across long time", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 502, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 505, + 513 + ], + "score": 1.0, + "content": "horizons (Duan et al., 2016). These recurrent strategies can be further subdivided into model-free", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 512, + 504, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 504, + 525 + ], + "score": 1.0, + "content": "methods (Hausknecht & Stone, 2015; Heess et al., 2015; Meng et al., 2021) , where the single objec-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 523, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 505, + 536 + ], + "score": 1.0, + "content": "tive is to maximize the return, and model-based methods (Freeman et al., 2019; Han et al., 2020; Igl", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 533, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 505, + 547 + ], + "score": 1.0, + "content": "et al., 2018; Watter et al., 2015) that have explicit objectives on modeling the belief states and use", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 544, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 505, + 558 + ], + "score": 1.0, + "content": "them as the inputs of memoryless policies. The recurrent model-free RL that we focus on belongs", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 556, + 362, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 362, + 569 + ], + "score": 1.0, + "content": "to the class of model-free off-policy memory-based algorithms.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 324, + 506, + 569 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 578, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 577, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 505, + 590 + ], + "score": 1.0, + "content": "Meta RL. Meta RL, also called “learning to learn” (Schmidhuber, 1987; Thrun & Pratt, 2012),", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 588, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 505, + 601 + ], + "score": 1.0, + "content": "focuses on POMDPs where some parameters in the rewards or (less commonly) dynamics are varied", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "score": 1.0, + "content": "from episode to episode, but remain fixed within a single episode, which represent different tasks", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "score": 1.0, + "content": "with different values (Humplik et al., 2019). The meta RL setting is almost the same as multi-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 622, + 505, + 633 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 505, + 633 + ], + "score": 1.0, + "content": "task RL (Wilson et al., 2007; Yu et al., 2019), but differs in that multi-task RL can observe the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "score": 1.0, + "content": "task parameters, making it an MDP instead of a POMDP. Algorithms for meta RL can be roughly", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 643, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 505, + 656 + ], + "score": 1.0, + "content": "categorized based on how the adaptation step is performed. Gradient-based algorithms (Fakoor", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 653, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 505, + 667 + ], + "score": 1.0, + "content": "et al., 2020; Finn et al., 2017; Hochreiter et al., 2001) run a few gradient steps on the pre-trained", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "models to adapt. Memory or context-based algorithms use RNNs to implicitly adapt, which can", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "be further subdivided into implicit and explicit task inference methods. Implicit task inference", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "methods (Duan et al., 2016; Wang et al., 2017) use RL objective only to learn recurrent policies.", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "Explicit task inference methods (Rakelly et al., 2019; Zintgraf et al., 2020) train an extra inference", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "model to explicitly estimate task embeddings (i.e., a representation of the unobserved parameters)", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "by variational inference. Task embeddings are then used as additional inputs to memoryless policies.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 42.5, + "bbox_fs": [ + 105, + 577, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 236 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 505, + 95 + ], + "score": 1.0, + "content": "Robust RL. The goal of robust RL is to find a policy that maximizes returns in the worst-case", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "environments. Early work in the control and operations research community (Khalil et al., 1996;", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "Nilim & Ghaoui, 2005) and RL community (Bagnell et al., 2001; Morimoto & Doya, 2005) focused", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "on linear or finite systems. Prior work designs deep RL algorithms that are robust against a variety of", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 138 + ], + "score": 1.0, + "content": "adversarial attacks, including attacks on the dynamics (Jiang et al., 2021; Rajeswaran et al., 2017a),", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 136, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 506, + 150 + ], + "score": 1.0, + "content": "observations (Huang et al., 2017; Pattanaik et al., 2018; Zhang et al., 2021), and actions (Gleave", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 148, + 505, + 160 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 505, + 160 + ], + "score": 1.0, + "content": "et al., 2020; Pinto et al., 2017; Tessler et al., 2019). Treating the robust RL problem as a POMDP,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 158, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 505, + 172 + ], + "score": 1.0, + "content": "rather than an MDP (as done in most prior work), unlocks a key capability for RL agents, because", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 171, + 505, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 505, + 182 + ], + "score": 1.0, + "content": "agents can use their memory to identify the hidden states of the current adversarial environment,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 180, + 505, + 193 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 505, + 193 + ], + "score": 1.0, + "content": "although previous work (Jiang et al., 2021; Rajeswaran et al., 2017a) only train Markovian policies", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 191, + 505, + 204 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 505, + 204 + ], + "score": 1.0, + "content": "on POMDPs. While some work find memory-based policies are more robust to the adversarial", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 203, + 505, + 215 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 505, + 215 + ], + "score": 1.0, + "content": "attacks than Markovian policies (Russo & Proutiere ` , 2021; Zhang et al., 2021), they train these", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 213, + 505, + 226 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 505, + 226 + ], + "score": 1.0, + "content": "baselines in a single MDP without adversaries, which differs from our training setting where the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 225, + 343, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 343, + 237 + ], + "score": 1.0, + "content": "recurrent model-free RL can have access to a set of MDPs.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 107, + 246, + 505, + 444 + ], + "lines": [ + { + "bbox": [ + 105, + 245, + 505, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 505, + 260 + ], + "score": 1.0, + "content": "Generalization in RL. The goal of generalization in RL is to make RL algorithms perform well", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 258, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 106, + 258, + 505, + 270 + ], + "score": 1.0, + "content": "in test domains that are unseen during training, which emphasizes the average case on the novel test", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 268, + 505, + 281 + ], + "spans": [ + { + "bbox": [ + 106, + 268, + 505, + 281 + ], + "score": 1.0, + "content": "domains instead of the worse case in the possibly seen test domains as in robust RL. Prior work", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 280, + 504, + 292 + ], + "spans": [ + { + "bbox": [ + 106, + 280, + 504, + 292 + ], + "score": 1.0, + "content": "have studied generalization to initial states in the same MDP (Rajeswaran et al., 2017b; White-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 290, + 504, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 504, + 303 + ], + "score": 1.0, + "content": "son et al., 2011; Zhang et al., 2018b), random disturbance in dynamics (Rajeswaran et al., 2017b),", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 300, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 104, + 300, + 505, + 315 + ], + "score": 1.0, + "content": "states (Stulp et al., 2011), observations (Song et al., 2020; Zhang et al., 2018a), and actions (Srouji", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 312, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 505, + 325 + ], + "score": 1.0, + "content": "et al., 2018), and different modes in procedurally generated games (Cobbe et al., 2019; Farebrother", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 324, + 504, + 336 + ], + "spans": [ + { + "bbox": [ + 106, + 324, + 504, + 336 + ], + "score": 1.0, + "content": "et al., 2018; Justesen et al., 2018). Among them, Packer et al. (2018) provides a benchmark on both", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 333, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 505, + 348 + ], + "score": 1.0, + "content": "in-distribution (ID) and out-of-distribution (OOD) generalization to different dynamics parameters,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 345, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 106, + 345, + 505, + 358 + ], + "score": 1.0, + "content": "and Zhao et al. (2019) extends the benchmark by introducing random noise in states, observations,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 356, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 505, + 369 + ], + "score": 1.0, + "content": "and actions. Algorithms for improving generalization in RL can be roughly divided into classic", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 367, + 505, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 505, + 380 + ], + "score": 1.0, + "content": "regularization methods such as weight decay, dropout, batch normalization, and entropy regulariza-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 378, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 505, + 390 + ], + "score": 1.0, + "content": "tion (Cobbe et al., 2020; Farebrother et al., 2018; Igl et al., 2019), model architectures (Raileanu", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 389, + 504, + 402 + ], + "spans": [ + { + "bbox": [ + 106, + 389, + 504, + 402 + ], + "score": 1.0, + "content": "& Fergus, 2021; Srouji et al., 2018), data augmentation through randomization (Lee et al., 2020;", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 399, + 505, + 413 + ], + "spans": [ + { + "bbox": [ + 106, + 399, + 505, + 413 + ], + "score": 1.0, + "content": "Tobin et al., 2017), Although introducing observational noise and the change in dynamics parame-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 412, + 504, + 423 + ], + "spans": [ + { + "bbox": [ + 106, + 412, + 504, + 423 + ], + "score": 1.0, + "content": "ters will transform MDPs to POMDPs, few work study memory-based policies such as model-free", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 422, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 505, + 435 + ], + "score": 1.0, + "content": "recurrent RL with mixed results. Same algorithm RL2 (Duan et al., 2016) was found to perform", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 432, + 373, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 373, + 447 + ], + "score": 1.0, + "content": "badly in Packer et al. (2018) but relatively well in Yu et al. (2019).", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 22.5 + }, + { + "type": "title", + "bbox": [ + 108, + 456, + 443, + 469 + ], + "lines": [ + { + "bbox": [ + 105, + 456, + 446, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 446, + 471 + ], + "score": 1.0, + "content": "4 DESIGN CONSIDERATIONS FOR RECURRENT MODEL-FREE RL", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 477, + 505, + 587 + ], + "lines": [ + { + "bbox": [ + 105, + 477, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 505, + 489 + ], + "score": 1.0, + "content": "Implementing a recurrent model-free RL algorithm requires making a number of design decisions.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 488, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 505, + 500 + ], + "score": 1.0, + "content": "This section describes the decisions that we found most important to make recurrent model-free RL", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 498, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 505, + 511 + ], + "score": 1.0, + "content": "competitive with more complex, recent algorithms. We will focus on continuous control problems", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 510, + 505, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 505, + 522 + ], + "score": 1.0, + "content": "with state-based inputs (i.e., not image-based inputs). Importantly, we assume that the policy can", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 520, + 505, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 505, + 533 + ], + "score": 1.0, + "content": "observe the reward and done signals (the end of one episode during one trial (Duan et al., 2016))", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 532, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 505, + 544 + ], + "score": 1.0, + "content": "from the environment during evaluation. This assumption is common in prior work (Han et al.,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 542, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 505, + 555 + ], + "score": 1.0, + "content": "2020; Zintgraf et al., 2020), but many recurrent model-free implementations do not provide the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 553, + 506, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 553, + 506, + 566 + ], + "score": 1.0, + "content": "agent with information. In the following paragraphs, we will describe the important decision factors", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 564, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 505, + 577 + ], + "score": 1.0, + "content": "in recurrent model-free RL. Table 2 summarizes how prior work and our method makes these design", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 576, + 330, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 330, + 588 + ], + "score": 1.0, + "content": "decisions when implementing recurrent model-free RL.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 37.5 + }, + { + "type": "text", + "bbox": [ + 107, + 597, + 505, + 685 + ], + "lines": [ + { + "bbox": [ + 106, + 598, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 598, + 505, + 609 + ], + "score": 1.0, + "content": "Recurrent Off-Policy Actor-Critic Architecture. The first important design decision is whether", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 609, + 505, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 505, + 620 + ], + "score": 1.0, + "content": "the recurrent policy (actor) and the recurrent Q-value function (critic) use shared RNN encoder (and", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 619, + 505, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 505, + 632 + ], + "score": 1.0, + "content": "embedders) or use separate ones. In the experiment section (Sec. 5.2) we will show that a shared", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 631, + 505, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 505, + 642 + ], + "score": 1.0, + "content": "encoder would cause large gradient norm in the (off-policy) recurrent actor-critic and thus hinder", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 641, + 505, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 641, + 505, + 653 + ], + "score": 1.0, + "content": "learning, while separate encoders can greatly mitigate this issue and learn efficiently. This echoes", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 652, + 506, + 665 + ], + "spans": [ + { + "bbox": [ + 105, + 652, + 506, + 665 + ], + "score": 1.0, + "content": "prior work (Fakoor et al., 2020; Meng et al., 2021; Sun et al., 2021; Wang et al., 2020) that also use", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 664, + 505, + 676 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 505, + 676 + ], + "score": 1.0, + "content": "separate encoders in their (off-policy) recurrent actor-critic. To avoid running an inordinate number", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 673, + 414, + 687 + ], + "spans": [ + { + "bbox": [ + 105, + 673, + 414, + 687 + ], + "score": 1.0, + "content": "of experiments, we will use the separate architecture in the rest of the paper.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 46.5 + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "Policy Input Space. The next consideration is the input space of the model-free policy. The", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 301, + 722 + ], + "score": 1.0, + "content": "maximal input space of policy to emit an action", + "type": "text" + }, + { + "bbox": [ + 302, + 711, + 312, + 721 + ], + "score": 0.83, + "content": "a _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 710, + 344, + 722 + ], + "score": 1.0, + "content": "at time", + "type": "text" + }, + { + "bbox": [ + 344, + 711, + 349, + 720 + ], + "score": 0.71, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 710, + 505, + 722 + ], + "score": 1.0, + "content": ", should be the history of all quantities", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 106, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 350, + 733 + ], + "score": 1.0, + "content": "that the policy has observed, namely the past observations", + "type": "text" + }, + { + "bbox": [ + 351, + 722, + 366, + 732 + ], + "score": 0.87, + "content": "o _ { 0 : t }", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 720, + 439, + 733 + ], + "score": 1.0, + "content": ", the past actions", + "type": "text" + }, + { + "bbox": [ + 439, + 722, + 465, + 732 + ], + "score": 0.9, + "content": "a _ { 0 : t - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 720, + 505, + 733 + ], + "score": 1.0, + "content": ", the past", + "type": "text" + } + ], + "index": 53 + } + ], + "index": 52 + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 309, + 39 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 309, + 39 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 759 + ], + "lines": [] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 236 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 505, + 95 + ], + "score": 1.0, + "content": "Robust RL. The goal of robust RL is to find a policy that maximizes returns in the worst-case", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "environments. Early work in the control and operations research community (Khalil et al., 1996;", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "Nilim & Ghaoui, 2005) and RL community (Bagnell et al., 2001; Morimoto & Doya, 2005) focused", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "on linear or finite systems. Prior work designs deep RL algorithms that are robust against a variety of", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 138 + ], + "score": 1.0, + "content": "adversarial attacks, including attacks on the dynamics (Jiang et al., 2021; Rajeswaran et al., 2017a),", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 136, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 506, + 150 + ], + "score": 1.0, + "content": "observations (Huang et al., 2017; Pattanaik et al., 2018; Zhang et al., 2021), and actions (Gleave", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 148, + 505, + 160 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 505, + 160 + ], + "score": 1.0, + "content": "et al., 2020; Pinto et al., 2017; Tessler et al., 2019). Treating the robust RL problem as a POMDP,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 158, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 505, + 172 + ], + "score": 1.0, + "content": "rather than an MDP (as done in most prior work), unlocks a key capability for RL agents, because", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 171, + 505, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 505, + 182 + ], + "score": 1.0, + "content": "agents can use their memory to identify the hidden states of the current adversarial environment,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 180, + 505, + 193 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 505, + 193 + ], + "score": 1.0, + "content": "although previous work (Jiang et al., 2021; Rajeswaran et al., 2017a) only train Markovian policies", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 191, + 505, + 204 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 505, + 204 + ], + "score": 1.0, + "content": "on POMDPs. While some work find memory-based policies are more robust to the adversarial", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 203, + 505, + 215 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 505, + 215 + ], + "score": 1.0, + "content": "attacks than Markovian policies (Russo & Proutiere ` , 2021; Zhang et al., 2021), they train these", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 213, + 505, + 226 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 505, + 226 + ], + "score": 1.0, + "content": "baselines in a single MDP without adversaries, which differs from our training setting where the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 225, + 343, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 343, + 237 + ], + "score": 1.0, + "content": "recurrent model-free RL can have access to a set of MDPs.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 6.5, + "bbox_fs": [ + 105, + 81, + 506, + 237 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 246, + 505, + 444 + ], + "lines": [ + { + "bbox": [ + 105, + 245, + 505, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 505, + 260 + ], + "score": 1.0, + "content": "Generalization in RL. The goal of generalization in RL is to make RL algorithms perform well", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 258, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 106, + 258, + 505, + 270 + ], + "score": 1.0, + "content": "in test domains that are unseen during training, which emphasizes the average case on the novel test", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 268, + 505, + 281 + ], + "spans": [ + { + "bbox": [ + 106, + 268, + 505, + 281 + ], + "score": 1.0, + "content": "domains instead of the worse case in the possibly seen test domains as in robust RL. Prior work", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 280, + 504, + 292 + ], + "spans": [ + { + "bbox": [ + 106, + 280, + 504, + 292 + ], + "score": 1.0, + "content": "have studied generalization to initial states in the same MDP (Rajeswaran et al., 2017b; White-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 290, + 504, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 504, + 303 + ], + "score": 1.0, + "content": "son et al., 2011; Zhang et al., 2018b), random disturbance in dynamics (Rajeswaran et al., 2017b),", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 300, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 104, + 300, + 505, + 315 + ], + "score": 1.0, + "content": "states (Stulp et al., 2011), observations (Song et al., 2020; Zhang et al., 2018a), and actions (Srouji", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 312, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 505, + 325 + ], + "score": 1.0, + "content": "et al., 2018), and different modes in procedurally generated games (Cobbe et al., 2019; Farebrother", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 324, + 504, + 336 + ], + "spans": [ + { + "bbox": [ + 106, + 324, + 504, + 336 + ], + "score": 1.0, + "content": "et al., 2018; Justesen et al., 2018). Among them, Packer et al. (2018) provides a benchmark on both", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 333, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 505, + 348 + ], + "score": 1.0, + "content": "in-distribution (ID) and out-of-distribution (OOD) generalization to different dynamics parameters,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 345, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 106, + 345, + 505, + 358 + ], + "score": 1.0, + "content": "and Zhao et al. (2019) extends the benchmark by introducing random noise in states, observations,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 356, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 505, + 369 + ], + "score": 1.0, + "content": "and actions. Algorithms for improving generalization in RL can be roughly divided into classic", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 367, + 505, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 505, + 380 + ], + "score": 1.0, + "content": "regularization methods such as weight decay, dropout, batch normalization, and entropy regulariza-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 378, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 505, + 390 + ], + "score": 1.0, + "content": "tion (Cobbe et al., 2020; Farebrother et al., 2018; Igl et al., 2019), model architectures (Raileanu", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 389, + 504, + 402 + ], + "spans": [ + { + "bbox": [ + 106, + 389, + 504, + 402 + ], + "score": 1.0, + "content": "& Fergus, 2021; Srouji et al., 2018), data augmentation through randomization (Lee et al., 2020;", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 399, + 505, + 413 + ], + "spans": [ + { + "bbox": [ + 106, + 399, + 505, + 413 + ], + "score": 1.0, + "content": "Tobin et al., 2017), Although introducing observational noise and the change in dynamics parame-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 412, + 504, + 423 + ], + "spans": [ + { + "bbox": [ + 106, + 412, + 504, + 423 + ], + "score": 1.0, + "content": "ters will transform MDPs to POMDPs, few work study memory-based policies such as model-free", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 422, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 505, + 435 + ], + "score": 1.0, + "content": "recurrent RL with mixed results. Same algorithm RL2 (Duan et al., 2016) was found to perform", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 432, + 373, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 373, + 447 + ], + "score": 1.0, + "content": "badly in Packer et al. (2018) but relatively well in Yu et al. (2019).", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 22.5, + "bbox_fs": [ + 104, + 245, + 505, + 447 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 456, + 443, + 469 + ], + "lines": [ + { + "bbox": [ + 105, + 456, + 446, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 446, + 471 + ], + "score": 1.0, + "content": "4 DESIGN CONSIDERATIONS FOR RECURRENT MODEL-FREE RL", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 477, + 505, + 587 + ], + "lines": [ + { + "bbox": [ + 105, + 477, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 505, + 489 + ], + "score": 1.0, + "content": "Implementing a recurrent model-free RL algorithm requires making a number of design decisions.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 488, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 505, + 500 + ], + "score": 1.0, + "content": "This section describes the decisions that we found most important to make recurrent model-free RL", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 498, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 505, + 511 + ], + "score": 1.0, + "content": "competitive with more complex, recent algorithms. We will focus on continuous control problems", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 510, + 505, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 505, + 522 + ], + "score": 1.0, + "content": "with state-based inputs (i.e., not image-based inputs). Importantly, we assume that the policy can", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 520, + 505, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 505, + 533 + ], + "score": 1.0, + "content": "observe the reward and done signals (the end of one episode during one trial (Duan et al., 2016))", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 532, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 505, + 544 + ], + "score": 1.0, + "content": "from the environment during evaluation. This assumption is common in prior work (Han et al.,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 542, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 505, + 555 + ], + "score": 1.0, + "content": "2020; Zintgraf et al., 2020), but many recurrent model-free implementations do not provide the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 553, + 506, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 553, + 506, + 566 + ], + "score": 1.0, + "content": "agent with information. In the following paragraphs, we will describe the important decision factors", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 564, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 505, + 577 + ], + "score": 1.0, + "content": "in recurrent model-free RL. Table 2 summarizes how prior work and our method makes these design", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 576, + 330, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 330, + 588 + ], + "score": 1.0, + "content": "decisions when implementing recurrent model-free RL.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 477, + 506, + 588 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 597, + 505, + 685 + ], + "lines": [ + { + "bbox": [ + 106, + 598, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 598, + 505, + 609 + ], + "score": 1.0, + "content": "Recurrent Off-Policy Actor-Critic Architecture. The first important design decision is whether", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 609, + 505, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 505, + 620 + ], + "score": 1.0, + "content": "the recurrent policy (actor) and the recurrent Q-value function (critic) use shared RNN encoder (and", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 619, + 505, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 505, + 632 + ], + "score": 1.0, + "content": "embedders) or use separate ones. In the experiment section (Sec. 5.2) we will show that a shared", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 631, + 505, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 505, + 642 + ], + "score": 1.0, + "content": "encoder would cause large gradient norm in the (off-policy) recurrent actor-critic and thus hinder", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 641, + 505, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 641, + 505, + 653 + ], + "score": 1.0, + "content": "learning, while separate encoders can greatly mitigate this issue and learn efficiently. This echoes", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 652, + 506, + 665 + ], + "spans": [ + { + "bbox": [ + 105, + 652, + 506, + 665 + ], + "score": 1.0, + "content": "prior work (Fakoor et al., 2020; Meng et al., 2021; Sun et al., 2021; Wang et al., 2020) that also use", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 664, + 505, + 676 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 505, + 676 + ], + "score": 1.0, + "content": "separate encoders in their (off-policy) recurrent actor-critic. To avoid running an inordinate number", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 673, + 414, + 687 + ], + "spans": [ + { + "bbox": [ + 105, + 673, + 414, + 687 + ], + "score": 1.0, + "content": "of experiments, we will use the separate architecture in the rest of the paper.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 46.5, + "bbox_fs": [ + 105, + 598, + 506, + 687 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "Policy Input Space. The next consideration is the input space of the model-free policy. 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Generally, the input space of optimal policy should only depend on the quantities that", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 360, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 505, + 375 + ], + "score": 1.0, + "content": "have connections with hidden states (defined in Sec. 2) (Izadi & Precup, 2005; Poupart & Boutilier,", + "type": "text", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 372, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 106, + 372, + 505, + 385 + ], + "score": 1.0, + "content": "2002). We show the policy input spaces that are connected with the hidden states for the discussed", + "type": "text", + "cross_page": true + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 384, + 505, + 396 + ], + "spans": [ + { + "bbox": [ + 106, + 384, + 505, + 396 + ], + "score": 1.0, + "content": "subareas in the “Inputs” column of Table 1. 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AlgorithmDomainArchEncoderInputsLenRL
Duan et al. (2016)Meta RLseparateGRUoard1000TRPO,PPO
Wang et al. (2017)Meta RLsharedLSTMoart5-150A2C
Baseline from Rakelly et al. (2019)Meta RLseparateGRUoard100PPO
Baseline from Zintgraf et al. (2020)Meta RLseparateGRUoardMaxA2C,PPO
Baseline from Fakoor et al. (2020)MetaRLseparateGRUoar10-25TD3
Baseline from Yu et al. (2019)Meta RLseparateGRUoard500PPO
Kostrikov (2018)POMDPsharedGRU05-2048PPO,A2C
Wang et al. (2020)POMDPseparateLSTMoa150TD3, SAC
Meng et al. (2021)POMDPseparateLSTMoa1-5TD3
Yang & Nguyen (2021)POMDPseparatebothoaMaxTD3, SAC
Baseline from Igl et al. (2018)POMDPsharedGRUoa25A2C
Baseline from Han et al. (2020)POMDPsharedLSTM064SAC
Baseline from Zhang et al. (2021)Robust RLseparateLSTM0100PPO
Baseline1 from Packer et al. (2018)GeneralizationsharedLSTM0128-512PPO,A2C
Baseline2 from Packer et al. (2018)GeneralizationseparateLSTMoard128-512PPO,A2C
Our methodMeta RLseparateLSTMoard64TD3
Our methodPOMDPseparateGRUoa64TD3
Our methodRobust RLseparateLSTM064TD3
Our methodGeneralizationseparateLSTM064TD3
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Recurrent model-free RL can be understood as applying an off-the-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 463, + 506, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 506, + 477 + ], + "score": 1.0, + "content": "shelf model-free RL algorithm with an actor and a Q function parametrized to take sequences of", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 475, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 506, + 488 + ], + "score": 1.0, + "content": "inputs. As such, the choice of the underlying model-free RL algorithm is paramount. Most prior", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 486, + 505, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 505, + 499 + ], + "score": 1.0, + "content": "work on continuous control POMDP problems used on-policy algorithms, such as A2C (Mnih et al.,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 497, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 106, + 497, + 505, + 509 + ], + "score": 1.0, + "content": "2016), TRPO (Schulman et al., 2015) or PPO (Schulman et al., 2017). While off-policy algorithms", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 507, + 505, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 505, + 521 + ], + "score": 1.0, + "content": "such as TD3 (Fujimoto et al., 2018) and SAC (Haarnoja et al., 2018a;b) greatly improve the per-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 519, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 106, + 519, + 505, + 532 + ], + "score": 1.0, + "content": "formance in continuous control MDP problems in terms of sample efficiency and asymptotic per-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 530, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 505, + 543 + ], + "score": 1.0, + "content": "formance, these methods are rarely used in recurrent model-free RL baselines (Rakelly et al., 2019;", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 541, + 506, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 506, + 554 + ], + "score": 1.0, + "content": "Zhang et al., 2020; Zintgraf et al., 2020). In the experiment section (Sec. 5.1), we will show that", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 552, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 505, + 565 + ], + "score": 1.0, + "content": "using these off-policy algorithms for recurrent model-free RL provides results that are better than", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 563, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 505, + 574 + ], + "score": 1.0, + "content": "using on-policy algorithms and are comparable to their specialized methods in POMDP. This echoes", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 573, + 505, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 505, + 587 + ], + "score": 1.0, + "content": "the finding that model-free off-policy TD3-Context (Fakoor et al., 2020) can be better than the spe-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 584, + 343, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 343, + 597 + ], + "score": 1.0, + "content": "cialized method PEARL (Rakelly et al., 2019) in meta RL.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 610, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 609, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 505, + 624 + ], + "score": 1.0, + "content": "RNN Variants and Context Length. RNN training is known to be unstable, especially with long", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 621, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 505, + 635 + ], + "score": 1.0, + "content": "sequences input (Bengio et al., 1994). The RNN variants like LSTM (Hochreiter & Schmidhuber,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 631, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 505, + 646 + ], + "score": 1.0, + "content": "1997) and GRU (Chung et al., 2014) mitigate the training issues, but still may fail to learn long-term", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 643, + 506, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 506, + 656 + ], + "score": 1.0, + "content": "dependencies (Trinh et al., 2018). In POMDP problems, these dependencies reflect the memory that", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 655, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 667 + ], + "score": 1.0, + "content": "an agent must have to solve a task. For example, a POMDP that hides velocities from observations", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 664, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 506, + 679 + ], + "score": 1.0, + "content": "theoretically requires a short memory length to infer velocities through consecutive positions (Meng", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "et al., 2021). Prior work in POMDPs choose a variety set of context lengths for RNNs from 1 to", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "2048 (see the “Len” column of Table 2), and we select three representatives of short (5), medium", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 699, + 504, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 504, + 711 + ], + "score": 1.0, + "content": "(64), and long length (larger than 100) in the experiments (Sec. 5) for comparison. We also try both", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "LSTM and GRU as RNN variants to compare their performance. We find that the optimal context", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 720, + 330, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 330, + 733 + ], + "score": 1.0, + "content": "length and RNN variant are task-specific (see Sec. 5.2).", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 35 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "score": 1.0, + "content": "5", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 113, + 109, + 496, + 326 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 50, + 505, + 105 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 49, + 505, + 61 + ], + "spans": [ + { + "bbox": [ + 106, + 49, + 505, + 61 + ], + "score": 1.0, + "content": "Table 2: How the prior work and our method implement the recurrent model-free RL as their own", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 61, + 505, + 72 + ], + "spans": [ + { + "bbox": [ + 106, + 61, + 505, + 72 + ], + "score": 1.0, + "content": "method or baseline. We can see that none of the prior work share the same set of decision variables, some of", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 72, + 505, + 84 + ], + "spans": [ + { + "bbox": [ + 106, + 72, + 505, + 84 + ], + "score": 1.0, + "content": "which have bad choices that may lead to the poor performance reported in the prior work. Our method covers", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "a range of choices in these decision factors and finds the combinations in the last rows that lead to the best", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 94, + 498, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 498, + 106 + ], + "score": 1.0, + "content": "performance in terms of the average performance across the experimented environments in each subarea.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "table_body", + "bbox": [ + 113, + 109, + 496, + 326 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 113, + 109, + 496, + 326 + ], + "spans": [ + { + "bbox": [ + 113, + 109, + 496, + 326 + ], + "score": 0.985, + "html": "
AlgorithmDomainArchEncoderInputsLenRL
Duan et al. (2016)Meta RLseparateGRUoard1000TRPO,PPO
Wang et al. (2017)Meta RLsharedLSTMoart5-150A2C
Baseline from Rakelly et al. (2019)Meta RLseparateGRUoard100PPO
Baseline from Zintgraf et al. (2020)Meta RLseparateGRUoardMaxA2C,PPO
Baseline from Fakoor et al. (2020)MetaRLseparateGRUoar10-25TD3
Baseline from Yu et al. (2019)Meta RLseparateGRUoard500PPO
Kostrikov (2018)POMDPsharedGRU05-2048PPO,A2C
Wang et al. (2020)POMDPseparateLSTMoa150TD3, SAC
Meng et al. (2021)POMDPseparateLSTMoa1-5TD3
Yang & Nguyen (2021)POMDPseparatebothoaMaxTD3, SAC
Baseline from Igl et al. (2018)POMDPsharedGRUoa25A2C
Baseline from Han et al. (2020)POMDPsharedLSTM064SAC
Baseline from Zhang et al. (2021)Robust RLseparateLSTM0100PPO
Baseline1 from Packer et al. (2018)GeneralizationsharedLSTM0128-512PPO,A2C
Baseline2 from Packer et al. (2018)GeneralizationseparateLSTMoard128-512PPO,A2C
Our methodMeta RLseparateLSTMoard64TD3
Our methodPOMDPseparateGRUoa64TD3
Our methodRobust RLseparateLSTM064TD3
Our methodGeneralizationseparateLSTM064TD3
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Recurrent model-free RL can be understood as applying an off-the-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 463, + 506, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 506, + 477 + ], + "score": 1.0, + "content": "shelf model-free RL algorithm with an actor and a Q function parametrized to take sequences of", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 475, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 506, + 488 + ], + "score": 1.0, + "content": "inputs. As such, the choice of the underlying model-free RL algorithm is paramount. Most prior", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 486, + 505, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 505, + 499 + ], + "score": 1.0, + "content": "work on continuous control POMDP problems used on-policy algorithms, such as A2C (Mnih et al.,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 497, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 106, + 497, + 505, + 509 + ], + "score": 1.0, + "content": "2016), TRPO (Schulman et al., 2015) or PPO (Schulman et al., 2017). While off-policy algorithms", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 507, + 505, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 505, + 521 + ], + "score": 1.0, + "content": "such as TD3 (Fujimoto et al., 2018) and SAC (Haarnoja et al., 2018a;b) greatly improve the per-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 519, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 106, + 519, + 505, + 532 + ], + "score": 1.0, + "content": "formance in continuous control MDP problems in terms of sample efficiency and asymptotic per-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 530, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 505, + 543 + ], + "score": 1.0, + "content": "formance, these methods are rarely used in recurrent model-free RL baselines (Rakelly et al., 2019;", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 541, + 506, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 506, + 554 + ], + "score": 1.0, + "content": "Zhang et al., 2020; Zintgraf et al., 2020). In the experiment section (Sec. 5.1), we will show that", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 552, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 505, + 565 + ], + "score": 1.0, + "content": "using these off-policy algorithms for recurrent model-free RL provides results that are better than", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 563, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 505, + 574 + ], + "score": 1.0, + "content": "using on-policy algorithms and are comparable to their specialized methods in POMDP. This echoes", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 573, + 505, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 505, + 587 + ], + "score": 1.0, + "content": "the finding that model-free off-policy TD3-Context (Fakoor et al., 2020) can be better than the spe-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 584, + 343, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 343, + 597 + ], + "score": 1.0, + "content": "cialized method PEARL (Rakelly et al., 2019) in meta RL.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 453, + 506, + 597 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 610, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 609, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 505, + 624 + ], + "score": 1.0, + "content": "RNN Variants and Context Length. RNN training is known to be unstable, especially with long", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 621, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 505, + 635 + ], + "score": 1.0, + "content": "sequences input (Bengio et al., 1994). The RNN variants like LSTM (Hochreiter & Schmidhuber,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 631, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 505, + 646 + ], + "score": 1.0, + "content": "1997) and GRU (Chung et al., 2014) mitigate the training issues, but still may fail to learn long-term", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 643, + 506, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 506, + 656 + ], + "score": 1.0, + "content": "dependencies (Trinh et al., 2018). In POMDP problems, these dependencies reflect the memory that", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 655, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 667 + ], + "score": 1.0, + "content": "an agent must have to solve a task. For example, a POMDP that hides velocities from observations", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 664, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 506, + 679 + ], + "score": 1.0, + "content": "theoretically requires a short memory length to infer velocities through consecutive positions (Meng", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "et al., 2021). Prior work in POMDPs choose a variety set of context lengths for RNNs from 1 to", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "2048 (see the “Len” column of Table 2), and we select three representatives of short (5), medium", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 699, + 504, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 504, + 711 + ], + "score": 1.0, + "content": "(64), and long length (larger than 100) in the experiments (Sec. 5) for comparison. We also try both", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "LSTM and GRU as RNN variants to compare their performance. We find that the optimal context", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 720, + 330, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 330, + 733 + ], + "score": 1.0, + "content": "length and RNN variant are task-specific (see Sec. 5.2).", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 35, + "bbox_fs": [ + 105, + 609, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 81, + 200, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 201, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 201, + 95 + ], + "score": 1.0, + "content": "5 EXPERIMENTS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 98, + 505, + 164 + ], + "lines": [ + { + "bbox": [ + 105, + 97, + 506, + 110 + ], + "spans": [ + { + "bbox": [ + 105, + 97, + 506, + 110 + ], + "score": 1.0, + "content": "Our experiments aim to answer two questions. First, how does a well-tuned implementation of", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 108, + 505, + 122 + ], + "spans": [ + { + "bbox": [ + 105, + 108, + 505, + 122 + ], + "score": 1.0, + "content": "recurrent model-free RL compare to specialized POMDP methods, such as purpose-designed meta", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 119, + 505, + 132 + ], + "spans": [ + { + "bbox": [ + 105, + 119, + 505, + 132 + ], + "score": 1.0, + "content": "RL and robust RL algorithms? To give these prior methods the strongest possible footing, we will", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 131, + 505, + 144 + ], + "spans": [ + { + "bbox": [ + 105, + 131, + 505, + 144 + ], + "score": 1.0, + "content": "compare prior methods on the specific problem types for which they were developed (i.e., meta RL", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 141, + 506, + 154 + ], + "spans": [ + { + "bbox": [ + 105, + 141, + 506, + 154 + ], + "score": 1.0, + "content": "algorithms were tested on meta RL tasks). Our second question studies which design decisions are", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 153, + 450, + 165 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 450, + 165 + ], + "score": 1.0, + "content": "essential for recurrent model-free RL. We put the environment details in Appendix D.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 107, + 177, + 504, + 232 + ], + "lines": [ + { + "bbox": [ + 107, + 177, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 107, + 177, + 505, + 189 + ], + "score": 1.0, + "content": "Code Implementation. We release a modular and highly-configurable implementation of recur-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "score": 1.0, + "content": "rent (off-policy) model-free RL: url. Our implementation is efficient in terms of computer memory", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 198, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 348, + 210 + ], + "score": 1.0, + "content": "compared to previous off-policy RL methods for POMDPs", + "type": "text" + }, + { + "bbox": [ + 348, + 199, + 370, + 209 + ], + "score": 0.56, + "content": "2 0 0 \\mathrm { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 198, + 505, + 210 + ], + "score": 1.0, + "content": "less RAM than Han et al. (2020)", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 209, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 106, + 209, + 123, + 222 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 123, + 210, + 135, + 220 + ], + "score": 0.42, + "content": "9 \\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 209, + 506, + 222 + ], + "score": 1.0, + "content": "less GPU memory than Dorfman et al. (2020)). Please see the appendix A for details, includ-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 219, + 464, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 464, + 234 + ], + "score": 1.0, + "content": "ing an explanation of why our implementation is more memory-efficient than prior work.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9 + }, + { + "type": "title", + "bbox": [ + 108, + 237, + 494, + 248 + ], + "lines": [ + { + "bbox": [ + 105, + 236, + 496, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 496, + 249 + ], + "score": 1.0, + "content": "5.1 RECURRENT MODEL-FREE RL IS COMPARABLE WITH PRIOR SPECIALIZED METHODS", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 255, + 505, + 321 + ], + "lines": [ + { + "bbox": [ + 105, + 254, + 505, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 505, + 267 + ], + "score": 1.0, + "content": "While prior work has studied a range of different POMDP settings (e.g., meta RL, occluded observa-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 265, + 505, + 278 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 505, + 278 + ], + "score": 1.0, + "content": "tions), recurrent model-free RL is a ubiquitous baseline (Han et al., 2020; Humplik et al., 2019; Igl", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 276, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 505, + 289 + ], + "score": 1.0, + "content": "et al., 2018; Rakelly et al., 2019; Zintgraf et al., 2020). However, prior work consistently report that", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 288, + 505, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 505, + 300 + ], + "score": 1.0, + "content": "this baseline is reported to be unperformed to more specialized methods. This section casts doubt", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 299, + 505, + 311 + ], + "spans": [ + { + "bbox": [ + 106, + 299, + 505, + 311 + ], + "score": 1.0, + "content": "on that claim, showing that a well-tuned implementation of recurrent model-free RL can perform at", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 311, + 277, + 321 + ], + "spans": [ + { + "bbox": [ + 106, + 311, + 277, + 321 + ], + "score": 1.0, + "content": "least as well as more specialized methods.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 107, + 326, + 505, + 425 + ], + "lines": [ + { + "bbox": [ + 106, + 327, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 106, + 327, + 505, + 339 + ], + "score": 1.0, + "content": "We study four subareas of POMDPs: the “standard” POMDP, meta RL, robust RL, and generaliza-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 338, + 505, + 350 + ], + "spans": [ + { + "bbox": [ + 106, + 338, + 505, + 350 + ], + "score": 1.0, + "content": "tion in RL. We tune a wide range of decision factors shown in Sec. 4 in our implemented recurrent", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 349, + 505, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 505, + 361 + ], + "score": 1.0, + "content": "model-free RL. Appendix A.3 shows the details of the tuning options. For each subarea, we show", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 360, + 505, + 372 + ], + "spans": [ + { + "bbox": [ + 106, + 360, + 505, + 372 + ], + "score": 1.0, + "content": "the performance of a single variant that works best across the environments in that subarea, com-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 370, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 506, + 384 + ], + "score": 1.0, + "content": "pared with the prior specialized methods in this subsection. In other words, the following plots of", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 381, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 505, + 394 + ], + "score": 1.0, + "content": "each subarea report the same model-free recurrent RL algorithm with the same hyperparame-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 393, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 393, + 505, + 405 + ], + "score": 1.0, + "content": "ters. The exact configurations of each subarea can be found in the last four rows of Table 2. Under", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 404, + 504, + 416 + ], + "spans": [ + { + "bbox": [ + 106, + 404, + 504, + 416 + ], + "score": 1.0, + "content": "this restricted setting, we find that our implementation can actually outperform prior (specialized)", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 415, + 314, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 314, + 426 + ], + "score": 1.0, + "content": "methods by a wide margin across the four subareas.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 432, + 505, + 519 + ], + "lines": [ + { + "bbox": [ + 105, + 430, + 506, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 506, + 444 + ], + "score": 1.0, + "content": "For each plot of learning curves, we show three approaches as reference. First, an Oracle policy has", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 442, + 505, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 505, + 454 + ], + "score": 1.0, + "content": "access to the POMDP hidden states, turning the POMDP into an MDP; this policy should therefore", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 453, + 504, + 465 + ], + "spans": [ + { + "bbox": [ + 106, + 453, + 504, + 465 + ], + "score": 1.0, + "content": "be treated as an upper bound on the performance that any POMDP method should receive. Sec-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 465, + 504, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 465, + 504, + 475 + ], + "score": 1.0, + "content": "ond, as a lower bound, we use a Markovian policy to solve the POMDP. Both Oracle policy and", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 475, + 504, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 504, + 487 + ], + "score": 1.0, + "content": "Markovian policy are trained with the same hyperparameters as our recurrent model-free RL imple-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 486, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 505, + 498 + ], + "score": 1.0, + "content": "mentation. Third, we add a Random policy, which represents a trivial lower bound. We show the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 496, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 506, + 509 + ], + "score": 1.0, + "content": "full learning curves in Appendix E.1 due to the space limit. See Appendix B for details about the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 508, + 261, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 508, + 261, + 520 + ], + "score": 1.0, + "content": "implementation of these comparisons.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 31.5 + }, + { + "type": "text", + "bbox": [ + 106, + 532, + 505, + 686 + ], + "lines": [ + { + "bbox": [ + 105, + 530, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 505, + 545 + ], + "score": 1.0, + "content": "“Standard” POMDP. Our first experiments look at the “standard” POMDPs that typically oc-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 543, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 543, + 506, + 556 + ], + "score": 1.0, + "content": "clude some part of states in the environment. We will compare against VRM (Han et al., 2020), a", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 553, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 553, + 505, + 568 + ], + "score": 1.0, + "content": "recent state-of-the-art model-based POMDP algorithm. We directly apply the environment design", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 564, + 504, + 577 + ], + "spans": [ + { + "bbox": [ + 106, + 564, + 504, + 577 + ], + "score": 1.0, + "content": "of VRM paper that occludes either positions&angles or velocities of the simulated robot in PyBul-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 576, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 476, + 590 + ], + "score": 1.0, + "content": "let (Coumans & Bai, 2016). There are 8 environments {Hopper, Ant, Walker, Cheetah}-", + "type": "text" + }, + { + "bbox": [ + 477, + 576, + 501, + 588 + ], + "score": 0.32, + "content": "\\{ \\mathrm { P } , \\mathrm { V } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 576, + 505, + 590 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 586, + 506, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 506, + 600 + ], + "score": 1.0, + "content": "where “-P” stands for observing positions&angles only, and “-V” stands for observing velocities", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 598, + 506, + 611 + ], + "spans": [ + { + "bbox": [ + 106, + 598, + 506, + 611 + ], + "score": 1.0, + "content": "only. Fig. 1 and Fig. 18 in appendix show that the best single variant of our model-free recurrent RL", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 608, + 504, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 487, + 622 + ], + "score": 1.0, + "content": "implementation outperform VRM in 6 out of 8 environments, especially in {Cheetah,Hopper}-", + "type": "text" + }, + { + "bbox": [ + 488, + 609, + 504, + 621 + ], + "score": 0.77, + "content": "\\{ \\mathrm { P } \\}", + "type": "inline_equation" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 620, + 505, + 633 + ], + "spans": [ + { + "bbox": [ + 106, + 620, + 130, + 633 + ], + "score": 1.0, + "content": "(over", + "type": "text" + }, + { + "bbox": [ + 131, + 620, + 151, + 631 + ], + "score": 0.88, + "content": "8 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 620, + 505, + 633 + ], + "score": 1.0, + "content": "of the Oracles). Our results suggest that, while the variational dynamics model used", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 631, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 505, + 644 + ], + "score": 1.0, + "content": "by VRM may be useful for some tasks, a simple recurrent model-free RL baseline can outperform", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 642, + 506, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 506, + 655 + ], + "score": 1.0, + "content": "VRM if properly tuned. While we are primarily interested in sample complexity, but not compute, it", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 653, + 505, + 665 + ], + "spans": [ + { + "bbox": [ + 106, + 653, + 505, + 665 + ], + "score": 1.0, + "content": "is worth noting that our recurrent model-free RL implementation is substantially more efficient than", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 663, + 506, + 676 + ], + "spans": [ + { + "bbox": [ + 105, + 663, + 303, + 676 + ], + "score": 1.0, + "content": "the open-source VRM implementation, training", + "type": "text" + }, + { + "bbox": [ + 303, + 664, + 317, + 674 + ], + "score": 0.85, + "content": "5 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 663, + 435, + 676 + ], + "score": 1.0, + "content": "faster and requiring at most", + "type": "text" + }, + { + "bbox": [ + 435, + 664, + 460, + 675 + ], + "score": 0.87, + "content": "2 0 0 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 663, + 506, + 676 + ], + "score": 1.0, + "content": "less RAM", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 675, + 208, + 687 + ], + "spans": [ + { + "bbox": [ + 105, + 675, + 208, + 687 + ], + "score": 1.0, + "content": "usage (see Appendix A).", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 42.5 + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "Meta RL. We next compare the recurrent model-free RL to the meta RL setting, where some", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 709, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 721 + ], + "score": 1.0, + "content": "indicator of the task is unobserved. We compare our implementation of recurrent model-free RL", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "to a specialized state-of-the-art method, VariBAD (Zintgraf et al., 2020) that explicitly learns the", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 51 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 81, + 200, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 201, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 201, + 95 + ], + "score": 1.0, + "content": "5 EXPERIMENTS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 98, + 505, + 164 + ], + "lines": [ + { + "bbox": [ + 105, + 97, + 506, + 110 + ], + "spans": [ + { + "bbox": [ + 105, + 97, + 506, + 110 + ], + "score": 1.0, + "content": "Our experiments aim to answer two questions. First, how does a well-tuned implementation of", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 108, + 505, + 122 + ], + "spans": [ + { + "bbox": [ + 105, + 108, + 505, + 122 + ], + "score": 1.0, + "content": "recurrent model-free RL compare to specialized POMDP methods, such as purpose-designed meta", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 119, + 505, + 132 + ], + "spans": [ + { + "bbox": [ + 105, + 119, + 505, + 132 + ], + "score": 1.0, + "content": "RL and robust RL algorithms? To give these prior methods the strongest possible footing, we will", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 131, + 505, + 144 + ], + "spans": [ + { + "bbox": [ + 105, + 131, + 505, + 144 + ], + "score": 1.0, + "content": "compare prior methods on the specific problem types for which they were developed (i.e., meta RL", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 141, + 506, + 154 + ], + "spans": [ + { + "bbox": [ + 105, + 141, + 506, + 154 + ], + "score": 1.0, + "content": "algorithms were tested on meta RL tasks). Our second question studies which design decisions are", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 153, + 450, + 165 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 450, + 165 + ], + "score": 1.0, + "content": "essential for recurrent model-free RL. We put the environment details in Appendix D.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3.5, + "bbox_fs": [ + 105, + 97, + 506, + 165 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 177, + 504, + 232 + ], + "lines": [ + { + "bbox": [ + 107, + 177, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 107, + 177, + 505, + 189 + ], + "score": 1.0, + "content": "Code Implementation. We release a modular and highly-configurable implementation of recur-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "score": 1.0, + "content": "rent (off-policy) model-free RL: url. Our implementation is efficient in terms of computer memory", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 198, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 348, + 210 + ], + "score": 1.0, + "content": "compared to previous off-policy RL methods for POMDPs", + "type": "text" + }, + { + "bbox": [ + 348, + 199, + 370, + 209 + ], + "score": 0.56, + "content": "2 0 0 \\mathrm { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 198, + 505, + 210 + ], + "score": 1.0, + "content": "less RAM than Han et al. (2020)", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 209, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 106, + 209, + 123, + 222 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 123, + 210, + 135, + 220 + ], + "score": 0.42, + "content": "9 \\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 209, + 506, + 222 + ], + "score": 1.0, + "content": "less GPU memory than Dorfman et al. (2020)). Please see the appendix A for details, includ-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 219, + 464, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 464, + 234 + ], + "score": 1.0, + "content": "ing an explanation of why our implementation is more memory-efficient than prior work.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9, + "bbox_fs": [ + 105, + 177, + 506, + 234 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 237, + 494, + 248 + ], + "lines": [ + { + "bbox": [ + 105, + 236, + 496, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 496, + 249 + ], + "score": 1.0, + "content": "5.1 RECURRENT MODEL-FREE RL IS COMPARABLE WITH PRIOR SPECIALIZED METHODS", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 255, + 505, + 321 + ], + "lines": [ + { + "bbox": [ + 105, + 254, + 505, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 505, + 267 + ], + "score": 1.0, + "content": "While prior work has studied a range of different POMDP settings (e.g., meta RL, occluded observa-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 265, + 505, + 278 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 505, + 278 + ], + "score": 1.0, + "content": "tions), recurrent model-free RL is a ubiquitous baseline (Han et al., 2020; Humplik et al., 2019; Igl", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 276, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 505, + 289 + ], + "score": 1.0, + "content": "et al., 2018; Rakelly et al., 2019; Zintgraf et al., 2020). However, prior work consistently report that", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 288, + 505, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 505, + 300 + ], + "score": 1.0, + "content": "this baseline is reported to be unperformed to more specialized methods. This section casts doubt", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 299, + 505, + 311 + ], + "spans": [ + { + "bbox": [ + 106, + 299, + 505, + 311 + ], + "score": 1.0, + "content": "on that claim, showing that a well-tuned implementation of recurrent model-free RL can perform at", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 311, + 277, + 321 + ], + "spans": [ + { + "bbox": [ + 106, + 311, + 277, + 321 + ], + "score": 1.0, + "content": "least as well as more specialized methods.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 254, + 505, + 321 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 326, + 505, + 425 + ], + "lines": [ + { + "bbox": [ + 106, + 327, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 106, + 327, + 505, + 339 + ], + "score": 1.0, + "content": "We study four subareas of POMDPs: the “standard” POMDP, meta RL, robust RL, and generaliza-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 338, + 505, + 350 + ], + "spans": [ + { + "bbox": [ + 106, + 338, + 505, + 350 + ], + "score": 1.0, + "content": "tion in RL. We tune a wide range of decision factors shown in Sec. 4 in our implemented recurrent", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 349, + 505, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 505, + 361 + ], + "score": 1.0, + "content": "model-free RL. Appendix A.3 shows the details of the tuning options. For each subarea, we show", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 360, + 505, + 372 + ], + "spans": [ + { + "bbox": [ + 106, + 360, + 505, + 372 + ], + "score": 1.0, + "content": "the performance of a single variant that works best across the environments in that subarea, com-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 370, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 506, + 384 + ], + "score": 1.0, + "content": "pared with the prior specialized methods in this subsection. In other words, the following plots of", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 381, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 505, + 394 + ], + "score": 1.0, + "content": "each subarea report the same model-free recurrent RL algorithm with the same hyperparame-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 393, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 393, + 505, + 405 + ], + "score": 1.0, + "content": "ters. The exact configurations of each subarea can be found in the last four rows of Table 2. Under", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 404, + 504, + 416 + ], + "spans": [ + { + "bbox": [ + 106, + 404, + 504, + 416 + ], + "score": 1.0, + "content": "this restricted setting, we find that our implementation can actually outperform prior (specialized)", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 415, + 314, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 314, + 426 + ], + "score": 1.0, + "content": "methods by a wide margin across the four subareas.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 327, + 506, + 426 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 432, + 505, + 519 + ], + "lines": [ + { + "bbox": [ + 105, + 430, + 506, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 506, + 444 + ], + "score": 1.0, + "content": "For each plot of learning curves, we show three approaches as reference. First, an Oracle policy has", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 442, + 505, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 505, + 454 + ], + "score": 1.0, + "content": "access to the POMDP hidden states, turning the POMDP into an MDP; this policy should therefore", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 453, + 504, + 465 + ], + "spans": [ + { + "bbox": [ + 106, + 453, + 504, + 465 + ], + "score": 1.0, + "content": "be treated as an upper bound on the performance that any POMDP method should receive. Sec-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 465, + 504, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 465, + 504, + 475 + ], + "score": 1.0, + "content": "ond, as a lower bound, we use a Markovian policy to solve the POMDP. Both Oracle policy and", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 475, + 504, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 504, + 487 + ], + "score": 1.0, + "content": "Markovian policy are trained with the same hyperparameters as our recurrent model-free RL imple-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 486, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 505, + 498 + ], + "score": 1.0, + "content": "mentation. Third, we add a Random policy, which represents a trivial lower bound. We show the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 496, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 506, + 509 + ], + "score": 1.0, + "content": "full learning curves in Appendix E.1 due to the space limit. See Appendix B for details about the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 508, + 261, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 508, + 261, + 520 + ], + "score": 1.0, + "content": "implementation of these comparisons.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 430, + 506, + 520 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 532, + 505, + 686 + ], + "lines": [ + { + "bbox": [ + 105, + 530, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 505, + 545 + ], + "score": 1.0, + "content": "“Standard” POMDP. Our first experiments look at the “standard” POMDPs that typically oc-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 543, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 543, + 506, + 556 + ], + "score": 1.0, + "content": "clude some part of states in the environment. We will compare against VRM (Han et al., 2020), a", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 553, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 553, + 505, + 568 + ], + "score": 1.0, + "content": "recent state-of-the-art model-based POMDP algorithm. We directly apply the environment design", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 564, + 504, + 577 + ], + "spans": [ + { + "bbox": [ + 106, + 564, + 504, + 577 + ], + "score": 1.0, + "content": "of VRM paper that occludes either positions&angles or velocities of the simulated robot in PyBul-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 576, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 476, + 590 + ], + "score": 1.0, + "content": "let (Coumans & Bai, 2016). There are 8 environments {Hopper, Ant, Walker, Cheetah}-", + "type": "text" + }, + { + "bbox": [ + 477, + 576, + 501, + 588 + ], + "score": 0.32, + "content": "\\{ \\mathrm { P } , \\mathrm { V } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 576, + 505, + 590 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 586, + 506, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 506, + 600 + ], + "score": 1.0, + "content": "where “-P” stands for observing positions&angles only, and “-V” stands for observing velocities", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 598, + 506, + 611 + ], + "spans": [ + { + "bbox": [ + 106, + 598, + 506, + 611 + ], + "score": 1.0, + "content": "only. Fig. 1 and Fig. 18 in appendix show that the best single variant of our model-free recurrent RL", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 608, + 504, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 487, + 622 + ], + "score": 1.0, + "content": "implementation outperform VRM in 6 out of 8 environments, especially in {Cheetah,Hopper}-", + "type": "text" + }, + { + "bbox": [ + 488, + 609, + 504, + 621 + ], + "score": 0.77, + "content": "\\{ \\mathrm { P } \\}", + "type": "inline_equation" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 620, + 505, + 633 + ], + "spans": [ + { + "bbox": [ + 106, + 620, + 130, + 633 + ], + "score": 1.0, + "content": "(over", + "type": "text" + }, + { + "bbox": [ + 131, + 620, + 151, + 631 + ], + "score": 0.88, + "content": "8 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 620, + 505, + 633 + ], + "score": 1.0, + "content": "of the Oracles). Our results suggest that, while the variational dynamics model used", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 631, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 505, + 644 + ], + "score": 1.0, + "content": "by VRM may be useful for some tasks, a simple recurrent model-free RL baseline can outperform", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 642, + 506, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 506, + 655 + ], + "score": 1.0, + "content": "VRM if properly tuned. While we are primarily interested in sample complexity, but not compute, it", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 653, + 505, + 665 + ], + "spans": [ + { + "bbox": [ + 106, + 653, + 505, + 665 + ], + "score": 1.0, + "content": "is worth noting that our recurrent model-free RL implementation is substantially more efficient than", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 663, + 506, + 676 + ], + "spans": [ + { + "bbox": [ + 105, + 663, + 303, + 676 + ], + "score": 1.0, + "content": "the open-source VRM implementation, training", + "type": "text" + }, + { + "bbox": [ + 303, + 664, + 317, + 674 + ], + "score": 0.85, + "content": "5 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 663, + 435, + 676 + ], + "score": 1.0, + "content": "faster and requiring at most", + "type": "text" + }, + { + "bbox": [ + 435, + 664, + 460, + 675 + ], + "score": 0.87, + "content": "2 0 0 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 663, + 506, + 676 + ], + "score": 1.0, + "content": "less RAM", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 675, + 208, + 687 + ], + "spans": [ + { + "bbox": [ + 105, + 675, + 208, + 687 + ], + "score": 1.0, + "content": "usage (see Appendix A).", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 42.5, + "bbox_fs": [ + 105, + 530, + 506, + 687 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "Meta RL. We next compare the recurrent model-free RL to the meta RL setting, where some", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 709, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 721 + ], + "score": 1.0, + "content": "indicator of the task is unobserved. We compare our implementation of recurrent model-free RL", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "to a specialized state-of-the-art method, VariBAD (Zintgraf et al., 2020) that explicitly learns the", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 106, + 223, + 504, + 235 + ], + "spans": [ + { + "bbox": [ + 106, + 223, + 504, + 235 + ], + "score": 1.0, + "content": "task embeddings by variational model-based objectives. As suggested by Dorfman et al. (2020),", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 233, + 505, + 245 + ], + "spans": [ + { + "bbox": [ + 106, + 233, + 505, + 245 + ], + "score": 1.0, + "content": "we modified VariBAD to use SAC instead of PPO. This change also allows us for fair comparison", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 244, + 505, + 257 + ], + "spans": [ + { + "bbox": [ + 106, + 244, + 505, + 257 + ], + "score": 1.0, + "content": "with our implementation of recurrent model-free RL, which uses SAC and TD3. We adopt the three", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 256, + 504, + 267 + ], + "spans": [ + { + "bbox": [ + 106, + 256, + 504, + 267 + ], + "score": 1.0, + "content": "environments used in Dorfman et al. (2020) for experiments, including Semi-Circle and Cheetah-", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 267, + 505, + 278 + ], + "spans": [ + { + "bbox": [ + 106, + 267, + 505, + 278 + ], + "score": 1.0, + "content": "Vel, and we also adapt Wind to make it harder to solve. Figure 2 shows that our best single variant", + "type": "text", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 278, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 106, + 278, + 505, + 289 + ], + "score": 1.0, + "content": "outperforms VariBAD and even reaches Oracles in the two meta RL environments, Cheetah-Vel and", + "type": "text", + "cross_page": true + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 288, + 504, + 300 + ], + "spans": [ + { + "bbox": [ + 106, + 288, + 504, + 300 + ], + "score": 1.0, + "content": "Wind, leaving Semi-Circle in the appendix. Prior work (Rakelly et al., 2019; Zintgraf et al., 2020)", + "type": "text", + "cross_page": true + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 299, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 505, + 312 + ], + "score": 1.0, + "content": "show that disentangling task inference and control can stabilize training. However, our experiments", + "type": "text", + "cross_page": true + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 311, + 506, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 506, + 322 + ], + "score": 1.0, + "content": "suggest that joint training of task inference and control could also have comparable performance if", + "type": "text", + "cross_page": true + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 321, + 505, + 334 + ], + "spans": [ + { + "bbox": [ + 106, + 321, + 505, + 334 + ], + "score": 1.0, + "content": "well implemented. 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The single best variant of our implementation", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 195, + 505, + 208 + ], + "spans": [ + { + "bbox": [ + 105, + 195, + 505, + 208 + ], + "score": 1.0, + "content": "on recurrent model-free RL can surpass the specialized meta RL method off-policy VariBAD (Dorfman et al.,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 206, + 444, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 206, + 444, + 217 + ], + "score": 1.0, + "content": "2020), and match the performance of an “Oracle” policy that gets to observe the hidden state.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 106, + 223, + 505, + 354 + ], + "lines": [ + { + "bbox": [ + 106, + 223, + 504, + 235 + ], + "spans": [ + { + "bbox": [ + 106, + 223, + 504, + 235 + ], + "score": 1.0, + "content": "task embeddings by variational model-based objectives. 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We show the average returns", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 465, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 465, + 505, + 475 + ], + "score": 1.0, + "content": "(left figure) and worst returns (right figure) of each method. The single best variant of our implementation on", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 475, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 505, + 487 + ], + "score": 1.0, + "content": "recurrent model-free RL can greatly outperform the specialized robust RL method MRPO (Jiang et al., 2021),", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 484, + 394, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 394, + 497 + ], + "score": 1.0, + "content": "and surpass the Oracle that are trained with same simulation and gradient steps.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22.5 + } + ], + "index": 20.75 + }, + { + "type": "text", + "bbox": [ + 106, + 500, + 505, + 675 + ], + "lines": [ + { + "bbox": [ + 105, + 498, + 505, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 505, + 512 + ], + "score": 1.0, + "content": "Robust RL. Thirdly, we focus on the robust RL that aims to maximize the worst returns over", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 510, + 505, + 522 + ], + "spans": [ + { + "bbox": [ + 106, + 510, + 505, + 522 + ], + "score": 1.0, + "content": "the tasks. We choose the recent specialized algorithm MRPO (Jiang et al., 2021) as the compared", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 522, + 503, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 503, + 533 + ], + "score": 1.0, + "content": "method, and adopt their used environments based on SunBlaze benchmark (Packer et al., 2018).", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 532, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 106, + 532, + 505, + 544 + ], + "score": 1.0, + "content": "These environments have hidden states that are fixed during one episode, including the density and", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 542, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 437, + 556 + ], + "score": 1.0, + "content": "the friction coefficients of the simulated robots, namely {Cheetah, Hopper, Walker", + "type": "text" + }, + { + "bbox": [ + 438, + 544, + 443, + 555 + ], + "score": 0.43, + "content": "\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 542, + 506, + 556 + ], + "score": 1.0, + "content": "-Robust. Fig. 3", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 555, + 505, + 566 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 505, + 566 + ], + "score": 1.0, + "content": "shows both the average return and worst return of our single best variant and MRPO on the three", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 564, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 104, + 564, + 454, + 578 + ], + "score": 1.0, + "content": "environments, where the worst return is measured by the average return in the worst", + "type": "text" + }, + { + "bbox": [ + 455, + 565, + 474, + 576 + ], + "score": 0.89, + "content": "1 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 564, + 505, + 578 + ], + "score": 1.0, + "content": "testing", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 576, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 505, + 589 + ], + "score": 1.0, + "content": "tasks following the practice in Jiang et al. (2021). The results are quite surprising: although our", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 587, + 506, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 506, + 599 + ], + "score": 1.0, + "content": "implementation, using average-case RL objective and without access to the hidden states, is not", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 597, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 505, + 610 + ], + "score": 1.0, + "content": "expected to surpass MRPO and Oracle with access to hidden states in worst return, we found that", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 609, + 505, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 505, + 621 + ], + "score": 1.0, + "content": "our best variant vastly outperforms the specialized MRPO and Oracle in both average return and", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 619, + 505, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 200, + 633 + ], + "score": 1.0, + "content": "worst return, with over", + "type": "text" + }, + { + "bbox": [ + 201, + 620, + 221, + 631 + ], + "score": 0.88, + "content": "8 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 619, + 505, + 633 + ], + "score": 1.0, + "content": "fewer simulation steps. Our implementation benefits from its memory", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 631, + 505, + 643 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 505, + 643 + ], + "score": 1.0, + "content": "and off-policy algorithms, while MRPO might suffer from its Markovian on-policy algorithm and", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 642, + 504, + 654 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 480, + 654 + ], + "score": 1.0, + "content": "a bit ideal Lipschitz assumption in dynamics. Nevertheless, our implementation is around", + "type": "text" + }, + { + "bbox": [ + 480, + 642, + 504, + 653 + ], + "score": 0.74, + "content": "1 7 . 5 \\mathrm { x }", + "type": "inline_equation" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 653, + 506, + 666 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 506, + 666 + ], + "score": 1.0, + "content": "slower than MRPO given the same simulation steps (see Appendix A), so we only run it with 3M", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 663, + 238, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 663, + 238, + 677 + ], + "score": 1.0, + "content": "steps with a limited time budget.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 32.5 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "Generalization in RL. 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The single best variant of our implementation", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 195, + 505, + 208 + ], + "spans": [ + { + "bbox": [ + 105, + 195, + 505, + 208 + ], + "score": 1.0, + "content": "on recurrent model-free RL can surpass the specialized meta RL method off-policy VariBAD (Dorfman et al.,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 206, + 444, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 206, + 444, + 217 + ], + "score": 1.0, + "content": "2020), and match the performance of an “Oracle” policy that gets to observe the hidden state.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 106, + 223, + 505, + 354 + ], + "lines": [], + "index": 11.5, + "bbox_fs": [ + 105, + 223, + 506, + 355 + ], + "lines_deleted": true + }, + { + "type": "image", + "bbox": [ + 127, + 361, + 481, + 455 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 127, + 361, + 481, + 455 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 127, + 361, + 481, + 455 + ], + "spans": [ + { + "bbox": [ + 127, + 361, + 481, + 455 + ], + "score": 0.968, + "type": "image", + "image_path": "473825d8305f612f2647b1794d5d4c29303a403a99877872238870f73e2911e4.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 127, + 361, + 481, + 392.3333333333333 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 127, + 392.3333333333333, + 481, + 423.66666666666663 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 127, + 423.66666666666663, + 481, + 454.99999999999994 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 456, + 505, + 496 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 105, + 455, + 506, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 506, + 468 + ], + "score": 1.0, + "content": "Figure 3: Learning curves on one robust RL environment, Cheetah-Robust. We show the average returns", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 465, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 465, + 505, + 475 + ], + "score": 1.0, + "content": "(left figure) and worst returns (right figure) of each method. The single best variant of our implementation on", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 475, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 505, + 487 + ], + "score": 1.0, + "content": "recurrent model-free RL can greatly outperform the specialized robust RL method MRPO (Jiang et al., 2021),", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 484, + 394, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 394, + 497 + ], + "score": 1.0, + "content": "and surpass the Oracle that are trained with same simulation and gradient steps.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22.5 + } + ], + "index": 20.75 + }, + { + "type": "text", + "bbox": [ + 106, + 500, + 505, + 675 + ], + "lines": [ + { + "bbox": [ + 105, + 498, + 505, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 505, + 512 + ], + "score": 1.0, + "content": "Robust RL. 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Fig. 3", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 555, + 505, + 566 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 505, + 566 + ], + "score": 1.0, + "content": "shows both the average return and worst return of our single best variant and MRPO on the three", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 564, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 104, + 564, + 454, + 578 + ], + "score": 1.0, + "content": "environments, where the worst return is measured by the average return in the worst", + "type": "text" + }, + { + "bbox": [ + 455, + 565, + 474, + 576 + ], + "score": 0.89, + "content": "1 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 564, + 505, + 578 + ], + "score": 1.0, + "content": "testing", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 576, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 505, + 589 + ], + "score": 1.0, + "content": "tasks following the practice in Jiang et al. 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The results are quite surprising: although our", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 587, + 506, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 506, + 599 + ], + "score": 1.0, + "content": "implementation, using average-case RL objective and without access to the hidden states, is not", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 597, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 505, + 610 + ], + "score": 1.0, + "content": "expected to surpass MRPO and Oracle with access to hidden states in worst return, we found that", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 609, + 505, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 505, + 621 + ], + "score": 1.0, + "content": "our best variant vastly outperforms the specialized MRPO and Oracle in both average return and", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 619, + 505, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 200, + 633 + ], + "score": 1.0, + "content": "worst return, with over", + "type": "text" + }, + { + "bbox": [ + 201, + 620, + 221, + 631 + ], + "score": 0.88, + "content": "8 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 619, + 505, + 633 + ], + "score": 1.0, + "content": "fewer simulation steps. Our implementation benefits from its memory", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 631, + 505, + 643 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 505, + 643 + ], + "score": 1.0, + "content": "and off-policy algorithms, while MRPO might suffer from its Markovian on-policy algorithm and", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 642, + 504, + 654 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 480, + 654 + ], + "score": 1.0, + "content": "a bit ideal Lipschitz assumption in dynamics. 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We show the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 154, + 506, + 166 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 506, + 166 + ], + "score": 1.0, + "content": "interpolation success rates (left figures) and extrapolation success rates (right figures) of each method. The", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 165, + 505, + 176 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 505, + 176 + ], + "score": 1.0, + "content": "single best variant of our implementation on recurrent model-free RL can be par with the specialized method", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 173, + 506, + 185 + ], + "spans": [ + { + "bbox": [ + 106, + 173, + 506, + 185 + ], + "score": 1.0, + "content": "EPOpt-PPO-FF (Rajeswaran et al., 2017a) in interpolation and outperform it in extrapolation. The data of", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 183, + 506, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 183, + 506, + 196 + ], + "score": 1.0, + "content": "EPOpt-PPO-FF and A2C-RC (a recurrent model-free on-policy RL method) are copied from the Table 7 & 8", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 194, + 190, + 205 + ], + "spans": [ + { + "bbox": [ + 106, + 194, + 190, + 205 + ], + "score": 1.0, + "content": "in Packer et al. 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We can see that our model-free method is on par", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 248, + 506, + 261 + ], + "spans": [ + { + "bbox": [ + 106, + 248, + 506, + 261 + ], + "score": 1.0, + "content": "with the EPOpt-PPO-FF in the interpolation benchmark, while EPOpt-PPO-FF requires access to", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 259, + 505, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 505, + 272 + ], + "score": 1.0, + "content": "the dynamics parameters but ours does not. In the extrapolation benchmark, our method greatly", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 270, + 465, + 283 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 465, + 283 + ], + "score": 1.0, + "content": "outperforms the previous method, although our objective does not consider extrapolation.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 108, + 287, + 503, + 309 + ], + "lines": [ + { + "bbox": [ + 106, + 287, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 106, + 287, + 505, + 299 + ], + "score": 1.0, + "content": "Overall, we can see that with careful tuning on recurrent model-free RL, it can at least perform as", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 298, + 441, + 309 + ], + "spans": [ + { + "bbox": [ + 106, + 298, + 441, + 309 + ], + "score": 1.0, + "content": "well as the specialized or more complicated methods, in various kinds of POMDPs.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15.5 + }, + { + "type": "title", + "bbox": [ + 107, + 319, + 415, + 331 + ], + "lines": [ + { + "bbox": [ + 105, + 318, + 416, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 416, + 333 + ], + "score": 1.0, + "content": "5.2 WHAT MATTERS IN RECURRENT MODEL-FREE RL ALGORITHMS?", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 108, + 339, + 505, + 372 + ], + "lines": [ + { + "bbox": [ + 106, + 339, + 505, + 352 + ], + "spans": [ + { + "bbox": [ + 106, + 339, + 505, + 352 + ], + "score": 1.0, + "content": "In the previous subsection, we showed that recurrent model-free RL can perform on par with the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 351, + 505, + 362 + ], + "spans": [ + { + "bbox": [ + 106, + 351, + 505, + 362 + ], + "score": 1.0, + "content": "specialized (state-of-the-art) methods, then a natural question comes: Why our implementation of", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 361, + 417, + 373 + ], + "spans": [ + { + "bbox": [ + 106, + 361, + 417, + 373 + ], + "score": 1.0, + "content": "recurrent model-free RL outperforms the implementation used in prior work?", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 378, + 505, + 466 + ], + "lines": [ + { + "bbox": [ + 106, + 378, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 106, + 378, + 505, + 390 + ], + "score": 1.0, + "content": "Our analysis will focus on ablating the five important design decisions introduced in Sec. 4: the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 389, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 505, + 401 + ], + "score": 1.0, + "content": "actor-critic architecture (Arch), the policy input space (Inputs), the underlying model-free RL", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 400, + 506, + 412 + ], + "spans": [ + { + "bbox": [ + 106, + 400, + 506, + 412 + ], + "score": 1.0, + "content": "algorithm (RL), the RNN encoder (Encoder), and the RNN context length (Len). See Table. 2 for", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 411, + 505, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 505, + 423 + ], + "score": 1.0, + "content": "a summary of how prior work made these design decisions. Due to the space limit, we show the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 422, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 422, + 505, + 434 + ], + "score": 1.0, + "content": "ablation results in some but not all the environments to compare the performance between the best", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 433, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 505, + 446 + ], + "score": 1.0, + "content": "single variant and the other variant that only differs in one decision factor. We also provide “single", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 444, + 506, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 506, + 456 + ], + "score": 1.0, + "content": "factor analysis” plots for each decision factor by averaging the performance over the other factors in", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 455, + 167, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 167, + 467 + ], + "score": 1.0, + "content": "Appendix E.2.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 24.5 + }, + { + "type": "text", + "bbox": [ + 107, + 484, + 276, + 626 + ], + "lines": [ + { + "bbox": [ + 106, + 483, + 276, + 495 + ], + "spans": [ + { + "bbox": [ + 106, + 483, + 276, + 495 + ], + "score": 1.0, + "content": "Recurrent Off-Policy Actor-Critic Ar-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 494, + 277, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 494, + 277, + 507 + ], + "score": 1.0, + "content": "chitecture. 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We", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 550, + 277, + 560 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 277, + 560 + ], + "score": 1.0, + "content": "can see that the shared architecture failed", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 560, + 277, + 572 + ], + "spans": [ + { + "bbox": [ + 106, + 560, + 277, + 572 + ], + "score": 1.0, + "content": "to learn, compared to the separate archi-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 572, + 277, + 583 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 277, + 583 + ], + "score": 1.0, + "content": "tecture. 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For", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 284, + 604, + 505, + 616 + ], + "spans": [ + { + "bbox": [ + 284, + 604, + 505, + 616 + ], + "score": 1.0, + "content": "the separate architecture, :critic and :actor refer to the", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 284, + 613, + 487, + 627 + ], + "spans": [ + { + "bbox": [ + 284, + 613, + 487, + 627 + ], + "score": 1.0, + "content": "separate RNN in critic and actor networks, respectively.", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 49 + } + ], + "index": 46.25 + }, + { + "type": "text", + "bbox": [ + 107, + 626, + 504, + 659 + ], + "lines": [ + { + "bbox": [ + 104, + 623, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 104, + 623, + 506, + 640 + ], + "score": 1.0, + "content": "2021; Sun et al., 2021) that only consider separate RNN encoders that can achieve high asymp-", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 637, + 506, + 649 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 506, + 649 + ], + "score": 1.0, + "content": "totic rewards, and also echo that (Han et al., 2020) shows poor results in the shared architecture of", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 105, + 648, + 160, + 659 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 160, + 659 + ], + "score": 1.0, + "content": "SAC-LSTM.", + "type": "text" + } + ], + "index": 55 + } + ], + "index": 54 + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "Policy Input Space. 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The data of", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 183, + 506, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 183, + 506, + 196 + ], + "score": 1.0, + "content": "EPOpt-PPO-FF and A2C-RC (a recurrent model-free on-policy RL method) are copied from the Table 7 & 8", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 194, + 190, + 205 + ], + "spans": [ + { + "bbox": [ + 106, + 194, + 190, + 205 + ], + "score": 1.0, + "content": "in Packer et al. 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See Table. 2 for", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 411, + 505, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 505, + 423 + ], + "score": 1.0, + "content": "a summary of how prior work made these design decisions. Due to the space limit, we show the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 422, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 422, + 505, + 434 + ], + "score": 1.0, + "content": "ablation results in some but not all the environments to compare the performance between the best", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 433, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 505, + 446 + ], + "score": 1.0, + "content": "single variant and the other variant that only differs in one decision factor. We also provide “single", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 444, + 506, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 506, + 456 + ], + "score": 1.0, + "content": "factor analysis” plots for each decision factor by averaging the performance over the other factors in", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 455, + 167, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 167, + 467 + ], + "score": 1.0, + "content": "Appendix E.2.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 378, + 506, + 467 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 484, + 276, + 626 + ], + "lines": [ + { + "bbox": [ + 106, + 483, + 276, + 495 + ], + "spans": [ + { + "bbox": [ + 106, + 483, + 276, + 495 + ], + "score": 1.0, + "content": "Recurrent Off-Policy Actor-Critic Ar-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 494, + 277, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 494, + 277, + 507 + ], + "score": 1.0, + "content": "chitecture. 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We", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 550, + 277, + 560 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 277, + 560 + ], + "score": 1.0, + "content": "can see that the shared architecture failed", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 560, + 277, + 572 + ], + "spans": [ + { + "bbox": [ + 106, + 560, + 277, + 572 + ], + "score": 1.0, + "content": "to learn, compared to the separate archi-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 572, + 277, + 583 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 277, + 583 + ], + "score": 1.0, + "content": "tecture. 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Change in one decision factorSubareaEnvPerformance comparison
Inputs:oa→oar“Standard”POMDPWalker-P981.6 → 1345.0 (1.3×)
RL: TD3 →SAC“Standard”POMDPAnt-P310.7 → 2123.5 (6.8×)
Encoder: LSTM→GRURobust RLWalker-Robust765.9 → 931.3 (1.2×)
Len: 64 →400Meta RLCheetah-Vel-85.2 -→ -74.6 (+14%)
Len: 64→5GeneralizationHopper-Generalize0.292 -→ 0.415 (1.4×)
Len: 64→5“Standard"POMDPWalker-V121.4 -→ 264.3 (2.2×)
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Table 2 shows that TD3 dominates all the best variants in each sub-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 242, + 506, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 506, + 253 + ], + "score": 1.0, + "content": "area, which may be due to dense reward setting in most environments. However, the 2nd row of", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 251, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 505, + 265 + ], + "score": 1.0, + "content": "Table 3 shows the effect of RL algorithm in a POMDP environment Ant-P. 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This is prominent mainly in Ant-P as it might be much harder than the others.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 106, + 312, + 505, + 444 + ], + "lines": [ + { + "bbox": [ + 105, + 311, + 506, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 506, + 325 + ], + "score": 1.0, + "content": "RNN Variants and Context Length. Generally, there is no significant difference between LSTM", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 323, + 506, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 506, + 335 + ], + "score": 1.0, + "content": "and GRU (see the single factor analysis in Appendix E.2), which is understandable as both are", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 334, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 334, + 505, + 347 + ], + "score": 1.0, + "content": "designed for general purpose. Howover, the 3rd row of Table 3 shows the effect of RNN encoder", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 346, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 106, + 346, + 505, + 357 + ], + "score": 1.0, + "content": "in a robust RL environment. We can see replacing LSTM with GRU can increase the worst-case", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 356, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 505, + 369 + ], + "score": 1.0, + "content": "metric in Walker-Robust. For the context length in RNNs, a medium length (64) dominates in", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 368, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 106, + 368, + 505, + 379 + ], + "score": 1.0, + "content": "all the best variants in each subarea (see Table 2), which could be viewed as a trade-off between", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 379, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 505, + 390 + ], + "score": 1.0, + "content": "memory capacity and computation costs. However, the remaining rows of Table 3 show the mixed", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 389, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 106, + 389, + 505, + 402 + ], + "score": 1.0, + "content": "effects of context length in RNNs. Both increasing and decreasing the context length can boost the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 400, + 506, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 506, + 412 + ], + "score": 1.0, + "content": "performance in different environments. Specifically, decreasing the length from 64 to 5 makes our", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 411, + 505, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 301, + 423 + ], + "score": 1.0, + "content": "method surpass VRM in Walker-V (increase by", + "type": "text" + }, + { + "bbox": [ + 302, + 412, + 324, + 422 + ], + "score": 0.87, + "content": "2 . 2 \\times \\phantom { }", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 411, + 505, + 423 + ], + "score": 1.0, + "content": ". This might explain why the prior methods", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 423, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 423, + 505, + 434 + ], + "score": 1.0, + "content": "adopt a wide range of context lengths from 1 to 2048 (see Table 2). Therefore, the choice of context", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 433, + 307, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 307, + 446 + ], + "score": 1.0, + "content": "length is problem-specific and can require tuning.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 107, + 460, + 392, + 471 + ], + "lines": [ + { + "bbox": [ + 105, + 458, + 394, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 394, + 475 + ], + "score": 1.0, + "content": "Summary. We now summarize the main findings of our experiments:", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 116, + 482, + 505, + 603 + ], + "lines": [ + { + "bbox": [ + 116, + 482, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 116, + 482, + 505, + 495 + ], + "score": 1.0, + "content": "1. Using separate weights for the recurrent actor and recurrent critic boosts performance, likely", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 128, + 494, + 409, + 506 + ], + "spans": [ + { + "bbox": [ + 128, + 494, + 409, + 506 + ], + "score": 1.0, + "content": "because it avoids gradient explosion (Fig. 5 and Fig. 17 in Appendix).", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 117, + 508, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 117, + 509, + 126, + 519 + ], + "score": 1.0, + "content": "2.", + "type": "text" + }, + { + "bbox": [ + 127, + 508, + 505, + 520 + ], + "score": 1.0, + "content": "Using state-of-the-art off-policy RL algorithms as the backbone in recurrent model-free RL", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 128, + 519, + 431, + 532 + ], + "spans": [ + { + "bbox": [ + 128, + 519, + 431, + 532 + ], + "score": 1.0, + "content": "can improve asymptotic performance (Fig. 1 and Figures in Appendix E.1).", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 117, + 532, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 117, + 532, + 505, + 547 + ], + "score": 1.0, + "content": "3. The context length for the recurrent actor and critic has a large influence on task performance,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 127, + 544, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 127, + 544, + 505, + 557 + ], + "score": 1.0, + "content": "but the optimal length is task-specific. Reasonable values are 5 to 500, and 64 is a good start", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 128, + 554, + 339, + 568 + ], + "spans": [ + { + "bbox": [ + 128, + 554, + 339, + 568 + ], + "score": 1.0, + "content": "(Rows 4–6 in Table 3 and Figures in Appendix E.2).", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 115, + 569, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 115, + 569, + 505, + 582 + ], + "score": 1.0, + "content": "4. It is important that the inputs to the recurrent actor and critic, such as past observations and", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 127, + 581, + 506, + 593 + ], + "spans": [ + { + "bbox": [ + 127, + 581, + 506, + 593 + ], + "score": 1.0, + "content": "past returns, contain enough information to infer the POMDP hidden states (Row 1 in Table 3", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 129, + 592, + 252, + 604 + ], + "spans": [ + { + "bbox": [ + 129, + 592, + 252, + 604 + ], + "score": 1.0, + "content": "and Figures in Appendix E.2).", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 32.5 + }, + { + "type": "text", + "bbox": [ + 106, + 614, + 503, + 625 + ], + "lines": [ + { + "bbox": [ + 106, + 613, + 504, + 626 + ], + "spans": [ + { + "bbox": [ + 106, + 613, + 504, + 626 + ], + "score": 1.0, + "content": "These findings may provide a useful initialization for researchers studying recurrent model-free RL.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "title", + "bbox": [ + 107, + 636, + 196, + 649 + ], + "lines": [ + { + "bbox": [ + 105, + 633, + 198, + 652 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 198, + 652 + ], + "score": 1.0, + "content": "6 CONCLUSION", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 106, + 654, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "In this paper, we show that a carefully-designed implementation of recurrent model-free RL can", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "perform well across a range of POMDP domains, often on par with (if not significantly better than)", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "prior methods that are specifically designed for specific types of POMDPs. Our ablation experiments", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "demonstrate the importance of key design decisions, such as the underlying RL algorithm and RNN", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "context length. While the best choices for some decisions (such as using separate RNNs for the", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 710, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 721 + ], + "score": 1.0, + "content": "actor and the critic) are consistent across domains, the best choices for other decisions (such as", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 721, + 288, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 288, + 732 + ], + "score": 1.0, + "content": "RNN context length) are problem-dependent.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 43 + } + ], + "page_idx": 8, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "9", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 109, + 117, + 500, + 213 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 54, + 505, + 115 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 53, + 506, + 65 + ], + "spans": [ + { + "bbox": [ + 105, + 53, + 506, + 65 + ], + "score": 1.0, + "content": "Table 3: Ablation results in our implementation of recurrent model-free RL. In this table, we show how a", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 65, + 504, + 75 + ], + "spans": [ + { + "bbox": [ + 106, + 65, + 504, + 75 + ], + "score": 1.0, + "content": "single change in one decision factor from the variant that is best on average in that subarea, could significantly", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 74, + 505, + 86 + ], + "spans": [ + { + "bbox": [ + 106, + 74, + 505, + 86 + ], + "score": 1.0, + "content": "increase the performance. 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Change in one decision factorSubareaEnvPerformance comparison
Inputs:oa→oar“Standard”POMDPWalker-P981.6 → 1345.0 (1.3×)
RL: TD3 →SAC“Standard”POMDPAnt-P310.7 → 2123.5 (6.8×)
Encoder: LSTM→GRURobust RLWalker-Robust765.9 → 931.3 (1.2×)
Len: 64 →400Meta RLCheetah-Vel-85.2 -→ -74.6 (+14%)
Len: 64→5GeneralizationHopper-Generalize0.292 -→ 0.415 (1.4×)
Len: 64→5“Standard"POMDPWalker-V121.4 -→ 264.3 (2.2×)
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Table 2 shows that TD3 dominates all the best variants in each sub-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 242, + 506, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 506, + 253 + ], + "score": 1.0, + "content": "area, which may be due to dense reward setting in most environments. However, the 2nd row of", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 251, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 505, + 265 + ], + "score": 1.0, + "content": "Table 3 shows the effect of RL algorithm in a POMDP environment Ant-P. SAC is significantly", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 263, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 226, + 276 + ], + "score": 1.0, + "content": "better than TD3 (increase by", + "type": "text" + }, + { + "bbox": [ + 227, + 264, + 248, + 275 + ], + "score": 0.87, + "content": "6 . 8 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 263, + 505, + 276 + ], + "score": 1.0, + "content": ", surpassing the PPO-GRU (Kostrikov, 2018) in Fig. 1), possi-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 273, + 506, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 506, + 288 + ], + "score": 1.0, + "content": "bly due to strategic exploration where the action noise conditions on the history instead of being", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 285, + 473, + 298 + ], + "spans": [ + { + "bbox": [ + 106, + 285, + 473, + 298 + ], + "score": 1.0, + "content": "independent. This is prominent mainly in Ant-P as it might be much harder than the others.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11.5, + "bbox_fs": [ + 105, + 230, + 506, + 298 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 312, + 505, + 444 + ], + "lines": [ + { + "bbox": [ + 105, + 311, + 506, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 506, + 325 + ], + "score": 1.0, + "content": "RNN Variants and Context Length. Generally, there is no significant difference between LSTM", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 323, + 506, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 506, + 335 + ], + "score": 1.0, + "content": "and GRU (see the single factor analysis in Appendix E.2), which is understandable as both are", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 334, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 334, + 505, + 347 + ], + "score": 1.0, + "content": "designed for general purpose. Howover, the 3rd row of Table 3 shows the effect of RNN encoder", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 346, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 106, + 346, + 505, + 357 + ], + "score": 1.0, + "content": "in a robust RL environment. We can see replacing LSTM with GRU can increase the worst-case", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 356, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 505, + 369 + ], + "score": 1.0, + "content": "metric in Walker-Robust. For the context length in RNNs, a medium length (64) dominates in", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 368, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 106, + 368, + 505, + 379 + ], + "score": 1.0, + "content": "all the best variants in each subarea (see Table 2), which could be viewed as a trade-off between", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 379, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 505, + 390 + ], + "score": 1.0, + "content": "memory capacity and computation costs. However, the remaining rows of Table 3 show the mixed", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 389, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 106, + 389, + 505, + 402 + ], + "score": 1.0, + "content": "effects of context length in RNNs. Both increasing and decreasing the context length can boost the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 400, + 506, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 506, + 412 + ], + "score": 1.0, + "content": "performance in different environments. Specifically, decreasing the length from 64 to 5 makes our", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 411, + 505, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 301, + 423 + ], + "score": 1.0, + "content": "method surpass VRM in Walker-V (increase by", + "type": "text" + }, + { + "bbox": [ + 302, + 412, + 324, + 422 + ], + "score": 0.87, + "content": "2 . 2 \\times \\phantom { }", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 411, + 505, + 423 + ], + "score": 1.0, + "content": ". This might explain why the prior methods", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 423, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 423, + 505, + 434 + ], + "score": 1.0, + "content": "adopt a wide range of context lengths from 1 to 2048 (see Table 2). Therefore, the choice of context", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 433, + 307, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 307, + 446 + ], + "score": 1.0, + "content": "length is problem-specific and can require tuning.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 311, + 506, + 446 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 460, + 392, + 471 + ], + "lines": [ + { + "bbox": [ + 105, + 458, + 394, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 394, + 475 + ], + "score": 1.0, + "content": "Summary. We now summarize the main findings of our experiments:", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 458, + 394, + 475 + ] + }, + { + "type": "list", + "bbox": [ + 116, + 482, + 505, + 603 + ], + "lines": [ + { + "bbox": [ + 116, + 482, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 116, + 482, + 505, + 495 + ], + "score": 1.0, + "content": "1. Using separate weights for the recurrent actor and recurrent critic boosts performance, likely", + "type": "text" + } + ], + "index": 28, + "is_list_start_line": true + }, + { + "bbox": [ + 128, + 494, + 409, + 506 + ], + "spans": [ + { + "bbox": [ + 128, + 494, + 409, + 506 + ], + "score": 1.0, + "content": "because it avoids gradient explosion (Fig. 5 and Fig. 17 in Appendix).", + "type": "text" + } + ], + "index": 29, + "is_list_end_line": true + }, + { + "bbox": [ + 117, + 508, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 117, + 509, + 126, + 519 + ], + "score": 1.0, + "content": "2.", + "type": "text" + }, + { + "bbox": [ + 127, + 508, + 505, + 520 + ], + "score": 1.0, + "content": "Using state-of-the-art off-policy RL algorithms as the backbone in recurrent model-free RL", + "type": "text" + } + ], + "index": 30, + "is_list_start_line": true + }, + { + "bbox": [ + 128, + 519, + 431, + 532 + ], + "spans": [ + { + "bbox": [ + 128, + 519, + 431, + 532 + ], + "score": 1.0, + "content": "can improve asymptotic performance (Fig. 1 and Figures in Appendix E.1).", + "type": "text" + } + ], + "index": 31, + "is_list_end_line": true + }, + { + "bbox": [ + 117, + 532, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 117, + 532, + 505, + 547 + ], + "score": 1.0, + "content": "3. The context length for the recurrent actor and critic has a large influence on task performance,", + "type": "text" + } + ], + "index": 32, + "is_list_start_line": true + }, + { + "bbox": [ + 127, + 544, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 127, + 544, + 505, + 557 + ], + "score": 1.0, + "content": "but the optimal length is task-specific. Reasonable values are 5 to 500, and 64 is a good start", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 128, + 554, + 339, + 568 + ], + "spans": [ + { + "bbox": [ + 128, + 554, + 339, + 568 + ], + "score": 1.0, + "content": "(Rows 4–6 in Table 3 and Figures in Appendix E.2).", + "type": "text" + } + ], + "index": 34, + "is_list_end_line": true + }, + { + "bbox": [ + 115, + 569, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 115, + 569, + 505, + 582 + ], + "score": 1.0, + "content": "4. 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Our ablation experiments", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "demonstrate the importance of key design decisions, such as the underlying RL algorithm and RNN", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "context length. 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A.3.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 107, + 242, + 505, + 396 + ], + "lines": [ + { + "bbox": [ + 105, + 241, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 505, + 254 + ], + "score": 1.0, + "content": "Memory-efficient replay buffer for sequences. Moreover, we design an efficient replay buffer", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 251, + 506, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 506, + 267 + ], + "score": 1.0, + "content": "for off-policy RL methods to cope with sequential inputs. 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MethodEnvironmentTime costRAMGPU memory
Ours (GRU)Hopper-V22.5 h0(1)1.2 GB
VRM (Han et al., 2020)Hopper-V102 h0(200)N/A
OursSemi-Circle12h0(1)1 GB
VariBAD (Dorfman et al., 2020)Semi-Circle2.3 h0(1)*9.5 GB
OursCheetah-Robust7h0(1)1.1 GB
MRPO (Jiang et al., 2021)Cheetah-Robust0.4 hN/AN/A
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MethodEnvironmentTime costRAMGPU memory
Ours (GRU)Hopper-V22.5 h0(1)1.2 GB
VRM (Han et al., 2020)Hopper-V102 h0(200)N/A
OursSemi-Circle12h0(1)1 GB
VariBAD (Dorfman et al., 2020)Semi-Circle2.3 h0(1)*9.5 GB
OursCheetah-Robust7h0(1)1.1 GB
MRPO (Jiang et al., 2021)Cheetah-Robust0.4 hN/AN/A
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The left", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 509, + 506, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 506, + 522 + ], + "score": 1.0, + "content": "part shows the actor network, and the right shows the critic network. Each block shows a trainable", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 521, + 505, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 505, + 533 + ], + "score": 1.0, + "content": "module, with independent weights. We italicize the previous action and reward encoders as they are", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 531, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 505, + 544 + ], + "score": 1.0, + "content": "optional. By default, each encoder has one hidden layer, each RNN is one-layer LSTM or GRU,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 543, + 241, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 241, + 555 + ], + "score": 1.0, + "content": "each MLP has two hidden layers.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23 + } + ], + "index": 21.0 + }, + { + "type": "text", + "bbox": [ + 107, + 566, + 503, + 600 + ], + "lines": [ + { + "bbox": [ + 105, + 565, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 505, + 578 + ], + "score": 1.0, + "content": "Fig. 6 shows our (separate) recurrent actor-critic architecture. Table 5 shows the main hyperparam-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 576, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 505, + 590 + ], + "score": 1.0, + "content": "eters we adopt for each subarea. We did not tune these hyperparameters, except that we adjusted the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 588, + 445, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 445, + 600 + ], + "score": 1.0, + "content": "number of gradient steps so that all the experiments could be completed in 36 hours.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 565, + 505, + 600 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 604, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 105, + 603, + 506, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 506, + 617 + ], + "score": 1.0, + "content": "For Markovian policies (SAC and TD3), we remove the encoders and RNNs from the actor-critic", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 616, + 504, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 504, + 628 + ], + "score": 1.0, + "content": "architecture, and train them with same hyperparameters as those of recurrent policies. For each task,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 626, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 626, + 505, + 639 + ], + "score": 1.0, + "content": "we report the results of either SAC or TD3, whichever achieves higher returns. For Oracle policies,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 637, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 637, + 505, + 650 + ], + "score": 1.0, + "content": "we use the well-tuned results from Table 1 (“SAC w/ unstructured row”) in Raffin et al. (2021),", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 648, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 106, + 648, + 505, + 661 + ], + "score": 1.0, + "content": "which is based on Stable Baseline3 (Raffin, 2020), for “standard” POMDPs. For the other subareas,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 660, + 504, + 672 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 504, + 672 + ], + "score": 1.0, + "content": "we have to run the Markovian policies (SAC and TD3) with access to the hidden states, using the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 670, + 506, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 506, + 683 + ], + "score": 1.0, + "content": "same training hyperparameters as those of recurrent policies. But these Oracle policies might be not", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 680, + 506, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 680, + 506, + 694 + ], + "score": 1.0, + "content": "well-tuned given same environment and gradient steps, especially in robust RL and generalization", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 691, + 135, + 705 + ], + "spans": [ + { + "bbox": [ + 104, + 691, + 135, + 705 + ], + "score": 1.0, + "content": "in RL.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 33, + "bbox_fs": [ + 104, + 603, + 506, + 705 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "We also show the settings of the specialized methods we compared in the main paper in Table 6. Note", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "score": 1.0, + "content": "that our recurrent model-free RL share the exactly same settings as (off-policy) VariBAD (Dorfman", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 505, + 506, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 506, + 518 + ], + "score": 1.0, + "content": "et al., 2020) and VRM (Han et al., 2020). For MRPO (Jiang et al., 2021) and EPOPT (Rajeswaran", + "type": "text", + "cross_page": true + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 516, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 104, + 516, + 506, + 529 + ], + "score": 1.0, + "content": "et al., 2017a), they adopt totally different settings, i.e. on-policy Markovian approaches to MDPs", + "type": "text", + "cross_page": true + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 528, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 505, + 541 + ], + "score": 1.0, + "content": "(with access to the ground-truth state (s) of environment). 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For", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 79, + 505, + 93 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 505, + 93 + ], + "score": 1.0, + "content": "each subarea, we report the hidden layer size of each module, RL and training hyperparameters.", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 90, + 505, + 104 + ], + "spans": [ + { + "bbox": [ + 105, + 90, + 505, + 104 + ], + "score": 1.0, + "content": "For Meta-RL, we take the model on Cheetah-Vel as example, which follows the architecture design", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 101, + 505, + 114 + ], + "spans": [ + { + "bbox": [ + 105, + 101, + 505, + 114 + ], + "score": 1.0, + "content": "of off-policy VariBAD (Dorfman et al., 2020). 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Meta-RL* POMDP Robust RL Gen. in RL
HiddenlayersizeObserv.encoderPrev. Action encoder*Reward encoder*RNNMLP[32][16][16][128][128,128,128] [256,256]
RLhparamsLearning rateDiscount factor ySmoothing coef T SAC temperatureTD3 noisesReplay buffer sizeBatch size3e-40.990.005 automatically updated by Haarnoja et al. (2018b)default values from Fujimoto et al. (2018)1e632 64
TraininghparamsEnvironment stepsGradient steps5M 1.5M 3M0.1M 1.5M 0.6M
PolicyinputsLargest input spaceBest input spaceoard oar oa oaroard oa 0 0
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Meta-RL*POMDPRobustRLGen. in RL
Approach(off-policy) VariBADVRMMRPOEPOPT
Memory-based?√√XX
Off-policy?√√XX
Input spaceoardoarSS
Access to hidden states?XX√√
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Meta-RL* POMDP Robust RL Gen. in RL
HiddenlayersizeObserv.encoderPrev. Action encoder*Reward encoder*RNNMLP[32][16][16][128][128,128,128] [256,256]
RLhparamsLearning rateDiscount factor ySmoothing coef T SAC temperatureTD3 noisesReplay buffer sizeBatch size3e-40.990.005 automatically updated by Haarnoja et al. (2018b)default values from Fujimoto et al. (2018)1e632 64
TraininghparamsEnvironment stepsGradient steps5M 1.5M 3M0.1M 1.5M 0.6M
PolicyinputsLargest input spaceBest input spaceoard oar oa oaroard oa 0 0
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Meta-RL*POMDPRobustRLGen. in RL
Approach(off-policy) VariBADVRMMRPOEPOPT
Memory-based?√√XX
Off-policy?√√XX
Input spaceoardoarSS
Access to hidden states?XX√√
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The goal state only appears in reward function:", + "type": "text" + }, + { + "bbox": [ + 349, + 318, + 505, + 331 + ], + "score": 0.93, + "content": "R ( s _ { t } ^ { o } , s _ { t + 1 } ^ { o } , a _ { t } , s ^ { h } ) \\equiv R ( p _ { t + 1 } , p _ { g } { \\bar { ) } } =", + "type": "inline_equation" + } + ], + "index": 15 + }, + { + "bbox": [ + 107, + 329, + 428, + 342 + ], + "spans": [ + { + "bbox": [ + 107, + 329, + 194, + 342 + ], + "score": 0.9, + "content": "\\mathbb { 1 } ( \\lVert p _ { t + 1 } - p _ { g } \\rVert _ { 2 } \\leq r )", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 329, + 289, + 341 + ], + "score": 1.0, + "content": ". The dynamic function", + "type": "text" + }, + { + "bbox": [ + 289, + 330, + 298, + 339 + ], + "score": 0.82, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 329, + 428, + 341 + ], + "score": 1.0, + "content": "is independent of the goal state.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 107, + 352, + 505, + 408 + ], + "lines": [ + { + "bbox": [ + 106, + 352, + 505, + 364 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 505, + 364 + ], + "score": 1.0, + "content": "Cheetah-Vel. We directly follow Dorfman et al. (2020) using MuJoCo (Todorov et al., 2012)", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 362, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 329, + 376 + ], + "score": 1.0, + "content": "simulator of HalfCheetah-v2. The hidden state", + "type": "text" + }, + { + "bbox": [ + 330, + 363, + 340, + 374 + ], + "score": 0.87, + "content": "s ^ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 362, + 434, + 376 + ], + "score": 1.0, + "content": "is the target velocity", + "type": "text" + }, + { + "bbox": [ + 434, + 365, + 445, + 376 + ], + "score": 0.85, + "content": "v _ { g }", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 362, + 505, + 376 + ], + "score": 1.0, + "content": "and observed", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 375, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 129, + 387 + ], + "score": 1.0, + "content": "state", + "type": "text" + }, + { + "bbox": [ + 129, + 375, + 140, + 385 + ], + "score": 0.84, + "content": "s ^ { o }", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 375, + 234, + 387 + ], + "score": 1.0, + "content": "includes the velocity", + "type": "text" + }, + { + "bbox": [ + 234, + 377, + 241, + 385 + ], + "score": 0.71, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 375, + 505, + 387 + ], + "score": 1.0, + "content": ". Reward function includes both the hidden state and action:", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 381, + 508, + 404 + ], + "spans": [ + { + "bbox": [ + 106, + 385, + 367, + 398 + ], + "score": 0.86, + "content": "R ( s _ { t } ^ { o } , s _ { t + 1 } ^ { o } , a _ { t } , s ^ { h } ) \\equiv R ( v _ { t } , v _ { g } , a _ { t } ) = - \\| v _ { t } - v _ { g } \\| _ { 1 } - 0 . 0 5 \\| a _ { t } \\| _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 381, + 462, + 404 + ], + "score": 1.0, + "content": ". The dynamic function", + "type": "text" + }, + { + "bbox": [ + 463, + 386, + 471, + 396 + ], + "score": 0.82, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 381, + 508, + 404 + ], + "score": 1.0, + "content": "is inde-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 397, + 209, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 209, + 409 + ], + "score": 1.0, + "content": "pendent of the goal state.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 420, + 505, + 489 + ], + "lines": [ + { + "bbox": [ + 106, + 419, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 506, + 432 + ], + "score": 1.0, + "content": "Wind. We modified the parameters of Wind environment in Dorfman et al. (2020) to make it", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 431, + 506, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 431, + 407, + 444 + ], + "score": 1.0, + "content": "harder to solve. The agent must navigate to a fixed (but unknown) goal", + "type": "text" + }, + { + "bbox": [ + 407, + 433, + 419, + 444 + ], + "score": 0.86, + "content": "p _ { g }", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 431, + 506, + 444 + ], + "score": 1.0, + "content": "within a distance of", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 107, + 442, + 505, + 454 + ], + "spans": [ + { + "bbox": [ + 107, + 442, + 135, + 452 + ], + "score": 0.89, + "content": "D = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 442, + 505, + 454 + ], + "score": 1.0, + "content": "from its fixed initial state. Similarly to Semi-Circle, the reward function is goal conditioned", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 452, + 506, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 210, + 468 + ], + "score": 1.0, + "content": "but without hidden state:", + "type": "text" + }, + { + "bbox": [ + 210, + 452, + 451, + 466 + ], + "score": 0.88, + "content": "R ( s _ { t } ^ { o } , s _ { t + 1 } ^ { o } , a _ { t } , \\bar { s ^ { h } } ) \\equiv R ( p _ { t + 1 } , p _ { g } ) = \\mathbb { 1 } ( \\| p _ { t + 1 } - p _ { g } \\| _ { 2 } \\bar { \\leq ^ { } { r } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 452, + 506, + 468 + ], + "score": 1.0, + "content": ". The hidden", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 463, + 507, + 481 + ], + "spans": [ + { + "bbox": [ + 104, + 463, + 127, + 481 + ], + "score": 1.0, + "content": "state", + "type": "text" + }, + { + "bbox": [ + 127, + 465, + 138, + 476 + ], + "score": 0.87, + "content": "s ^ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 463, + 368, + 481 + ], + "score": 1.0, + "content": "appear in the deterministic dynamics as a noise term, i.e.", + "type": "text" + }, + { + "bbox": [ + 369, + 466, + 453, + 479 + ], + "score": 0.9, + "content": "s _ { t + 1 } ^ { o } = s _ { t } ^ { o } + a _ { t } + s ^ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 463, + 483, + 481 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 484, + 465, + 495, + 476 + ], + "score": 0.88, + "content": "s ^ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 463, + 507, + 481 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 477, + 398, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 164, + 489 + ], + "score": 1.0, + "content": "sampled from", + "type": "text" + }, + { + "bbox": [ + 164, + 477, + 226, + 489 + ], + "score": 0.87, + "content": "U [ - 0 . 0 8 , 0 . 0 8 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 477, + 398, + 489 + ], + "score": 1.0, + "content": "at the initial time-step and then kept fixed.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24.5 + }, + { + "type": "title", + "bbox": [ + 108, + 501, + 185, + 513 + ], + "lines": [ + { + "bbox": [ + 105, + 500, + 187, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 187, + 515 + ], + "score": 1.0, + "content": "D.3 ROBUST RL", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 522, + 505, + 600 + ], + "lines": [ + { + "bbox": [ + 106, + 521, + 506, + 536 + ], + "spans": [ + { + "bbox": [ + 106, + 521, + 506, + 536 + ], + "score": 1.0, + "content": "{Hopper,Walker,Cheetah}-Robust. We directly adopt the environments used in MRPO (Jiang", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 532, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 505, + 546 + ], + "score": 1.0, + "content": "et al., 2021). In each environment, the hidden state is the dynamics parameters including the density", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 544, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 506, + 556 + ], + "score": 1.0, + "content": "and friction coefficients of the simulated robot in roboschool, adapted from the SunBlaze (Packer", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 555, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 505, + 567 + ], + "score": 1.0, + "content": "et al., 2018). The exact ranges of the hidden states in each environment can be found in Table 1 of", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 567, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 567, + 505, + 578 + ], + "score": 1.0, + "content": "MRPO (Jiang et al., 2021). We evaluate the algorithms with 100 tasks in each environment, and use", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 577, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 381, + 590 + ], + "score": 1.0, + "content": "the average of them as average returns, and the average of the worst", + "type": "text" + }, + { + "bbox": [ + 382, + 577, + 402, + 588 + ], + "score": 0.88, + "content": "1 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 577, + 506, + 590 + ], + "score": 1.0, + "content": "of them as worst returns,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 587, + 205, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 205, + 601 + ], + "score": 1.0, + "content": "following MRPO paper.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 32 + }, + { + "type": "title", + "bbox": [ + 107, + 613, + 239, + 624 + ], + "lines": [ + { + "bbox": [ + 106, + 613, + 240, + 626 + ], + "spans": [ + { + "bbox": [ + 106, + 613, + 240, + 626 + ], + "score": 1.0, + "content": "D.4 GENERALIZATION IN RL", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 633, + 505, + 700 + ], + "lines": [ + { + "bbox": [ + 107, + 633, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 107, + 633, + 506, + 645 + ], + "score": 1.0, + "content": "{Hopper,Cheetah}-Generalize. We directly adopt the environments used in SunBlaze (Packer", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "score": 1.0, + "content": "et al., 2018). In each environment, the hidden state is the dynamics parameters including the den-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "score": 1.0, + "content": "sity, friction coefficients, and the power of the simulated robot in roboschool. The exact ranges of", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "both interpolation and extrapolation in the hidden state distribution for each environment can be", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "found in Table 1 of SunBlaze (Packer et al., 2018). We follow the practice of SunBlaze to evaluate", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 689, + 288, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 288, + 700 + ], + "score": 1.0, + "content": "interpolation and extrapolation success rates.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 39.5 + } + ], + "page_idx": 17, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "18", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 254, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 256, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 256, + 96 + ], + "score": 1.0, + "content": "D ENVIRONMENT DETAILS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "title", + "bbox": [ + 107, + 105, + 231, + 118 + ], + "lines": [ + { + "bbox": [ + 105, + 104, + 232, + 120 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 232, + 120 + ], + "score": 1.0, + "content": "D.1 “STANDARD” POMDP", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 107, + 127, + 505, + 171 + ], + "lines": [ + { + "bbox": [ + 106, + 127, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 505, + 138 + ], + "score": 1.0, + "content": "Except for the classic Pendulum environment, we use PyBullet (Coumans & Bai, 2016) as the sim-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 136, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 505, + 150 + ], + "score": 1.0, + "content": "ulator for “standard” POMDP environments. As the practice in VRM (Han et al., 2020), we remove", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 148, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 505, + 162 + ], + "score": 1.0, + "content": "all the position/angle-related entries in the observation space for “-V” environments and velocity-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 159, + 438, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 438, + 172 + ], + "score": 1.0, + "content": "related entries for “-P” environments, to transform the original MDP into POMDP.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3.5, + "bbox_fs": [ + 105, + 127, + 505, + 172 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 183, + 505, + 216 + ], + "lines": [ + { + "bbox": [ + 106, + 181, + 506, + 197 + ], + "spans": [ + { + "bbox": [ + 106, + 181, + 506, + 197 + ], + "score": 1.0, + "content": "{Pendulum,Ant,Cheetah,Hopper,Walker}-P. The “-P” stands for the environments that keep", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 194, + 505, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 457, + 205 + ], + "score": 1.0, + "content": "position-related entries by removal of velocity-related entries. Thus, the observed state", + "type": "text" + }, + { + "bbox": [ + 458, + 195, + 468, + 204 + ], + "score": 0.85, + "content": "s ^ { o }", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 194, + 505, + 205 + ], + "score": 1.0, + "content": "includes", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 204, + 333, + 218 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 145, + 218 + ], + "score": 1.0, + "content": "positions", + "type": "text" + }, + { + "bbox": [ + 145, + 207, + 151, + 217 + ], + "score": 0.75, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 204, + 244, + 218 + ], + "score": 1.0, + "content": ", while the hidden state", + "type": "text" + }, + { + "bbox": [ + 245, + 205, + 255, + 215 + ], + "score": 0.88, + "content": "s ^ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 204, + 321, + 218 + ], + "score": 1.0, + "content": "is the velocities", + "type": "text" + }, + { + "bbox": [ + 321, + 208, + 327, + 215 + ], + "score": 0.7, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 204, + 333, + 218 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 181, + 506, + 218 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 228, + 505, + 262 + ], + "lines": [ + { + "bbox": [ + 105, + 227, + 506, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 227, + 506, + 243 + ], + "score": 1.0, + "content": "{Pendulum,Ant,Cheetah,Hopper,Walker}-V. The “-V” stands for the environments that keep", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 239, + 505, + 252 + ], + "spans": [ + { + "bbox": [ + 106, + 239, + 457, + 252 + ], + "score": 1.0, + "content": "velocity-related entries by removal of position-related entries. 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We directly follow off-policy version of VariBAD (Dorfman et al., 2020). 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The goal state only appears in reward function:", + "type": "text" + }, + { + "bbox": [ + 349, + 318, + 505, + 331 + ], + "score": 0.93, + "content": "R ( s _ { t } ^ { o } , s _ { t + 1 } ^ { o } , a _ { t } , s ^ { h } ) \\equiv R ( p _ { t + 1 } , p _ { g } { \\bar { ) } } =", + "type": "inline_equation" + } + ], + "index": 15 + }, + { + "bbox": [ + 107, + 329, + 428, + 342 + ], + "spans": [ + { + "bbox": [ + 107, + 329, + 194, + 342 + ], + "score": 0.9, + "content": "\\mathbb { 1 } ( \\lVert p _ { t + 1 } - p _ { g } \\rVert _ { 2 } \\leq r )", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 329, + 289, + 341 + ], + "score": 1.0, + "content": ". The dynamic function", + "type": "text" + }, + { + "bbox": [ + 289, + 330, + 298, + 339 + ], + "score": 0.82, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 329, + 428, + 341 + ], + "score": 1.0, + "content": "is independent of the goal state.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14.5, + "bbox_fs": [ + 104, + 295, + 506, + 342 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 352, + 505, + 408 + ], + "lines": [ + { + "bbox": [ + 106, + 352, + 505, + 364 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 505, + 364 + ], + "score": 1.0, + "content": "Cheetah-Vel. We directly follow Dorfman et al. (2020) using MuJoCo (Todorov et al., 2012)", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 362, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 329, + 376 + ], + "score": 1.0, + "content": "simulator of HalfCheetah-v2. The hidden state", + "type": "text" + }, + { + "bbox": [ + 330, + 363, + 340, + 374 + ], + "score": 0.87, + "content": "s ^ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 362, + 434, + 376 + ], + "score": 1.0, + "content": "is the target velocity", + "type": "text" + }, + { + "bbox": [ + 434, + 365, + 445, + 376 + ], + "score": 0.85, + "content": "v _ { g }", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 362, + 505, + 376 + ], + "score": 1.0, + "content": "and observed", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 375, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 129, + 387 + ], + "score": 1.0, + "content": "state", + "type": "text" + }, + { + "bbox": [ + 129, + 375, + 140, + 385 + ], + "score": 0.84, + "content": "s ^ { o }", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 375, + 234, + 387 + ], + "score": 1.0, + "content": "includes the velocity", + "type": "text" + }, + { + "bbox": [ + 234, + 377, + 241, + 385 + ], + "score": 0.71, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 375, + 505, + 387 + ], + "score": 1.0, + "content": ". Reward function includes both the hidden state and action:", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 381, + 508, + 404 + ], + "spans": [ + { + "bbox": [ + 106, + 385, + 367, + 398 + ], + "score": 0.86, + "content": "R ( s _ { t } ^ { o } , s _ { t + 1 } ^ { o } , a _ { t } , s ^ { h } ) \\equiv R ( v _ { t } , v _ { g } , a _ { t } ) = - \\| v _ { t } - v _ { g } \\| _ { 1 } - 0 . 0 5 \\| a _ { t } \\| _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 381, + 462, + 404 + ], + "score": 1.0, + "content": ". The dynamic function", + "type": "text" + }, + { + "bbox": [ + 463, + 386, + 471, + 396 + ], + "score": 0.82, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 381, + 508, + 404 + ], + "score": 1.0, + "content": "is inde-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 397, + 209, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 209, + 409 + ], + "score": 1.0, + "content": "pendent of the goal state.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 352, + 508, + 409 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 420, + 505, + 489 + ], + "lines": [ + { + "bbox": [ + 106, + 419, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 506, + 432 + ], + "score": 1.0, + "content": "Wind. We modified the parameters of Wind environment in Dorfman et al. (2020) to make it", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 431, + 506, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 431, + 407, + 444 + ], + "score": 1.0, + "content": "harder to solve. The agent must navigate to a fixed (but unknown) goal", + "type": "text" + }, + { + "bbox": [ + 407, + 433, + 419, + 444 + ], + "score": 0.86, + "content": "p _ { g }", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 431, + 506, + 444 + ], + "score": 1.0, + "content": "within a distance of", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 107, + 442, + 505, + 454 + ], + "spans": [ + { + "bbox": [ + 107, + 442, + 135, + 452 + ], + "score": 0.89, + "content": "D = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 442, + 505, + 454 + ], + "score": 1.0, + "content": "from its fixed initial state. Similarly to Semi-Circle, the reward function is goal conditioned", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 452, + 506, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 210, + 468 + ], + "score": 1.0, + "content": "but without hidden state:", + "type": "text" + }, + { + "bbox": [ + 210, + 452, + 451, + 466 + ], + "score": 0.88, + "content": "R ( s _ { t } ^ { o } , s _ { t + 1 } ^ { o } , a _ { t } , \\bar { s ^ { h } } ) \\equiv R ( p _ { t + 1 } , p _ { g } ) = \\mathbb { 1 } ( \\| p _ { t + 1 } - p _ { g } \\| _ { 2 } \\bar { \\leq ^ { } { r } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 452, + 506, + 468 + ], + "score": 1.0, + "content": ". The hidden", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 463, + 507, + 481 + ], + "spans": [ + { + "bbox": [ + 104, + 463, + 127, + 481 + ], + "score": 1.0, + "content": "state", + "type": "text" + }, + { + "bbox": [ + 127, + 465, + 138, + 476 + ], + "score": 0.87, + "content": "s ^ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 463, + 368, + 481 + ], + "score": 1.0, + "content": "appear in the deterministic dynamics as a noise term, i.e.", + "type": "text" + }, + { + "bbox": [ + 369, + 466, + 453, + 479 + ], + "score": 0.9, + "content": "s _ { t + 1 } ^ { o } = s _ { t } ^ { o } + a _ { t } + s ^ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 463, + 483, + 481 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 484, + 465, + 495, + 476 + ], + "score": 0.88, + "content": "s ^ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 463, + 507, + 481 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 477, + 398, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 164, + 489 + ], + "score": 1.0, + "content": "sampled from", + "type": "text" + }, + { + "bbox": [ + 164, + 477, + 226, + 489 + ], + "score": 0.87, + "content": "U [ - 0 . 0 8 , 0 . 0 8 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 477, + 398, + 489 + ], + "score": 1.0, + "content": "at the initial time-step and then kept fixed.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24.5, + "bbox_fs": [ + 104, + 419, + 507, + 489 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 501, + 185, + 513 + ], + "lines": [ + { + "bbox": [ + 105, + 500, + 187, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 187, + 515 + ], + "score": 1.0, + "content": "D.3 ROBUST RL", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 522, + 505, + 600 + ], + "lines": [ + { + "bbox": [ + 106, + 521, + 506, + 536 + ], + "spans": [ + { + "bbox": [ + 106, + 521, + 506, + 536 + ], + "score": 1.0, + "content": "{Hopper,Walker,Cheetah}-Robust. We directly adopt the environments used in MRPO (Jiang", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 532, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 505, + 546 + ], + "score": 1.0, + "content": "et al., 2021). In each environment, the hidden state is the dynamics parameters including the density", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 544, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 506, + 556 + ], + "score": 1.0, + "content": "and friction coefficients of the simulated robot in roboschool, adapted from the SunBlaze (Packer", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 555, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 505, + 567 + ], + "score": 1.0, + "content": "et al., 2018). The exact ranges of the hidden states in each environment can be found in Table 1 of", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 567, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 567, + 505, + 578 + ], + "score": 1.0, + "content": "MRPO (Jiang et al., 2021). We evaluate the algorithms with 100 tasks in each environment, and use", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 577, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 381, + 590 + ], + "score": 1.0, + "content": "the average of them as average returns, and the average of the worst", + "type": "text" + }, + { + "bbox": [ + 382, + 577, + 402, + 588 + ], + "score": 0.88, + "content": "1 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 577, + 506, + 590 + ], + "score": 1.0, + "content": "of them as worst returns,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 587, + 205, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 205, + 601 + ], + "score": 1.0, + "content": "following MRPO paper.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 521, + 506, + 601 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 613, + 239, + 624 + ], + "lines": [ + { + "bbox": [ + 106, + 613, + 240, + 626 + ], + "spans": [ + { + "bbox": [ + 106, + 613, + 240, + 626 + ], + "score": 1.0, + "content": "D.4 GENERALIZATION IN RL", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 633, + 505, + 700 + ], + "lines": [ + { + "bbox": [ + 107, + 633, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 107, + 633, + 506, + 645 + ], + "score": 1.0, + "content": "{Hopper,Cheetah}-Generalize. We directly adopt the environments used in SunBlaze (Packer", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "score": 1.0, + "content": "et al., 2018). In each environment, the hidden state is the dynamics parameters including the den-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "score": 1.0, + "content": "sity, friction coefficients, and the power of the simulated robot in roboschool. The exact ranges of", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "both interpolation and extrapolation in the hidden state distribution for each environment can be", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "found in Table 1 of SunBlaze (Packer et al., 2018). 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DPFRL (Ma et al.,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 160, + 505, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 505, + 173 + ], + "score": 1.0, + "content": "2020b) applies PF-GRU to POMDP tasks, using the mean of particles and MGF features of parti-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 171, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 505, + 183 + ], + "score": 1.0, + "content": "cles as the belief state. 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Our implementation", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 265, + 411, + 280 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 411, + 280 + ], + "score": 1.0, + "content": "still outperforms VariBAD-BOReL, supporting our claim in the main paper.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 243, + 506, + 280 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 283, + 505, + 328 + ], + "lines": [ + { + "bbox": [ + 105, + 282, + 505, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 505, + 295 + ], + "score": 1.0, + "content": "Finally, in the newly added and more challenging Ant-Dir, we found our recurrent model-free RL", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 295, + 504, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 504, + 306 + ], + "score": 1.0, + "content": "can even greatly surpass Oracle (trained with same environment and gradient steps), while VariBAD-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 305, + 505, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 305, + 505, + 318 + ], + "score": 1.0, + "content": "BOReL somehow has extremely low performance (worse than Random) and suffers from numerical", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 316, + 203, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 203, + 329 + ], + "score": 1.0, + "content": "issue to terminate early.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5, + "bbox_fs": [ + 105, + 282, + 505, + 329 + ] + }, + { + "type": "image", + "bbox": [ + 113, + 340, + 505, + 577 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 113, + 340, + 505, + 577 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 113, + 340, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 113, + 340, + 505, + 577 + ], + "score": 0.974, + "type": "image", + "image_path": "2bee8e27956779c2058d9477678f8a6064ff9f8f6cd810dfb28714dc2934413c.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 113, + 340, + 505, + 419.0 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 113, + 419.0, + 505, + 498.0 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 113, + 498.0, + 505, + 577.0 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 581, + 505, + 626 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 580, + 505, + 592 + ], + "spans": [ + { + "bbox": [ + 106, + 580, + 505, + 592 + ], + "score": 1.0, + "content": "Figure 24: Additional learning curves on meta RL environments. We show the results from the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 591, + 505, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 505, + 604 + ], + "score": 1.0, + "content": "single variant of our implementation on recurrent model-free RL (sac-gru-oar-400), and the spe-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 603, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 106, + 603, + 505, + 614 + ], + "score": 1.0, + "content": "cialized meta RL method off-policy VariBAD (Dorfman et al., 2020) (their official implementation", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 613, + 339, + 626 + ], + "spans": [ + { + "bbox": [ + 106, + 613, + 339, + 626 + ], + "score": 1.0, + "content": "VariBAD-BOReL and our re-implementation VariBAD).", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5 + } + ], + "index": 21.75 + } + ] + } + ], + "_backend": "pipeline", + "_version_name": "2.2.2" +} \ No newline at end of file diff --git a/parse/dev/KLaDXLAzzFT/KLaDXLAzzFT_content_list.json b/parse/dev/KLaDXLAzzFT/KLaDXLAzzFT_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..92e76e4d894d655fe7f656bd2c91650dd792f6aa --- /dev/null +++ b/parse/dev/KLaDXLAzzFT/KLaDXLAzzFT_content_list.json @@ -0,0 +1,6235 @@ +[ + { + "type": "text", + "text": "NEAR-OPTIMAL OFFLINE REINFORCEMENT LEARNING WITH LINEAR REPRESENTATION: LEVERAGING VARIANCE INFORMATION WITH PESSIMISM ", + "text_level": 1, + "bbox": [ + 173, + 98, + 826, + 171 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Ming Yin∗ Yaqi Duan† ∗Department of Computer Science University of California, Santa Barbara {ming yin,yuxiangw}@cs.ucsb.edu ", + "bbox": [ + 184, + 195, + 478, + 252 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Mengdi Wang† Yu-Xiang Wang∗ †Department of Electrical and Computer Engineering Princeton University {yaqid,mengdiw}@princeton.edu ", + "bbox": [ + 477, + 194, + 844, + 252 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 289, + 544, + 304 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Offline reinforcement learning, which seeks to utilize offline/historical data to optimize sequential decision-making strategies, has gained surging prominence in recent studies. Due to the advantage that appropriate function approximators can help mitigate the sample complexity burden in modern reinforcement learning problems, existing endeavors usually enforce powerful function representation models (e.g. neural networks) to learn the optimal policies. However, a precise understanding of the statistical limits with function representations, remains elusive, even when such a representation is linear. ", + "bbox": [ + 233, + 319, + 766, + 429 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Towards this goal, we study the statistical limits of offline reinforcement learning with linear model representations. To derive the tight offline learning bound, we design the variance-aware pessimistic value iteration (VAPVI), which adopts the conditional variance information of the value function for time-inhomogeneous episodic linear Markov decision processes (MDPs). VAPVI leverages estimated variances of the value functions to reweight the Bellman residuals in the leastsquare pessimistic value iteration and provides improved offline learning bounds over the best-known existing results (whereas the Bellman residuals are equally weighted by design). More importantly, our learning bounds are expressed in terms of system quantities, which provide natural instance-dependent characterizations that previous results are short of. We hope our results draw a clearer picture of what offline learning should look like when linear representations are provided. ", + "bbox": [ + 233, + 431, + 766, + 598 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 622, + 336, + 637 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Offline reinforcement learning (offline RL or batch RL Lange et al. (2012); Levine et al. (2020)) is the framework for learning a reward-maximizing policy in an unknown environment (Markov Decision Process or MDP)1 using the logged data coming from some behavior policy $\\mu$ . Function approximations, on the other hand, are well-known for generalization in the standard supervised learning. Offline RL with function representation/approximation, as a result, provides generalization across large state-action spaces for the challenging sequential decision-making problems when no iteration is allowed (as opposed to online learning). This paradigm is crucial to the success of modern RL problems as many deep RL algorithms find their prototypes in the literature of offline RL. For example, Xie and Jiang (2020) provides a view that Fitted $Q$ -Iteration (Gordon, 1999; Ernst et al., 2005) can be considered as the theoretical prototype of the deep $Q$ -networks algorithm (DQN) Mnih et al. (2015) with neural networks being the function representors. On the empirical side, there are a huge body of deep RL-based algorithms (Mnih et al., 2015; Silver et al., 2017; Fujimoto et al., 2019; Kumar et al., 2019; Wu et al., 2019; Kidambi et al., 2020; Yu et al., 2020; Kumar et al., 2020; Janner et al., 2021; Chen et al., 2021a; Kostrikov et al., 2022) that utilize function approximations to achieve respective successes in the offline regime. However, it is also realized that practical function approximation schemes can be quite sample inefficient (e.g. millions of samples are needed for deep $Q$ -network to solve certain Atari games Mnih et al. (2015)). ", + "bbox": [ + 174, + 654, + 825, + 888 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "To understand this phenomenon, there are numerous studies consider how to achieve sample efficiency with function approximation from the theoretical side, as researchers find sample efficient algorithms are possible with particular model representations, in either online RL (e.g. Yang and Wang (2019; 2020); Modi et al. (2020); Jin et al. (2020); Ayoub et al. (2020); Jiang et al. (2017); Du et al. (2019); Sun et al. (2019); Zanette et al. (2020); Zhou et al. (2021a); Jin et al. (2021a); Du et al. (2021)) or offline RL (e.g. Munos (2003); Chen and Jiang (2019); Xie and Jiang (2020); Jin et al. (2021b); Xie et al. (2021a); Min et al. (2021); Duan et al. (2021); Nguyen-Tang et al. (2021); Zanette et al. (2021)). ", + "bbox": [ + 174, + 103, + 825, + 202 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Among them, the linear MDP model (Yang and Wang, 2020; Jin et al., 2020), where the transition is represented as a linear combinations of the given $d$ -dimensional feature, is (arguably) the most studied setting in function approximation and there are plenty of extensions based upon it (e.g. generalized linear model (Wang et al., 2021b), reward-free RL (Wang et al., 2020), gap-dependent analysis (He et al., 2021) or generative adversarial learning (Liu et al., 2021)). Given its prosperity, however, there are still unknowns for understanding function representations in RL, especially in the offline case. ", + "bbox": [ + 174, + 208, + 825, + 291 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "• While there are surging researches in showing provable sample efficiency (polynomial sample complexity is possible) under a variety of function approximation schemes, how to improve the sample efficiency for a given class of function representations remains understudied. For instance, given a neural network approximation class, an algorithm that learns the optimal policy with complexity ${ \\cal O } ( H ^ { 1 0 } )$ is far worse than the one that can learn in $O ( H ^ { 3 } )$ sample complexity, despite that both algorithms are considered sample efficient. Therefore, how to achieve the optimal/tight sample complexity when function approximation is provided is a valuable question to consider. On the other hand, it is known that tight sample complexity, due to the limit of the existing statistical analysis tools, can be very tough to establish when function representation has a very complicated form. However, does this mean tight analysis is not hopeful even when the representation is linear? ", + "bbox": [ + 217, + 304, + 825, + 455 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "• Second, in the existing analysis of offline RL (with function approximation or simply the tabular MDPs), the learning bounds depend either explicitly on the data-coverage quantities (e.g. uniform concentrability coefficients Chen and Jiang (2019); Xie and Jiang (2020), uniform visitation measure Yin et al. (2021); Yin and Wang (2021a) and single concentrability Rashidinejad et al. (2021); Xie et al. (2021b)) or the horizon length $H$ (Jin et al., 2021b; Uehara and Sun, 2021). While those results are valuable as they do not depend on the structure of the particular problem (therefore, remain valid even for pathological MDPs), in practice, the empirical performances of offline reinforcement learning are often far better than those non-adaptive bounds would indicate. Can the learning bounds reflect the nature of individual MDP instances when the MDP model has a certain function representation? ", + "bbox": [ + 218, + 460, + 825, + 598 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In this work, we think about offline RL from the above two aspects. In particular, we consider the fundamental linear model representations and ask the following question of interest: ", + "bbox": [ + 176, + 611, + 820, + 638 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Can we achieve the statistical limits for offline RL when models have linear representations? ", + "bbox": [ + 186, + 650, + 808, + 665 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "1.1 RELATED WORKS ", + "text_level": 1, + "bbox": [ + 176, + 683, + 334, + 695 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Offline RL with general function representations. The finite sample analysis of offline RL with function approximation is initially conducted by Fitted $Q$ -Iteration (FQI) type algorithms and can be dated back to (Munos, 2003; Szepesvari and Munos ´ , 2005; Antos et al., 2008a;b). Later, Chen and Jiang (2019); Le et al. (2019); Xie and Jiang (2020) follow this line of research and derive the improved learning results. However, owing to the aim for tackling general function approximation, those learning bounds are expressed in terms of the stringent concentrability coefficients (therefore, are less adaptive to individual instances) and are usually only information-theoretical, due to the computational intractability of the optimization procedure over the general function classes. Other works impose weaker assumptions (e.g. partial coverage (Liu et al., 2020; Kidambi et al., 2020; Uehara and Sun, 2021)), and their finite sample analysis are generally suboptimal in terms of $H$ or the effective horizon $( 1 - \\gamma ) ^ { - 1 }$ . ", + "bbox": [ + 174, + 708, + 825, + 861 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Offline RL with tabular models. For tabular MDPs, tight learning bounds can be achieved under several data-coverage assumptions. For the class of problems with uniform data-visitation measure $d _ { m }$ , the near-optimal sample complexity bound has the rate $O ( H ^ { 3 } / d _ { m } \\epsilon ^ { 2 } )$ for time-inhomogeneous MDPs (Yin et al., 2021) and $O ( H ^ { 2 } / d _ { m } \\dot { \\epsilon } )$ for time-homogeneous MDPs (Yin and Wang, 2021a; Ren et al., 2021). Under the single concentrability assumption, the tight rate ${ \\cal O } ( H ^ { 3 } S C ^ { \\star } / \\epsilon ^ { 2 } )$ is obtained by Xie et al. (2021b). In particular, the recent study Yin and Wang (2021b) introduces the intrinsic offline learning bound that is not only instance-dependent but also subsumes previous optimal results. ", + "bbox": [ + 174, + 867, + 823, + 924 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 823, + 145 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Offline RL with linear model representations. Recently, there is more focus on studying the provable efficient offline RL under the linear model representations. Jin et al. (2021b) first shows offline RL with linear MDP is provably efficient by the pessimistic value iteration. Their analysis deviates from their lower bound by a factor of $d \\cdot H$ (check their Theorem 4.4 and 4.6). Later, Xie et al. (2021a) considers function approximation under the Bellman-consistent assumptions, and, when realized to linear MDP setting, improves the sample complexity guarantee of Jin et al. (2021b) by an order $O ( d )$ (Theorem 3.2).2 However, their improvement only holds for finite action space (due to the dependence $\\log \\left| { \\mathcal { A } } \\right| )$ and by the direct reduction (from Theorem 3.1) their result does not imply a computationally tractable algorithm with the same guarantee. Concurrently, Zanette et al. (2021) considers the Linear Bellman Complete model and designs the actor-critic style algorithm that achieves tight result under the assumption that the value function is bounded by 1. While their algorithm is efficient (which is based on solving a sequence of second-order cone programs), the resulting learning bound requires the action space to be finite due to the mirror descent updates in the Actor procedure (Agarwal et al., 2021). Besides, assuming the value function to be less than 1 simplifies the challenges in dealing with horizon $H$ since when rescaling their result to $[ 0 , H ]$ , there is a $H$ factor blow-up, which makes no horizon improvement comparing to Jin et al. (2021b). As a result, none of the existing algorithms can achieve the statistical limit for the well-structured linear MDP model with the general (infinite or continuous) state-action spaces. On the other hand, Wang et al. (2021a); Zanette (2021) study the statistical hardness of offline RL with linear representations by proving the exponential lower bounds. Recently, Foster et al. (2021) shows realizability and concentrability are not sufficient for offline learning when state space is arbitrary large. ", + "bbox": [ + 173, + 152, + 825, + 444 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Variance-aware studies. Talebi and Maillard (2018) first incorporates the variance structure in online tabular MDPs and Zanette and Brunskill (2019) tightens the result. For linear mixture MDPs, Zhou et al. (2021a) first uses variance structure to achieve near-optimal result and the Weighted OFUL incorporates the variance structure explicitly in the regret bound. Recently, Variance-awareness is also considered in Zhang et al. (2021) for horizon-free setting and for OPE problem (Min et al., 2021). In particular, We point out that Min et al. (2021) is the first work that uses variance reweighting for policy evaluation in offline RL, which inspires our study for policy optimization problem. The guarantee of Min et al. (2021) strictly improves over Duan et al. (2020) for OPE problem. ", + "bbox": [ + 174, + 450, + 825, + 563 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "1.2 OUR CONTRIBUTION ", + "text_level": 1, + "bbox": [ + 176, + 585, + 357, + 599 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In this work, we study offline RL for time-inhomogeneous episodic linear Markov decision processes. Linear MDPs serve as one critical step towards understanding function approximation in RL since: 1. unlike general function representation, linear MDP representation has the well-structured form by the given feature representors, which makes delicate statistical analysis hopeful; 2. unlike tabular representation, which only works for finite models, linear MDP provides generalization as it adapts to infinite or continuous state-action spaces. Especially, we design the variance-aware pessimistic value iteration (VAPVI, Algorithm 1) which incorporates the conditional variance information of the value function and, by the variance structure, Theorem 3.2 is able to improve over the aforementioned stateof-the-art guarantees. In addition, we further improve the state-action guarantee by designing an even tighter bonus (4). VAPVI-Improved (Theorem 3.3) is near-minimax optimal as indicated by our lower bound (Theorem 3.5). Importantly, the resulting learning bounds from VAPVI/VAPVI-Improved are able to characterize the adaptive nature of individual instances and yield different convergence rates for different problems. Algorithmically, our algorithm builds upon the nice Min et al. (2021) with pessimism as we use the estimated variances to reweight the Bellman residual learning objective so that the (training) samples with high uncertainty get less attention (Section 3). This is the key to obtaining instance-adaptive guarantees. ", + "bbox": [ + 174, + 614, + 826, + 835 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2 PRELIMINARIES ", + "text_level": 1, + "bbox": [ + 176, + 863, + 338, + 878 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.1 PROBLEM SETTINGS ", + "text_level": 1, + "bbox": [ + 176, + 103, + 354, + 117 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Episodic time-inhomogeneous linear Markov decision process. A finite-horizon Markov Decision Process (MDP) is denoted as $M = \\left( S , A , P , r , H , d _ { 1 } \\right)$ (Sutton and Barto, 2018), where $s$ is the arbitrary state space and $\\mathcal { A }$ is the arbitrary action space which can be infinite or even continuous. A time-inhomogeneous transition kernel $\\begin{array} { r } { \\dot { P } _ { h } : S \\times \\bar { \\mathcal { A } } \\mapsto \\Delta ^ { \\mathcal { S } } } \\end{array}$ ( $\\Delta ^ { s }$ represents a probability simplex) maps each state action $\\left( s _ { h } , a _ { h } \\right)$ to a probability distribution $P _ { h } ( \\cdot | s _ { h } , a _ { h } )$ and $P _ { h }$ can be different across time. In addition, $r : \\mathcal { S } \\times A \\mapsto \\mathbb { R }$ is the mean reward function satisfying $0 \\leq r \\leq 1$ . $d _ { 1 }$ is the initial state distribution. $H$ is the horizon. A policy $\\pi = ( \\pi _ { 1 } , \\ldots , \\pi _ { H } ) $ assigns each state $s _ { h } \\in S$ a probability distribution over actions according to the map $s _ { h } \\mapsto \\pi _ { h } \\bigl ( \\cdot | s _ { h } \\bigr ) \\bar { \\forall } h \\in [ H ]$ and induces a random trajectory $s _ { 1 } , a _ { 1 } , r _ { 1 } , \\ldots , s _ { H } , a _ { H } , r _ { H } , s _ { H + 1 }$ with $s _ { 1 } \\sim d _ { 1 } , a _ { h } \\sim \\pi ( \\cdot | s _ { h } ) , s _ { h + 1 } \\sim$ $P _ { h } ( \\cdot | s _ { h } , a _ { h } ) , \\forall h \\in [ \\bar { H } ]$ . In particular, we adopts the linear MDP protocol from Jin et al. (2020; 2021b), meaning that the transition kernel and the mean reward function admit linear structures in the feature map. ", + "bbox": [ + 173, + 128, + 826, + 296 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Definition 2.1 (Linear MDPs). 3 An episodic MDP $( S , { \\mathcal { A } } , H , P , r )$ is called a linear MDP with a known (unsigned) feature map $\\phi : \\mathcal { S } \\times \\mathcal { A } \\mathbb { R } ^ { d }$ if there exist $d$ unknown (unsigned) measures $\\nu _ { h } = ( \\nu _ { h } ^ { ( 1 ) } , \\dots , \\nu _ { h } ^ { ( d ) } )$ over $s$ and an unknown vector $\\theta _ { h } \\in \\mathbb { R } ^ { d }$ such that ", + "bbox": [ + 174, + 299, + 825, + 347 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "$P _ { h } \\left( \\boldsymbol { s } ^ { \\prime } \\mid \\boldsymbol { s } , \\boldsymbol { a } \\right) = \\left. \\phi ( \\boldsymbol { s } , \\boldsymbol { a } ) , \\nu _ { h } \\left( \\boldsymbol { s } ^ { \\prime } \\right) \\right. , \\quad r _ { h } \\left( \\boldsymbol { s } , \\boldsymbol { a } \\right) = \\left. \\phi ( \\boldsymbol { x } , \\boldsymbol { a } ) , \\theta _ { h } \\right. , \\quad \\forall \\boldsymbol { s } ^ { \\prime } , \\boldsymbol { s } \\in \\mathcal { S } , \\boldsymbol { a } \\in \\mathcal { A } , \\ : h \\in [ H ] .$ . where $\\| \\nu _ { h } ( S ) \\| _ { 2 } \\leq \\sqrt { d }$ and $\\operatorname* { m a x } ( \\left\\| \\phi ( s , a ) \\right\\| _ { 2 } , \\left\\| \\theta _ { h } \\right\\| _ { 2 } ) \\leq 1$ for all $h \\in [ H ]$ and $\\forall s , a \\in S \\times A$ . $\\begin{array} { r } { \\| \\mu _ { h } ( S ) \\| = \\int _ { S } \\| \\mu _ { h } ( s ) \\| d s } \\end{array}$ . ", + "bbox": [ + 174, + 352, + 826, + 409 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "$V$ $Q _ { h } ^ { \\pi } ( \\cdot , \\cdot ) \\in \\mathbb { R } ^ { S \\times A }$ -values and $Q$ -values. For any policy fined as: . The perf $\\begin{array} { r } { V _ { h } ^ { \\pi } ( s ) = \\mathbb { E } _ { \\pi } [ \\sum _ { t = h } ^ { H } r _ { t } | s _ { h } = s ] , \\quad Q _ { h } ^ { \\pi } ( s , a ) = \\mathbb { E } _ { \\pi } [ \\sum _ { t = h } ^ { H } r _ { \\frac { t } { s } } | s _ { h } , a _ { h } = \\frac { 1 } { s } ] } \\end{array}$ $\\pi$ , the $V$ -value functions $V _ { h } ^ { \\pi } ( \\cdot ) \\in \\mathbb { R } ^ { S }$ and Q-value functions $s , a ] , \\forall s , a , h \\in S , { \\mathcal { A } } , [ H ]$ $\\begin{array} { r } { \\boldsymbol { v } ^ { \\pi } : = \\mathbb { E } _ { d _ { 1 } } \\left[ V _ { 1 } ^ { \\pi } \\right] = \\mathbb { E } _ { \\pi , d _ { 1 } } \\left[ \\sum _ { t = 1 } ^ { H } \\boldsymbol { r } _ { t } \\right| } \\end{array}$ The Bellman (optimality) equations follow $\\iota \\in [ H ] \\colon Q _ { h } ^ { \\pi } = r _ { h } + P _ { h } V _ { h + 1 } ^ { \\pi } , ~ V _ { h } ^ { \\pi } = \\mathbb { E } _ { a \\sim \\pi _ { h } } [ Q _ { h } ^ { \\bar { \\pi } } ] , ~ Q _ { h } ^ { \\star } \\stackrel { = } { = }$ $r _ { h } + P _ { h } V _ { h + 1 } ^ { \\star }$ , $V _ { h } ^ { \\star } = \\operatorname* { m a x } _ { a } Q _ { h } ^ { \\star } ( \\cdot , a )$ (where $Q _ { h } , V _ { h } , P _ { h }$ are vectors). By Definition 2.1, the $Q$ -values also admit linear structures, i.e. $Q _ { h } ^ { \\pi } = \\langle \\phi , w _ { h } ^ { \\pi } \\rangle$ for some $w _ { h } ^ { \\pi } \\in \\mathbb { R } ^ { d }$ (Lemma H.9). Lastly, for a policy $\\pi$ , we denote the induced occupancy measure over the state-action space at any time $h \\in [ H ]$ to be: for any $E \\subseteq S \\times A$ , $d _ { h } ^ { \\pi } ( E ) : = \\mathbb { E } [ ( s _ { h } , a _ { h } ) \\in E | s _ { 1 } \\sim d _ { 1 } , a _ { i } \\sim \\pi ( \\cdot | s _ { i } ) , s _ { i } \\sim P _ { i - 1 } ( \\cdot | s _ { i - 1 } , \\bar { a _ { i - 1 } } ) , 1 \\le h | s _ { i - 1 } \\sim \\pi ( \\cdot | s _ { i - 1 } , \\bar { a _ { i - 1 } } ) ]$ $i \\leq h \\}$ and $\\begin{array} { r } { \\mathbb { E } _ { \\pi , h } [ f ( s , a ) ] : = \\int _ { \\mathcal { S } \\times \\mathcal { A } } f ( s , a ) d _ { h } ^ { \\pi } ( s , a ) d s d a } \\end{array}$ . Here for notation simplicity we abuse $d _ { h } ^ { \\pi } ( \\cdot )$ to denote either probability measure or density function. ", + "bbox": [ + 173, + 417, + 826, + 570 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Offline learning setting. Offlprovided with the historical data learn the policy rolled out from s $\\pi$ that maximizes me behavior polic $v ^ { \\pi }$ ,. $\\mathcal { D } = \\{ ( s _ { h } ^ { \\tau } , \\bar { a } _ { h } ^ { \\tau } , r _ { h } ^ { \\tau } , s _ { h + 1 } ^ { \\tau } ) \\} _ { \\tau \\in [ K ] } ^ { h \\in [ H ] }$ $\\mu$ The offline nature requires we cannot change $\\mu$ and in particular we do not know the data generating distribution of $\\mu$ . To sum up, the agent seeks to find a policy $\\pi _ { \\mathrm { a l g } }$ such that $v ^ { \\star } - v ^ { \\pi _ { \\mathrm { a l g } } } \\leq \\epsilon$ for the given batch data $\\mathcal { D }$ and a given targeted accuracy $\\epsilon > 0$ . ", + "bbox": [ + 173, + 577, + 826, + 651 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "2.2 ASSUMPTIONS ", + "text_level": 1, + "bbox": [ + 174, + 667, + 316, + 681 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "It is known that learning a near-optimal policy from the offline data $\\mathcal { D }$ cannot be sample efficient without certain data-coverage assumptions (Wang et al., 2021a; Yin and Wang, 2021b). To begin with, we define the population covariance matrix under the behavior policy $\\mu$ for all $h \\in [ H ]$ : ", + "bbox": [ + 174, + 693, + 825, + 734 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/a392cf2fac77b1af545c26a2481b647c07aaca80dc834e4153aada685a1f50ae.jpg", + "text": "$$\n\\Sigma _ { h } ^ { p } : = \\mathbb { E } _ { \\mu , h } \\left[ \\phi ( s , a ) \\phi ( s , a ) ^ { \\top } \\right] ,\n$$", + "text_format": "latex", + "bbox": [ + 393, + 741, + 601, + 761 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "since $\\Sigma _ { h } ^ { p }$ measure the coverage of state-action space for data $\\mathcal { D }$ , we make the following assumption. ", + "bbox": [ + 181, + 765, + 823, + 781 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Assumption 2.2 (Feature Coverage). The data distributions $\\mu$ satisfy the minimum eigenvalue condition: $\\forall h \\in [ H ]$ , $\\kappa _ { h } : = \\lambda _ { \\operatorname* { m i n } } ( \\bar { \\Sigma } _ { h } ^ { p } ) > 0$ and denote $\\kappa = \\operatorname* { m i n } _ { h } \\kappa _ { h }$ . Note $\\kappa$ is a system-dependent (non-universal) quantity as it is upper bounded by $1 / d$ (Assumption 2 in Wang et al. (2021a)). ", + "bbox": [ + 174, + 784, + 825, + 827 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We make this assumption for the following reasons. First of all, our offline learning guarantee (Theorem 3.2) provides simultaneously comparison to all the policies, which is stronger than only competing with the optimal policy (whereas relaxed assumption suffices, e.g. $\\begin{array} { r } { \\operatorname* { s u p } _ { x \\in \\mathbb { R } ^ { d } } \\frac { \\bar { x } \\Sigma _ { \\pi ^ { \\star } } x ^ { \\top } } { x \\Sigma _ { \\mu } x ^ { \\top } } < \\infty } \\end{array}$ (Uehara and Sun, 2021)). As a consequence, the behavior distribution $\\mu$ must be able to explore each feature dimension for the result to be valid. Second, even if Assumption 2.2 does not hold, we can always restrict our algorithmic design to the effective subspan of $\\bar { \\Sigma _ { h } ^ { p } }$ , which causes the alternative notion of $\\begin{array} { r } { \\kappa : = \\operatorname* { m i n } _ { h \\in [ H ] } \\left\\{ \\kappa _ { h } : s . t . \\right. } \\end{array}$ $\\kappa _ { h } =$ smallest positive eigenvalue at time $h \\}$ (see Appendix G.1 for detailed discussions). In this scenario, learning the optimal policy cannot be guaranteed as a constant suboptimality gap needs to be suffered due to the lack of coverage and this is formed as assumption-free $R L$ in Yin and Wang (2021b). Lastly, previous works analyzing the linear MDPs impose very similar assumptions, e.g. Xie et al. (2021a) Theorem 3.2 where $\\dot { \\Sigma _ { \\mathcal { D } } } ^ { - 1 }$ exists and Min et al. (2021) for the OPE problem. ", + "bbox": [ + 173, + 837, + 825, + 887 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 229 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Next, for any function $V _ { h + 1 } ( \\cdot ) \\in [ 0 , H - h ]$ , we define the conditional variance $\\sigma _ { V _ { h + 1 } } : S \\times \\mathcal { A } \\to \\mathbb { R } _ { + }$ as $\\sigma _ { V _ { h + 1 } } ( s , a ) ^ { 2 } : = \\operatorname* { m a x } \\{ 1 , \\operatorname { V a r } _ { P _ { h } } ( V _ { h + 1 } ) ( s , a ) \\}$ .4 Based on this definition, we can define the variance-involved population covariance matrices as: $\\mathbf { \\Sigma } _ { h } ^ { p } : = \\mathbb { E } _ { \\mu , h } \\left[ \\sigma _ { V _ { h + 1 } } ( s , a ) ^ { - 2 } \\phi ( s , a ) \\phi ( s , a ) ^ { \\top } \\right]$ . In particular, when ${ \\cal V } _ { h } = { \\cal V } _ { h } ^ { \\star }$ , we use the notation $\\Lambda _ { h } ^ { \\star p }$ instead. ", + "bbox": [ + 174, + 234, + 825, + 299 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3 ALGORITHM ", + "text_level": 1, + "bbox": [ + 174, + 319, + 312, + 335 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Least square regression is usually considered as one of the “default” tools for handling problems with linear structures (e.g. LinUCB algorithm for linear Bandits) and finds its popularity in RL as well since Least-Square Value Iteration (LSVI, Jin et al. (2020)) is shown to be provably efficient for linear MDPs, due to that $V _ { h + 1 } ( s ^ { \\prime } )$ is an unbiased estimator of $[ P _ { h } V _ { h + 1 } ] ( s , a )$ . Concretely, it solves the ridge regression problems at each time steps (with $\\lambda > 0$ being the regularization parameter): ", + "bbox": [ + 173, + 351, + 825, + 416 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/d9ccac91acff31ad29c316c70376d5b706363272ac3b901cc9e157b0a337bde8.jpg", + "text": "$$\n\\widehat { w } _ { h } : = \\underset { w \\in \\mathbb { R } ^ { d } } { \\mathrm { a r g m i n } } \\ \\lambda \\| w \\| _ { 2 } ^ { 2 } + \\sum _ { k = 1 } ^ { K } \\left[ \\langle \\phi ( s _ { h } ^ { k } , a _ { h } ^ { k } ) , w \\rangle - r _ { h } ^ { k } - V _ { h + 1 } \\big ( s _ { h + 1 } ^ { \\prime k } \\big ) \\right] ^ { 2 }\n$$", + "text_format": "latex", + "bbox": [ + 290, + 424, + 705, + 464 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "and has the closed-form solution $\\begin{array} { r } { \\widehat { w } _ { h } \\ = \\ \\Sigma _ { h } ^ { - 1 } \\sum _ { k = 1 } ^ { K } \\phi ( s _ { h } ^ { k } , a _ { h } ^ { k } ) [ r _ { k , h } + V _ { h + 1 } ( s _ { h } ^ { \\prime k } ) ] } \\end{array}$ with $\\Sigma _ { h } ^ { - 1 } ~ =$ $\\begin{array} { r } { \\sum _ { k = 1 } ^ { K } \\phi ( s _ { h } ^ { k } , a _ { h } ^ { k } ) \\phi ( s _ { h } ^ { k } , a _ { h } ^ { k } ) ^ { \\top } + \\lambda I } \\end{array}$ b. In offline RL, this has also been leveraged in pessimistic value iteration (Jin et al., 2021b) and fitted $Q$ -evaluation (Duan et al., 2020). Nevertheless, LSVI could only yield suboptimal guarantees, as illustrated by the following example. ", + "bbox": [ + 173, + 473, + 825, + 536 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Example 3.1. Instantiate PEVI (Theorem 4.4 in Jin et al. (2021b)) with $\\phi ( s , a ) = \\mathbf { 1 } _ { s , a }$ (i.e. tabular $M D P s ) ^ { 5 }$ , by direct calculation the learning bound has the form $\\begin{array} { r l } & { O ( d H \\cdot \\sum _ { \\boldsymbol { h } , \\boldsymbol { s } , \\boldsymbol { a } } { d _ { \\boldsymbol { h } } ^ { \\pi } } ^ { \\star } ( \\boldsymbol { s } , \\boldsymbol { a } ) \\sqrt { \\frac { 1 } { K \\cdot d _ { \\boldsymbol { h } } ^ { \\mu } ( \\boldsymbol { s } , \\boldsymbol { a } ) } } ) } \\end{array}$ q 1K·dµh(s,a) ) and the optimal result (Yin and Wang (2021b) Theorem 4.1) gives $\\begin{array} { r l } & { O ( \\sum _ { h , s , a } { d _ { h } ^ { \\pi ^ { \\star } } ( s , a ) \\sqrt { \\frac { \\operatorname { V a r } _ { P _ { s , a } } ( r + V _ { h + 1 } ^ { \\star } ) } { K \\cdot d _ { h } ^ { \\mu } ( s , a ) } } } ) } \\end{array}$ The former has the horizon dependence $H ^ { 2 }$ and the latter is $H ^ { 3 / 2 }$ by law of total variance. ", + "bbox": [ + 173, + 540, + 826, + 623 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Motivation. By comparing the above two expressions, it can be seen that PEVI cannot get rid of the explicit $H$ factor due to missing the variance information $( w . r . t ~ V ^ { \\star } )$ . If we go deeper, one could find that it might not be all that ideal to put equal weights on all the training samples in the least square objective (2), since, unlike linear regression where the randomness coming from one source distribution, we are regressing over a sequence of distributions in RL (i.e. each $s _ { h } , a _ { h }$ corresponds to a different distribution $P ( \\cdot | s _ { h } , a _ { h } )$ and there are possibly infinite many of them). Therefore, conceptually, the sample piece $( s _ { h } , a _ { h } , s _ { h + 1 } )$ that has higher variance distribution $P ( \\cdot | s _ { h } , a _ { h } )$ tends to be less “reliable” than the one $( s _ { h } ^ { \\prime } , a _ { h } ^ { \\prime } , s _ { h + 1 } ^ { \\prime } )$ with lower variance (hence should not have equal weight in (2)). This suggests reweighting scheme might help improve the learning guarantee and reweighting over the variance of the value function stands as a natural choice. ", + "bbox": [ + 173, + 635, + 826, + 775 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.1 VARIANCE-AWARE PESSIMISTIC VALUE ITERATION ", + "text_level": 1, + "bbox": [ + 173, + 792, + 573, + 808 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Now we explain our framework that incorporates the variance information. Our design is motivated by previous Zhou et al. (2021a) (for online learning) and MinBy the offline nature, we can use the independent episodic data $\\mathcal { D } ^ { \\prime } = \\{ ( \\bar { s } _ { h } ^ { \\tau } , \\bar { a } _ { h } ^ { \\tau } , \\bar { r } _ { h } ^ { \\tau } , \\bar { s } _ { h } ^ { \\tau \\prime } ) \\} _ { \\tau \\in [ K ] } ^ { h \\in [ H ] }$ ation).(from $\\mu _ { . }$ ) to estimate the conditional variance of any -values $V _ { h + 1 }$ via the definition $[ \\mathrm { V a r } _ { h } V _ { h + 1 } ] ( s , a ) =$ ", + "bbox": [ + 174, + 819, + 825, + 881 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "$[ P _ { h } ( V _ { h + 1 } ) ^ { 2 } ] ( s , a ) - ( [ P _ { h } V _ { h + 1 } ] ( s , a ) ) ^ { 2 } .$ . For the second order moment, by Definition 2.1, it holds ", + "bbox": [ + 169, + 102, + 810, + 119 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/eb85e05a81a380bfe0f0495cbc25786cae5ad599c5a5a82768ed53bae10cd749.jpg", + "text": "$$\n\\left[ P _ { h } V _ { h + 1 } ^ { 2 } \\right] ( s , a ) = \\int _ { S } V _ { h + 1 } ^ { 2 } \\left( s ^ { \\prime } \\right) \\mathrm { d } { \\cal P } _ { h } \\left( s ^ { \\prime } \\mid s , a \\right) = \\phi ( s , a ) ^ { \\top } \\int _ { S } V _ { h + 1 } ^ { 2 } \\left( s ^ { \\prime } \\right) \\mathrm { d } \\nu _ { h } \\left( s ^ { \\prime } \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 243, + 121, + 754, + 151 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Denote $\\begin{array} { r } { \\beta _ { h } : = \\int _ { S } V _ { h + 1 } ^ { 2 } \\left( s ^ { \\prime } \\right) \\mathrm { d } \\nu _ { h } \\left( s ^ { \\prime } \\right) } \\end{array}$ , then $P _ { h } V _ { h + 1 } ^ { 2 } = \\left. \\phi , \\beta _ { h } \\right.$ and we can estimator it via: ", + "bbox": [ + 174, + 152, + 759, + 171 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/ce73e95e1738e745e5014aff6a07fdb08e0fb63ba1af129801736b80d1b75d2e.jpg", + "text": "$$\n\\bar { \\beta } _ { h } = \\underset { \\beta \\in \\mathbb { R } ^ { d } } { \\mathrm { a r g m i n } } \\sum _ { k = 1 } ^ { K } \\left[ \\left. \\phi ( \\bar { s } _ { h } ^ { k } , \\bar { a } _ { h } ^ { k } ) , \\beta \\right. - V _ { h + 1 } ^ { 2 } \\left( \\bar { s } _ { h + 1 } ^ { k } \\right) \\right] ^ { 2 } + \\lambda \\| \\beta \\| _ { 2 } ^ { 2 } = \\bar { \\Sigma } _ { h } ^ { - 1 } \\sum _ { k = 1 } ^ { K } \\phi ( \\bar { s } _ { h } ^ { k } , \\bar { a } _ { h } ^ { k } ) V _ { h + 1 } ^ { 2 } \\left( \\bar { s } _ { h + 1 } ^ { k } \\right)\n$$", + "text_format": "latex", + "bbox": [ + 197, + 172, + 799, + 213 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "and, similarly, the first order moment $P _ { h } V _ { h + 1 } : = \\left. \\phi , \\theta _ { h } \\right.$ can be estimated via: ", + "bbox": [ + 174, + 214, + 692, + 231 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/5181d298ba4d99bca1c9f238c2a8de43778265b249b79d9b927947b6bd45cf1e.jpg", + "text": "$$\n\\bar { \\theta } _ { h } = \\underset { \\theta \\in \\mathbb { R } ^ { d } } { \\operatorname { a r g m i n } } \\sum _ { k = 1 } ^ { K } \\left[ \\left. \\phi ( \\bar { s } _ { h } ^ { k } , \\bar { a } _ { h } ^ { k } ) , \\theta \\right. - V _ { h + 1 } \\left( \\bar { s } _ { h + 1 } ^ { k } \\right) \\right] ^ { 2 } + \\lambda \\| \\theta \\| _ { 2 } ^ { 2 } = \\bar { \\Sigma } _ { h } ^ { - 1 } \\sum _ { k = 1 } ^ { K } \\phi ( \\bar { s } _ { h } ^ { k } , \\bar { a } _ { h } ^ { k } ) V _ { h + 1 } \\left( \\bar { s } _ { h + 1 } ^ { k } \\right)\n$$", + "text_format": "latex", + "bbox": [ + 200, + 231, + 797, + 271 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The final estimator is defined as $\\widehat { \\sigma } _ { V _ { h } } ^ { 2 } ( \\cdot , \\cdot ) \\ : = \\ \\operatorname* { m a x } \\{ 1 , \\widehat { \\mathrm { V a r } } _ { h } V _ { h + 1 } ( \\cdot , \\cdot ) \\}$ with $\\widehat { \\mathrm { V a r } _ { h } } V _ { h + 1 } ( \\cdot , \\cdot ) \\ =$ $\\langle \\phi ( \\cdot , \\cdot ) , \\bar { \\beta } _ { h } \\rangle _ { [ 0 , ( H - h + 1 ) ^ { 2 } ] } - \\left[ \\langle \\phi ( \\cdot , \\cdot ) , \\bar { \\theta } _ { h } \\rangle _ { [ 0 , H - h + 1 ] } \\right] ^ { 2 }$ .6 In particular, when setting $V _ { h + 1 } = \\widehat { V } _ { h + 1 }$ , it recovers $\\widehat { \\sigma } _ { h }$ in Algorithm 1 line 8. Here $\\begin{array} { r } { \\bar { \\Sigma } _ { h } = \\sum _ { \\tau = 1 } ^ { K } \\phi ( \\bar { s } _ { h } ^ { \\tau } , \\bar { a } _ { h } ^ { \\tau } ) \\phi ( \\bar { s } _ { h } ^ { \\tau } , \\bar { a } _ { h } ^ { \\tau } ) ^ { \\top } + \\lambda I _ { d } . } \\end{array}$ . ", + "bbox": [ + 173, + 275, + 826, + 332 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Variance-weighted LSVI. The idea of LSVI (2) is based on approximate the Bellman updates: ${ \\mathcal { T } } _ { h } ( V ) ( s , a ) { \\stackrel { \\smile } { = } } r _ { h } ( s , a ) + ( P _ { h } V ) ( s , a )$ . With variance estimator $\\bar { \\widehat { \\sigma } } _ { h }$ at hand, we can modify (2) to bsolve the variance-weighted LSVI instead (Line 10 of Algorithm 1) ", + "bbox": [ + 174, + 335, + 825, + 377 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/49aa8bea12f8c76be121be4caf0974c4e6d9ea60f56e192cc41a1844db85c256.jpg", + "text": "$$\n\\widehat { w } _ { h } : = \\operatorname { a r g m i n } _ { w \\in \\mathbb { R } ^ { d } } \\lambda \\| w \\| _ { 2 } ^ { 2 } + \\sum _ { k = 1 } ^ { K } \\frac { \\left[ \\langle \\phi ( s _ { h } ^ { k } , a _ { h } ^ { k } ) , w \\rangle - r _ { h } ^ { k } - \\widehat { V } _ { h + 1 } ( s _ { h + 1 } ^ { \\prime k } ) \\right] ^ { 2 } } { \\widehat { \\sigma } _ { h } ^ { 2 } ( s _ { h } ^ { k } , a _ { h } ^ { k } ) } = \\widehat { \\Lambda } _ { h } ^ { - 1 } \\sum _ { k = 1 } ^ { K } \\frac { \\phi \\left( s _ { h } ^ { k } , a _ { h } ^ { k } \\right) \\cdot \\left[ r _ { h } ^ { k } + \\widehat { V } _ { h + 1 } \\left( s _ { h + 1 } ^ { k } \\right) \\right] } { \\widehat { \\sigma } ^ { 2 } ( s _ { h } ^ { k } , a _ { h } ^ { k } ) }\n$$", + "text_format": "latex", + "bbox": [ + 181, + 378, + 813, + 416 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "where $\\begin{array} { r } { \\widehat { \\Lambda } _ { h } = \\sum _ { k = 1 } ^ { K } \\phi \\big ( s _ { h } ^ { k } , a _ { h } ^ { k } \\big ) \\phi \\big ( s _ { h } ^ { k } , a _ { h } ^ { k } \\big ) ^ { \\top } \\big / \\widehat \\sigma _ { h } ^ { 2 } \\big ( s _ { h } ^ { k } , a _ { h } ^ { k } \\big ) + \\lambda I _ { d } } \\end{array}$ . The estimated Bellman update $\\widehat { \\mathcal { T } } _ { h }$ (acts on $\\widehat { V } _ { h + 1 } )$ is defined as: $( \\widehat { T } _ { h } \\widehat { V } _ { h + 1 } ) ( \\cdot , \\cdot ) = \\phi ( \\cdot , \\cdot ) ^ { \\top } \\widehat { w } _ { h }$ and the pessimism $\\Gamma _ { h }$ is assigned to update $\\widehat { Q } _ { h } \\approx \\widehat { \\mathcal { T } } _ { h } \\widehat { V } _ { h + 1 } - \\Gamma _ { h }$ , i.e. Bellman update $^ +$ Pessimism (Line 10-12 in Algorithm 1). ", + "bbox": [ + 173, + 419, + 825, + 470 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Tighter Pessimistic Design. To improve the learning guarantee, we create a tighter penalty design that includes $\\widehat { \\Lambda } _ { h } ^ { - 1 }$ rather than $\\bar { \\Sigma } _ { h } ^ { - 1 }$ and an extra higher order $O ( \\textstyle { \\frac { 1 } { K } } )$ term: ", + "bbox": [ + 173, + 474, + 823, + 507 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/3227629b1006c5cb87f22be640151763c4a09875a521327d6eab049b38a7ce55.jpg", + "text": "$$\n\\Gamma _ { h } O ( \\sqrt { d } \\cdot ( \\phi ( \\cdot , \\cdot ) ^ { \\top } \\widehat { \\Lambda } _ { h } ^ { - 1 } \\phi ( \\cdot , \\cdot ) ) ^ { 1 / 2 } ) + \\frac { 2 H ^ { 3 } \\sqrt { d } } { K }\n$$", + "text_format": "latex", + "bbox": [ + 339, + 511, + 656, + 540 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Note such a design admits no explicit factor in $H$ in the main term (as opposed to Jin et al. (2021b)) therefore is the key for achieving adaptive/problem-dependent results (as we shall discuss later). The full algorithm VAindependent parts with $\\mathcal { D } = \\{ ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } , r _ { h } ^ { \\tau } , s _ { h } ^ { \\tau \\prime } ) \\} _ { \\tau \\in [ K ] } ^ { h \\in [ H ] }$ In pand $\\mathcal { D } ^ { \\prime } = \\{ ( \\bar { s } _ { h } ^ { \\tau } , \\bar { a } _ { h } ^ { \\tau } , \\bar { r } _ { h } ^ { \\tau } , \\bar { s } _ { h } ^ { \\tau \\prime } ) \\} _ { \\tau \\in [ K ] } ^ { h \\in [ H ] }$ data into twofor different purposes (estimating variance and updating $Q$ -values). ", + "bbox": [ + 173, + 541, + 826, + 618 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3.2 MAIN RESULT ", + "text_level": 1, + "bbox": [ + 174, + 632, + 313, + 647 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We denote quantities $\\mathcal { M } _ { 1 } , \\mathcal { M } _ { 2 } , \\mathcal { M } _ { 3 } , \\mathcal { M } _ { 4 }$ as in the notation list A. Then VAPVI provides the following result. The complete proof is provided in Appendix C. ", + "bbox": [ + 174, + 659, + 825, + 688 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Theorem 3.2. Let $K$ be the number of episodes. If $K > \\operatorname* { m a x } \\{ \\mathcal { M } _ { 1 } , \\mathcal { M } _ { 2 } , \\mathcal { M } _ { 3 } , \\mathcal { M } _ { 4 } \\}$ and $\\sqrt { d } > \\xi$ , $\\begin{array} { r } { \\xi : = \\operatorname* { s u p } _ { V \\in [ 0 , H ] , \\ : s ^ { \\prime } \\sim P _ { h } ( s , a ) , \\ : h \\in [ H ] } \\left| \\frac { r _ { h } + V \\left( s ^ { \\prime } \\right) - ( \\mathcal T _ { h } V ) ( s , a ) } { \\sigma _ { V } ( s , a ) } \\right| . } \\end{array}$ Then for any $0 < \\lambda < \\kappa$ , with probability $1 - \\delta ,$ , for all policy $\\pi$ simultaneously, the output $\\widehat { \\pi }$ of Algorithm 1 satisfies ", + "bbox": [ + 173, + 691, + 826, + 752 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/c196ec7edcba08a2d1b311144741788cc3c74e1c099af5706c9436f5a54a5cd1.jpg", + "text": "$$\nv ^ { \\pi } - v ^ { \\widehat { \\pi } } \\leq \\widetilde { O } \\big ( \\sqrt { d } \\cdot \\sum _ { h = 1 } ^ { H } \\mathbb { E } _ { \\pi } \\bigg [ \\sqrt { \\phi ( \\cdot , \\cdot ) ^ { \\top } \\Lambda _ { h } ^ { - 1 } \\phi ( \\cdot , \\cdot ) } \\bigg ] \\big ) + \\frac { 2 H ^ { 4 } \\sqrt { d } } { K }\n$$", + "text_format": "latex", + "bbox": [ + 305, + 755, + 692, + 792 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "$\\begin{array} { r } { \\Lambda _ { h } = \\sum _ { k = 1 } ^ { K } \\frac { \\phi ( s _ { h } ^ { k } , a _ { h } ^ { k } ) \\cdot \\phi ( s _ { h } ^ { k } , a _ { h } ^ { k } ) ^ { \\top } } { \\sigma _ { \\hat { V } _ { h + 1 } ( s _ { h } ^ { k } , a _ { h } ^ { k } ) } ^ { 2 } } + \\lambda I _ { d } } \\end{array}$ . In particular, we have with probability $1 - \\delta$ , ", + "bbox": [ + 173, + 795, + 772, + 827 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/62410ec3d465354818426b31b1b0c371e81728db16189bf149a77c4f1f7f4d6e.jpg", + "text": "$$\n\\boldsymbol { v } ^ { \\star } - \\boldsymbol { v } ^ { \\widehat { \\pi } } \\leq \\widetilde { O } \\big ( \\sqrt { d } \\cdot \\sum _ { h = 1 } ^ { H } \\mathbb { E } _ { \\pi ^ { \\star } } \\bigg [ \\sqrt { \\phi ( \\cdot , \\cdot ) ^ { \\top } \\Lambda _ { h } ^ { \\star - 1 } \\phi ( \\cdot , \\cdot ) } \\bigg ] \\big ) + \\frac { 2 H ^ { 4 } \\sqrt { d } } { K }\n$$", + "text_format": "latex", + "bbox": [ + 299, + 829, + 699, + 867 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "$\\begin{array} { r } { \\Lambda _ { h } ^ { \\star } = \\sum _ { k = 1 } ^ { K } \\frac { \\phi ( s _ { h } ^ { k } , a _ { h } ^ { k } ) \\cdot \\phi ( s _ { h } ^ { k } , a _ { h } ^ { k } ) ^ { \\top } } { \\sigma _ { V _ { h + 1 } ^ { \\star } ( s _ { h } ^ { k } , a _ { h } ^ { k } ) } ^ { 2 } } + \\lambda I _ { d } } \\end{array}$ and $\\widetilde O$ hides universal constants and the Polylog terms. ", + "bbox": [ + 173, + 869, + 823, + 902 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Algorithm 1 Variance-Aware Pessimistic Value Iteration (VAPVI) ", + "text_level": 1, + "bbox": [ + 173, + 103, + 611, + 117 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "1: Input: Dataset $\\begin{array} { r } { \\mathcal { D } = \\{ \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } , r _ { h } ^ { \\tau } \\right) \\} _ { \\tau , h = 1 } ^ { K , H } \\mathcal { D } ^ { \\prime } = \\{ \\left( \\bar { s } _ { h } ^ { \\tau } , \\bar { a } _ { h } ^ { \\tau } , \\bar { r } _ { h } ^ { \\tau } \\right) \\} _ { \\tau , h = 1 } ^ { K , H } } \\end{array}$ . Universal constant $C$ . \n2: Initialization: Set $\\widehat { V } _ { H + 1 } ( \\cdot ) 0$ . \n3: for $h = H , H - 1 , \\ldots , 1$ do \n4: $\\diamond$ Phase1: Regular Least-square Value Iteration for conditional variances \n5: Set $\\begin{array} { r } { \\bar { \\Sigma } _ { h } \\sum _ { \\tau = 1 } ^ { K } \\phi ( \\bar { s } _ { h } ^ { \\tau } , \\bar { a } _ { h } ^ { \\tau } ) \\phi ( \\bar { s } _ { h } ^ { \\tau } , \\bar { a } _ { h } ^ { \\tau } ) ^ { \\top } + \\lambda I } \\end{array}$ \n6: Set $\\begin{array} { r } { \\bar { \\beta } _ { h } \\gets \\bar { \\Sigma } _ { h } ^ { - 1 } \\sum _ { \\tau = 1 } ^ { K } \\phi ( \\bar { s } _ { h } ^ { \\tau } , \\bar { a } _ { h } ^ { \\tau } ) \\cdot \\widehat { V } _ { h + 1 } ( \\bar { s } _ { h + 1 } ^ { \\tau } ) ^ { 2 } } \\end{array}$ \n7: Set $\\begin{array} { r } { \\bar { \\theta } _ { h } \\gets \\bar { \\Sigma } _ { h } ^ { - 1 } \\sum _ { \\tau = 1 } ^ { K } \\phi ( \\bar { s } _ { h } ^ { \\tau } , \\bar { a } _ { h } ^ { \\tau } ) \\cdot \\widehat { V } _ { h + 1 } \\big ( \\bar { s } _ { h + 1 } ^ { \\tau } \\big ) } \\end{array}$ \n8: Set $\\begin{array} { r } { \\left[ \\widehat { \\mathrm { V a r } _ { h } } \\widehat { V } _ { h + 1 } \\right] ( \\cdot , \\cdot ) = \\left. \\phi ( \\cdot , \\cdot ) , \\bar { \\beta } _ { h } \\right. _ { \\left[ 0 , ( H - h + 1 ) ^ { 2 } \\right] } - \\left[ \\left. \\phi ( \\cdot , \\cdot ) , \\bar { \\theta } _ { h } \\right. _ { \\left[ 0 , H - h + 1 \\right] } \\right] ^ { 2 } } \\end{array}$ \n9: Set $\\widehat { \\sigma } _ { h } ( \\cdot , \\cdot ) ^ { 2 } \\gets \\operatorname* { m a x } \\{ 1 , \\widehat { \\mathrm { V a r } } _ { P _ { h } } \\widehat { V } _ { h + 1 } ( \\cdot , \\cdot ) \\}$ \n10: $\\diamond$ bPhase2: Weighted Least-square Value Iteration for pessimistic updates \n11: $\\begin{array} { r l } & { \\mathrm { S e t } \\widehat { \\Lambda } _ { h } \\gets \\sum _ { \\tau = 1 } ^ { \\overline { { K } } } \\phi ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) ^ { \\widehat { \\circ } } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) ^ { \\top } / \\widehat { \\sigma } ^ { 2 } \\big ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\big ) ^ { \\widehat { \\mathbf { \\phi } } } + \\lambda \\cdot I , } \\\\ & { \\mathrm { S e t } \\widehat { w } _ { h } \\gets \\widehat { \\Lambda } _ { h } ^ { - 1 } ( \\sum _ { \\tau = 1 } ^ { K } \\phi ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) \\cdot ( r _ { h } ^ { \\tau } + \\widehat { V } _ { h + 1 } ( s _ { h + 1 } ^ { \\tau } ) ) / \\widehat { \\sigma } ^ { 2 } \\big ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\big ) ) } \\\\ & { \\mathrm { S e t } \\Gamma _ { h } ( \\cdot , \\cdot ) \\gets C \\sqrt { d } \\cdot ( \\phi ( \\cdot , \\cdot ) ^ { \\top } \\widehat { \\Lambda } _ { h } ^ { - 1 } \\phi ( \\cdot , \\cdot ) ) ^ { 1 / 2 } + \\frac { 2 H ^ { 3 } \\sqrt { d } } { K } \\qquad \\mathrm { ( U s e ~ \\Gamma } _ { h } ^ { I } \\mathrm { ~ f o r ~ t h e ~ i m p r o v e d ~ v e r s i o n ) } } \\\\ & { \\mathrm { S e t } \\widehat { \\bar { Q } } _ { h } ( \\cdot , \\cdot ) \\gets \\phi \\big ( \\cdot , \\cdot \\big ) ^ { \\top } \\widehat { w } _ { h } - \\Gamma _ { h } ( \\cdot , \\cdot ) } \\\\ & { \\mathrm { S e t } \\widehat { \\widehat { \\hat { Q } } } _ { h } ( \\cdot , \\cdot ) \\gets \\operatorname* { m i n } \\{ \\widehat { \\bar { Q } } _ { h } ( \\cdot , \\cdot ) , H - h + 1 \\} ^ { + } } \\\\ & \\mathrm { S e t } \\widehat { \\hat { \\mathcal { R } } _ { h } } ( \\cdot , \\cdot ) \\gets \\operatorname* { a r g m a x } _ { \\pi _ { h } } \\widehat { Q } _ { h } ( \\cdot , \\cdot ) , \\pi _ { h } ( \\cdot \\cdot ) _ { A } , \\widehat { V } _ { h } ( \\cdot ) \\gets \\operatorname* { m a x } _ { \\pi _ { h } } \\widehat { Q } _ { h } ( \\cdot , \\cdot ) , \\pi _ { h } \\end{array}$ \n12: \n13: \n14: \n15: \n16: \n17: end for18: Output: $\\left\\{ \\widehat { \\pi } _ { h } \\right\\} _ { h = 1 } ^ { H }$ . ", + "bbox": [ + 174, + 119, + 795, + 411 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Theorem 3.2 provides improvements over the existing best-known results and we now explain it. \nHowever, before that, we first discuss about our theorem condition. ", + "bbox": [ + 173, + 440, + 823, + 469 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Comparing to Zhou et al. (2021a). In the online regime, Zhou et al. (2021a) is the first result that achieves optimal regret rate with $O ( d H \\sqrt { T } )$ in the linear (mixture) MDPs. However, this result requires the condition $d \\geq H$ (their Theorem 6 and Remark 7). In offline RL, VAPVI only requires a milder condition $\\sqrt { d } > \\xi$ comparing to $d \\ge H$ (since for any fixed $V \\in [ 0 , H ]$ , the standardized quantity $\\frac { r + V ( s ^ { \\prime } ) - ( \\mathcal { T } _ { h } V ) ( s , a ) } { \\sigma _ { V } ( s , a ) }$ is bounded by constant with high probability, e.g. by chebyshev inequality), which makes our result apply to a wider range of linear MDPs. ", + "bbox": [ + 173, + 476, + 825, + 570 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Comparing to Jin et al. (2021b). Jin et al. (2021b) first shows pessimistic value iteration (PEVI) is provably efficient for Linear MDPs in offline RL. VAPVI improves PEVI over $O ( { \\sqrt { d } } )$ on the feature dimension, and improves the horizon dependence as $\\begin{array} { r } { \\Lambda _ { h } \\ \\succcurlyeq \\ \\frac { 1 } { H ^ { 2 } } \\sum _ { h } } \\end{array}$ implies $\\Lambda _ { h } ^ { - 1 } \\preccurlyeq$ $H ^ { 2 } \\Sigma _ { h } ^ { - 1 }$ . In addition, when instantiate to the tabular case, i.e. $\\phi ( s , a ) \\ : = \\ : \\mathbf { 1 } _ { s , a }$ , VAPVI gives $\\begin{array} { r l } & { O ( \\sqrt { d } \\sum _ { h , s , a } { d _ { h } ^ { \\pi } } ^ { \\star } ( s , a ) \\sqrt { \\frac { \\operatorname { V a r } _ { P _ { s , a } } ( r + V _ { h + 1 } ^ { \\star } ) } { K \\cdot d _ { h } ^ { \\mu } ( s , a ) } } ) } \\end{array}$ r VarPs,a (r+V ?h+1)K·dµ(s,a) ), which enjoys O(√H) improvement over PEVI (recall Example 3.1) and the order ${ \\cal O } ( H ^ { 3 / 2 } )$ is tight (check Section G for the detailed derivation). ", + "bbox": [ + 173, + 575, + 826, + 685 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Comparing to Xie et al. (2021a). Their linear MDP guarantee in Theorem 3.2. enjoys the same rate as VAPVI in feature dimension but the horizon dependence is essentially the same as Jin et al. (2021b) (by translating $\\begin{array} { r } { H \\approx O \\bigl ( \\frac { 1 } { 1 - \\gamma } \\bigr ) , } \\end{array}$ ) therefore is not optimal. The general function approximation scheme in Xie et al. (2021a) provides elegant characterizations for on-support error and off-support error, but the algorithmic framework is information-theoretical only (and the practical version PSPI will not yield the same learning guarantee). Also, due to the use finite function class and policy class, the reduction to linear MDP only works with finite action space. As a comparison, VAPVI has no constraints on any of these. ", + "bbox": [ + 173, + 690, + 826, + 804 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Comparing to Zanette et al. (2021). Concurrently, Zanette et al. (2021) considers offline RL with the linear Bellman complete model, which is more general than linear MDPs and, with the assumption $Q ^ { \\pi } \\leq 1$ , their PACLE algorithm provides near-minimax optimal guarantee in this setting. However, when recovering to the standard setting $Q ^ { \\pi } \\in [ 0 , H ]$ , their bound will rescale by an $H$ factor,7 which could be suboptimal due to the variance-unawareness. The reason behind this is: when $Q ^ { \\pi } \\leq 1$ , lack of variance information encoding will not matter, since in this case $\\mathrm { V a r } _ { P } ( V ^ { \\pi } ) \\leq 1$ has constant order (therefore will not affect the optimal rate); when $Q ^ { \\pi } \\in [ 0 , H ]$ , $\\operatorname { V a r } _ { P } ( V ^ { \\pi } )$ can be as large as $H ^ { 2 }$ , effectively leveraging the variance information can help improve the sample efficiency, e.g. via law of total variances, just like VAPVI does. On the other hand, their guarantee also requires finite action space, due to the mirror descent style analysis. Nevertheless, we do point out Zanette et al. (2021) has improved state-action measure than VAPVI, as $\\| E _ { \\pi } [ \\phi ( \\cdot , \\cdot ) ] \\| _ { M ^ { - 1 } } \\leq \\mathbb { E } _ { \\pi } [ \\| \\phi ( \\cdot , \\cdot ) \\| _ { M ^ { - 1 } } ]$ by Jensen’s inequality and that norm $\\lVert \\cdot \\rVert _ { M ^ { - 1 } }$ is convex for some positive-definite matrix $M$ . ", + "bbox": [ + 173, + 810, + 825, + 895 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 102, + 825, + 189 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Adaptive characterization and faster convergence. Comparing to existing works, one major improvement is that the main term for VAPVI $\\begin{array} { r l } { { \\sqrt { d } \\sum _ { h = 1 } ^ { H } \\mathbb { E } _ { \\pi ^ { \\star } } \\bigl [ \\sqrt { \\phi ( \\cdot , \\cdot ) ^ { \\top } \\Lambda _ { h } ^ { \\star - 1 } \\phi ( \\cdot , \\cdot ) } \\bigr ] } \\quad } & { { } } \\end{array}$ admits no explicit dependence on $H$ , which provides a more adaptive/instance-dependent characterization. For instance, if we ignore the technical treatment by taking $\\lambda = 0$ and $\\sigma _ { h } ^ { \\star } \\widetilde { \\approx } \\mathrm { V a r } _ { P } ( V _ { h + 1 } ^ { \\star } )$ , then for the partially deterministic systems (where there are $t$ stochastic $P _ { h }$ ’s and $H - t$ deterministic $P _ { h }$ ’s), the main term diminishes to $\\begin{array} { r l } { \\sqrt { d } \\sum _ { i = 1 } ^ { t } \\mathbb { E } _ { \\pi ^ { \\star } } \\left[ \\sqrt { \\phi ( \\cdot , \\cdot ) ^ { \\top } \\Lambda _ { h _ { i } } ^ { \\star - 1 } \\phi ( \\cdot , \\cdot ) } \\right] } & { { } } \\end{array}$ with $h _ { i } \\in \\{ h : ~ s . t . ~ P _ { h }$ is stochastic $\\}$ and can be a much smaller quantity when $t \\ll H$ . Furthermore, for the fully deterministic system, VAPVI automatically provides faster convergence rate $O ( \\textstyle { \\frac { 1 } { K } } )$ from the higher order term, given that the main term degenerates to 0. Those adaptive/instance-dependent features are not enjoyed by (Xie et al., 2021a; Zanette et al., 2021), as they always provide the standard statistical rate $\\scriptstyle { \\dot { O ( } } { \\frac { 1 } { \\sqrt { K } } } )$ (also check Remark C.9 for a related discussion). ", + "bbox": [ + 173, + 193, + 826, + 368 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "3.3 VAPVI-IMPROVED: FURTHER IMPROVEMENT IN STATE-ACTION DIMENSION ", + "text_level": 1, + "bbox": [ + 176, + 385, + 740, + 400 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Can we further improve the VAPVI? Indeed, by deploying a carefully tuned tighter penalty, we are able to further improve the state-action dependence if the feature is non-negative $( \\phi \\ge 0 )$ ). Concretely, we replace the following $\\Gamma _ { h } ^ { I }$ in Algorithm 1 instead, and call the algorithm VAPVI-Improved (or VAPVI-I for short). The proof can be found in Appendix D. ", + "bbox": [ + 174, + 410, + 825, + 465 + ], + "page_idx": 7 + }, + { + "type": "equation", + "img_path": "images/7631a8a82c712d14a00c564fc4509f76e658c972c927b874120be0ca5bf51fe5.jpg", + "text": "$$\n\\Gamma _ { h } ^ { I } ( s , a ) \\gets \\phi ( s , a ) ^ { \\top } \\left| \\widehat { \\Lambda } _ { h } ^ { - 1 } \\sum _ { \\tau = 1 } ^ { K } \\frac { \\phi \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\cdot \\left( r _ { h } ^ { \\tau } + \\widehat { V } _ { h + 1 } \\left( s _ { h + 1 } ^ { \\tau } \\right) - \\left( \\widehat { T } _ { h } \\widehat { V } _ { h + 1 } \\right) \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\right) } { \\widehat { \\sigma } _ { h } ^ { 2 } \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) } \\right| + \\widetilde { O } ( \\frac { H ^ { 3 } d / \\kappa } { K } )\n$$", + "text_format": "latex", + "bbox": [ + 179, + 470, + 802, + 515 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Theorem 3.3. Suppose the feature is non-negative √ $\\left( \\phi \\geq 0 \\right)$ ). Let $K$ be the number of episodes. If $K > \\operatorname* { m a x } \\{ \\mathcal { M } _ { 1 } , \\mathcal { M } _ { 2 } , \\mathcal { M } _ { 3 } , \\mathcal { M } _ { 4 } \\}$ and $\\sqrt { d } > \\xi$ . Deploying $\\Gamma _ { h } ^ { I }$ (4) in Algorithm $^ { l }$ . Then for any $0 < \\lambda < \\kappa$ , with probability $1 - \\delta$ , for all policy $\\pi$ simultaneously, the output $\\widehat { \\pi }$ of Algorithm $\\boldsymbol { l }$ (VAPVI-I) satisfies ", + "bbox": [ + 173, + 529, + 826, + 587 + ], + "page_idx": 7 + }, + { + "type": "equation", + "img_path": "images/657dc53c523f18ef592ea469bd94629f3f49c99e6d3f5fc110c83d5f0282e20f.jpg", + "text": "$$\nv ^ { \\pi } - v ^ { \\widehat { \\pi } } \\leq \\widetilde { O } \\big ( \\sqrt { d } \\cdot \\sum _ { h = 1 } ^ { H } \\sqrt { \\mathbb { E } _ { \\pi } [ \\phi ( \\cdot , \\cdot ) ] ^ { \\top } \\Lambda _ { h } ^ { - 1 } \\mathbb { E } _ { \\pi } [ \\phi ( \\cdot , \\cdot ) ] } \\big ) + \\widetilde { O } ( \\frac { H ^ { 4 } d / \\kappa } { K } )\n$$", + "text_format": "latex", + "bbox": [ + 287, + 593, + 710, + 632 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In particular, when choosing $\\pi = \\pi ^ { \\star }$ , the above guarantee holds true with $\\Lambda _ { h } ^ { - 1 }$ replaced by $\\Lambda _ { h } ^ { \\star - 1 }$ Here $\\Lambda _ { h } ^ { - 1 }$ , $\\Lambda _ { h } ^ { \\star - 1 }$ , $\\xi$ are defined the same as Theorem 3.2. ", + "bbox": [ + 169, + 640, + 823, + 672 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Theorem 3.3 maintains nearly all the features of Theorem 3.2 (except higher order term is slightly worse) and the dominate term evolves from $\\mathbb { E } _ { \\pi } \\left. \\phi \\right. _ { \\Lambda _ { h } ^ { - 1 } }$ to $\\| \\mathbb { E } _ { \\pi } [ \\phi ] \\| _ { \\Lambda _ { h } ^ { - 1 } }$ . Clearly, the two bounds differ by the magnitude of Jensen’s inequality. To provide a concrete view of how much improvement is made, we check the parameter dependence in the context of tabular MDPs (where we ignore the higher order term for conciseness). In particular, we compare the results under the single-policy concentrability. ", + "bbox": [ + 173, + 683, + 825, + 770 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Assumption 3.4 (Rashidinejad et al. (2021); Xie et al. (2021b)). There exists a optimal policy $\\pi ^ { \\star }$ , s.t. \n$\\begin{array} { r } { \\operatorname* { s u p } _ { h , s , a } \\bar { d } _ { h } ^ { \\pi ^ { \\star } } ( s , a ) / d _ { h } ^ { \\mu } ( s , a ) : = C ^ { \\star } < \\infty } \\end{array}$ , where $d ^ { \\pi }$ is the marginal state-action probability under $\\pi$ . ", + "bbox": [ + 173, + 773, + 826, + 806 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In tabular RL, $\\phi ( s , a ) = \\mathbf { 1 } _ { s , a }$ and $d = S \\cdot A \\left( S , A \\right.$ be the finite state, action cardinality), then ", + "bbox": [ + 171, + 816, + 782, + 833 + ], + "page_idx": 7 + }, + { + "type": "equation", + "img_path": "images/7308f6e7368f68683b5e9d15ddc915c4aa01c2cb21749620c10da7ef171bfde5.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathrm { T h e o r e m ~ 3 . 2 } \\sqrt { S A } \\displaystyle \\sum _ { h } ^ { H } \\sum _ { s , a } d _ { h } ^ { \\pi ^ { \\star } } ( s , a ) \\sqrt { \\frac { \\mathrm { V a r } _ { P s , a } ( r + V _ { h + 1 } ^ { \\star } ) } { K \\cdot d _ { h } ^ { \\mu } ( s , a ) } } \\leq \\sqrt { \\frac { H ^ { 3 } C ^ { \\star } S ^ { 2 } A } { K } } ; } \\\\ & { \\mathrm { T h e o r e m ~ 3 . 3 } \\sqrt { S A } \\displaystyle \\sum _ { h } ^ { H } \\sqrt { \\displaystyle \\sum _ { s , a } d _ { h } ^ { \\pi ^ { \\star } } ( s , a ) ^ { 2 } \\frac { \\mathrm { V a r } _ { P s , a } ( r + V _ { h + 1 } ^ { \\star } ) } { K \\cdot d _ { h } ^ { \\mu } ( s , a ) } } \\leq \\sqrt { \\frac { H ^ { 3 } C ^ { \\star } S A } { K } } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 248, + 840, + 746, + 929 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Theorem 3.3 enjoys a $S$ state improvement over Theorem 3.2 and nearly recovers the minimax rate $\\sqrt { \\frac { H ^ { 3 } C ^ { \\star } { \\cal S } } { K } }$ (Xie et al., 2021b). The detailed derivation can be found in Appendix G. Also, to show our result is near-optimal, we provide the corresponding lower bound. The proof is in Appendix E. ", + "bbox": [ + 173, + 102, + 825, + 155 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Theorem 3.5 (Minimax lower bound). There exist a pair of universal constants $c , c ^ { \\prime } > 0$ such that given dimension $d$ , horizon $H$ and sample size $K > c ^ { \\prime } d ^ { 3 }$ , one can always find a family of linear MDP instances M such that (where Λ?h = PKk=1 φ(s h,a h)·φ(s h,a h) Varh(V ? )(sk ,ak ) satisfies $( \\Lambda _ { h } ^ { \\star } ) ^ { - 1 }$ exists and ", + "bbox": [ + 174, + 156, + 825, + 223 + ], + "page_idx": 8 + }, + { + "type": "equation", + "img_path": "images/b810657ad689c435f0bcbbd8d7bed61d8304acd9da1fbff77bfcc2ba7f28750c.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathrm { V a r } _ { h } ( V _ { h + 1 } ^ { \\star } ) ( s _ { h } ^ { k } , a _ { h } ^ { k } ) > 0 \\forall M \\in \\mathcal { M } ) } \\\\ & { \\qquad \\quad \\frac { \\operatorname* { i n f } } { \\pi } \\underset { M \\in \\mathcal { M } } { \\operatorname* { s u p } } \\mathbb { E } _ { M } \\left[ v ^ { \\star } - v ^ { \\hat { \\pi } } \\right] / \\left( \\sqrt { d } \\cdot \\displaystyle \\sum _ { h = 1 } ^ { H } \\sqrt { \\mathbb { E } _ { \\pi ^ { \\star } } \\left[ \\phi \\right] ^ { \\top } ( \\Lambda _ { h } ^ { \\star } ) ^ { - 1 } \\mathbb { E } _ { \\pi ^ { \\star } } \\left[ \\phi \\right] } \\right) \\geq c . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 171, + 207, + 712, + 261 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Theorem 3.5 nearly matches the main term in VAPVI-I (Theorem 3.3) and certifies it is near-optimal. ", + "bbox": [ + 173, + 261, + 823, + 276 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "4 PROOF OVERVIEW ", + "text_level": 1, + "bbox": [ + 176, + 295, + 359, + 311 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Due to the space constraint, we could only provide a brief overview of the key proving ideas of the theorems. We begin with Theorem 3.2. First, by the extended value difference lemma (Lemma H.7), we can convert bounding the suboptimality gap of $\\boldsymbol { v } ^ { \\star } - \\boldsymbol { v } ^ { \\widehat { \\pi } }$ to bounding ${ \\begin{array} { r } { \\sum _ { h = 1 } ^ { H } 2 \\cdot \\mathbb { E } _ { \\pi } \\left[ \\Gamma _ { h } { \\left( s _ { h } , a _ { h } \\right) } \\right] } \\end{array} }$ , given that $| ( \\mathcal T _ { h } \\widehat { V } _ { h + 1 } - \\widehat { \\mathcal T } _ { h } \\widehat { V } _ { h + 1 } ) ( s , a ) | \\leq \\Gamma _ { h } ( s , a )$ for all $s , a , h$ . To bound $\\mathcal { T } _ { h } \\widehat { V } _ { h + 1 } - \\widehat { \\mathcal { T } } _ { h } \\widehat { V } _ { h + 1 }$ , by decomposing it reduces to bounding the key quantity ", + "bbox": [ + 173, + 325, + 826, + 402 + ], + "page_idx": 8 + }, + { + "type": "equation", + "img_path": "images/30c75e721e4d8cdd6fd373e0d3f390dfccfb1a8916332669093166d67337d475.jpg", + "text": "$$\n\\phi ( s , a ) ^ { \\top } \\widehat { \\Lambda } _ { h } ^ { - 1 } \\left[ \\sum _ { \\tau = 1 } ^ { K } \\phi \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\cdot \\left( r _ { h } ^ { \\tau } + \\widehat { V } _ { h + 1 } \\left( s _ { h + 1 } ^ { \\tau } \\right) - \\left( \\mathcal { T } _ { h } \\widehat { V } _ { h + 1 } \\right) \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\right) / \\widehat { \\sigma } _ { h } ^ { 2 } \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\right]\n$$", + "text_format": "latex", + "bbox": [ + 197, + 404, + 781, + 446 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "The term is treated in two steps. First, we bound the gap of $\\left\\| \\sigma _ { \\widehat { V } _ { h + 1 } } ^ { 2 } - \\widehat { \\sigma } _ { h } ^ { 2 } \\right\\|$ so we can convert $\\widehat { \\sigma } _ { h } ^ { 2 }$ to $\\sigma _ { \\widehat { V } _ { h + 1 } } ^ { 2 }$ . Next, since $\\begin{array} { r } { \\mathrm { V a r } \\left[ r _ { h } ^ { \\tau } + \\widehat { V } _ { h + 1 } \\left( s _ { h + 1 } ^ { \\tau } \\right) - \\left( \\mathcal { T } _ { h } \\widehat { V } _ { h + 1 } \\right) \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\mid s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right] \\approx \\sigma _ { \\widehat { V } _ { h + 1 } } ^ { 2 } } \\end{array}$ , therefore by the variance-weighted scheme in (equation 7), we can leverage the recent technical development Bernstein inequality for self-normalized martingale (Lemma H.3) for acquiring the tight result, in contrast to the previous treatment of Hoeffding inequality for self-normalized martingale $^ +$ Covering.8 For the second part, one needs to further convert $\\sigma _ { \\widehat { V } _ { h + 1 } } ^ { 2 ^ { - } }$ to $\\sigma _ { h } ^ { \\star 2 }$ $\\cdot \\Lambda _ { h } ^ { - 1 }$ to $\\Lambda _ { h } ^ { \\star - 1 }$ ) with appropriate concentrations. The proof of Theorem 3.3 is similar but with more complicated computations and relies on using the linear representation of $\\phi$ in $\\Gamma _ { h } ^ { I }$ (4), so that the expectation over $\\pi$ is inside the square root by taking expectation over the linear representation at the beginning. The lower bound proof uses a simple modification of Zanette et al. (2021) which consists of the reduction from learning to testing with Assouad’s method, and the use of standard information inequalities (e.g. from total variation to KL divergence). For completeness, we provide the full proof in Appendix E. ", + "bbox": [ + 173, + 446, + 826, + 638 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "5 DISCUSSION AND CONCLUSION ", + "text_level": 1, + "bbox": [ + 178, + 657, + 468, + 672 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "This work studies offline RL with linear MDP representation and contributes Variance Aware Pessimistic Value Iteration (VAPVI) which adopts the conditional variance information of the value function. VAPVI uses the estimated variances to reweight the Bellman residuals in the least-square pessimistic value iteration and provides improved offline learning bounds over the existing best-known results. VAPVI-I further improves over VAPVI in the state-action dimension and is near-minimax optimal. One highlight of the theorems is that our learning bounds are expressed in terms of system quantities, which automatically provide natural instance-dependent characterizations that previous results are short of. ", + "bbox": [ + 173, + 688, + 825, + 799 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "On the other hand, while VAPVI/VAPVI-I close the existing gap from previous literature (Jin et al., 2021b; Xie et al., 2021a), the optimal guarantee is in the minimax sense. Although our upper bounds possess instance-dependent characterizations, the lower bound only holds true for a class of hard instances. In this sense, whether “instance-dependent optimality” can be achieved remains elusive in the current linear MDP setting (such a discussion is recently initiated in MAB problems (Xiao et al., 2021)). We leave this as future work. ", + "bbox": [ + 174, + 806, + 825, + 890 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "ACKNOWLEDGMENTS ", + "text_level": 1, + "bbox": [ + 176, + 104, + 326, + 117 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "The authors would like to thank Quanquan Gu for explaining Min et al. 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", + "bbox": [ + 173, + 155, + 825, + 196 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Appendix ", + "text_level": 1, + "bbox": [ + 444, + 99, + 555, + 123 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "A NOTATION LIST ", + "text_level": 1, + "bbox": [ + 176, + 140, + 343, + 156 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "$\\Sigma _ { h } ^ { p }$ $\\begin{array} { r l } & { \\mathbb { E } _ { \\mu , h } \\left[ \\phi ( s , a ) \\phi ( s , a ) ^ { \\top } \\right] } \\\\ & { \\mathbb { E } _ { \\mu , h } \\left[ \\sigma _ { V _ { h + 1 } } ( s , a ) ^ { - 2 } \\phi ( s , a ) \\phi ( s , a ) ^ { \\top } \\right] } \\\\ & { \\operatorname* { m i n } _ { h } \\lambda _ { \\operatorname* { m i n } } ( \\Sigma _ { h } ^ { p } ) } \\end{array}$ \n$\\Lambda _ { h } ^ { p }$ \n$\\kappa$ \nι $\\operatorname* { m i n } _ { h } \\lambda _ { \\operatorname* { m i n } } ( \\Lambda _ { h } ^ { p } ) \\ge \\kappa / H ^ { 2 }$ for any $V _ { h }$ \n$\\sigma _ { V } ^ { 2 } ( s , a )$ $\\operatorname* { m a x } \\{ 1 , \\operatorname { V a r } _ { P _ { h } } ( V ) ( s , a ) \\}$ for any $V$ \nσ2Vh+1 (s, a) σ2Vh (s, a) $\\begin{array} { r l } & { \\operatorname* { m a x } \\{ 1 , \\nabla \\mathrm { a r } _ { h } ( V _ { h + 1 } ) ( s , a ) \\} } \\\\ & { \\operatorname* { m a x } \\{ 1 , \\widehat { \\mathrm { V a r } } _ { h } h _ { h + 1 } ( s , a ) \\} } \\\\ & { \\sum _ { \\tau = 1 } ^ { K } \\phi ( \\tilde { s } _ { h } ^ { \\tau } , \\tilde { a } _ { h } ^ { \\tau } ) \\phi ( \\tilde { s } _ { h } ^ { \\tau } , \\tilde { a } _ { h } ^ { \\tau } ) ^ { \\top } + \\lambda I _ { d } } \\\\ & { \\sum _ { k = 1 } ^ { K } \\phi ( s _ { h } ^ { k } , a _ { h } ^ { k } ) \\phi ( s _ { h } ^ { k } , a _ { h } ^ { k } ) ^ { \\top } / \\widehat { \\sigma } _ { h } ^ { 2 } ( s _ { h } ^ { k } , a _ { h } ^ { k } ) + \\lambda I _ { d } } \\\\ & { \\operatorname* { m a x } \\{ 2 \\lambda , 1 2 8 \\log ( 2 d / \\delta ) , 1 2 8 H ^ { 4 } \\log ( 2 d / \\delta ) / \\kappa ^ { 2 } \\} } \\\\ & { \\operatorname* { m a x } \\{ \\frac { \\lambda ^ { 2 } } { \\kappa \\log ( ( \\lambda + K ) H / \\lambda \\delta ) } , 9 6 ^ { 2 } H ^ { 1 2 } d \\log ( ( \\lambda + K ) H / \\lambda \\delta ) / \\kappa ^ { 5 } \\} } \\\\ & { \\operatorname* { m a x } \\{ 5 1 2 H ^ { 4 } / \\kappa ^ { 2 } \\log \\left( \\frac { 2 d } { \\delta } \\right) , 4 \\lambda H ^ { 2 } / \\kappa \\} } \\\\ & { 1 2 \\sqrt { H ^ { 4 } d \\log ( ( \\lambda + K ) H / \\lambda \\delta ) / \\kappa } } \\end{array}$ \nΣ¯ h \n$\\widehat { \\Lambda } _ { h }$ \n$\\mathcal { M } _ { 1 }$ \n$\\mathcal { M } _ { 2 }$ \nM3 \n$\\mathcal { M } _ { 4 }$ \nδ Failure probability \n$\\xi$ $\\begin{array} { r l } & { \\operatorname* { s u p } _ { V \\in [ 0 , H ] , \\ s ^ { \\prime } \\sim P _ { h } ( s , a ) , \\ h \\in [ H ] } \\left| \\frac { r _ { h } + V \\left( s ^ { \\prime } \\right) - ( \\mathcal { T } _ { h } V ) ( s , a ) } { \\sigma _ { V } ( s , a ) } \\right| } \\\\ & { 3 6 \\sqrt { \\frac { H ^ { 4 } d ^ { 3 } } { \\kappa } \\log \\left( \\frac { ( \\lambda + K ) 2 K d H ^ { 2 } } { \\lambda \\delta } \\right) } + 1 2 \\lambda \\frac { H ^ { 2 } \\sqrt { d } } { \\kappa } } \\end{array}$ \n$C _ { H , d , \\kappa , K }$ ", + "bbox": [ + 179, + 170, + 736, + 523 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "B EXTENDED LITERATURE REVIEW ", + "text_level": 1, + "bbox": [ + 174, + 546, + 488, + 563 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "B.1 LINEAR MODEL REPRESENTATION AND ITS EXTENSION IN ONLINE RL ", + "text_level": 1, + "bbox": [ + 176, + 578, + 700, + 592 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "There are numerous works in online RL that study linear model representations. Yang and Wang (2019; 2020); Jin et al. (2020) propose Linear MDP, which assumes the transition kernel and the reward are linear in given features. Cai et al. (2020); Ayoub et al. (2020); Modi et al. (2020); Zhou et al. (2021b) propose Linear mixture MDP, which assumes the transition probability is a linear combination of some base kernels. Linear Bellman Complete model (Zanette et al., 2020) generalizes linear MDP model by allowing linear functions to approximate the $Q$ -function and the function class is closed under the Bellman update. The notion of low Bellman rank (Jiang et al., 2017) subsumes not only linear MDPs but also models including linear quadratic regulator (LQR), Reactive POMDP (Krishnamurthy et al., 2016) and Block MDP (Du et al., 2019). There are also other models, e.g. factored MDP (Sun et al., 2019), Bellman Eluder dimension (Jin et al., 2021a) and Bilinear class (Du et al., 2021). With the linear MDP model itself, there are also fruitful extensions, e.g. gap-dependent analysis with logarithmic regret (He et al., 2021), low-switching cost RL (Gao et al., 2021), safe RL (Amani et al., 2021), reward-free RL (Wang et al., 2020), generalized linear model (GLM) (Wang et al., 2021b), two-player markov game (Chen et al., 2021b) and generative adversarial learning (Liu et al., 2021). In particular, Zhou et al. (2021a) shows UCRL- $\\mathrm { { V T R + } }$ is near-minimax optimal when feature dimension $d \\geq H$ . ", + "bbox": [ + 173, + 604, + 825, + 825 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "B.2 EXISTING RESULTS IN OFFLINE RL WITH MODEL REPRESENTATIONS ", + "text_level": 1, + "bbox": [ + 174, + 842, + 689, + 856 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Offline RL with general function representations. The finite sample analysis of offline RL with function approximation is initially conducted by Fitted $Q$ -Iteration (FQI) type algorithms and can be dated back to (Munos, 2003; Szepesvari and Munos ´ , 2005; Antos et al., 2008a;b). Later, Chen and Jiang (2019); Le et al. (2019); Xie and Jiang (2020) follow this line of research and derive the improved learning results. However, owing to the aim for tackling general function approximation, those learning bounds are expressed in terms of the stringent concentrability coefficients (therefore, are less adaptive to individual instances) and are usually only information-theoretical, due to the computational intractability of the optimization procedure over the general function classes. Other works impose weaker assumptions (e.g. partial coverage (Liu et al., 2020; Kidambi et al., 2020; Uehara and Sun, 2021)), and their finite sample analysis are generally suboptimal in terms of $H$ or the effective horizon $( 1 - \\gamma ) ^ { - 1 }$ . ", + "bbox": [ + 176, + 868, + 823, + 924 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 202 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Offline RL with tabular models. For tabular MDPs, tight learning bounds can be achieved under several data-coverage assumptions. For the class of problems with uniform data-visitation measure $d _ { m }$ , the near-optimal sample complexity bound has the rate $O ( H ^ { 3 } / d _ { m } \\epsilon ^ { 2 } )$ for time-inhomogeneous MDPs (Yin et al., 2021) and $O ( H ^ { 2 } / d _ { m } \\dot { \\epsilon } )$ for time-homogeneous MDPs (Yin and Wang, 2021a; Ren et al., 2021). Under the single concentrability assumption, the tight rate ${ \\cal O } ( H ^ { 3 } S C ^ { \\star } / \\epsilon ^ { \\top } )$ is obtained by Xie et al. (2021b). In particular, the recent study Yin and Wang (2021b) introduces the intrinsic offline learning bound that is not only instance-dependent but also subsumes previous optimal results. More recently, Shi et al. (2022) uses model-free approach to achieve minimax rate with a larger $\\epsilon$ -range. ", + "bbox": [ + 174, + 208, + 825, + 333 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Offline RL with linear model representations. Recently, there are more focus on studying the provable efficient offline RL under the linear model representations. Jin et al. (2021b) first shows offline RL with linear MDP is provably efficient by the pessimistic value iteration (PEVI), which is an offline counterpart of LSVI-UCB in Jin et al. (2020). Their analysis deviates from their lower bound by a factor of $d \\cdot H$ (check their Theorem 4.4 and 4.6). Later, Xie et al. (2021a) considers function approximation under the Bellman-consistent assumptions, and, when realized to linear MDP setting, improve the sample complexity guarantee of Jin et al. (2021b) by a order $O ( d )$ (Theorem 3.2). However, their improvement only holds for finite action space (due to the dependence $\\log \\left| { \\mathcal { A } } \\right| )$ and by the direct reduction (from Theorem 3.1) their result does not imply a computationally tractable algorithm. In addition, there is no improvement on the horizon dependence. Concurrently, Zanette et al. (2021) considers the Linear Bellman Complete model (which originates from its online version Zanette et al. (2020)) and designs the actor-critic style algorithm that achieves tight result under the assumption that the value function is bounded by 1. While their algorithm is efficient (which is based on solving a sequence of second-order cone programs), the resulting learning bound requires the action space to be finite due to the mirror descent/natural policy gradient updates in the Actor procedure (Agarwal et al., 2021). Besides, assuming the value function to be less than 1 simplifies the challenges in dealing with horizon $H$ since when rescale their result to $[ 0 , H ]$ , there is a $H$ factor blow-up, which makes no improvement in the horizon dependence comparing to Jin et al. (2021b). On the other hand, Wang et al. (2021a); Zanette (2021) study the statistical hardness of offline RL with linear representations by proofing the exponential lower bounds. As a result, none of the existing algorithms can achieve the statistical limit for the well-structured linear MDP model with the general (infinite or continuous) state-action spaces in the offline regime. ", + "bbox": [ + 173, + 339, + 825, + 645 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "C PROOFS IN SECTION 3.2 ", + "text_level": 1, + "bbox": [ + 176, + 667, + 410, + 684 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Instead of proofing the result for $v ^ { \\star } - v ^ { \\widehat { \\pi } }$ , in most parts of the proof we deal with $V _ { 1 } ^ { \\star } - V _ { 1 } ^ { \\widehat \\pi }$ , which is more general. ", + "bbox": [ + 176, + 700, + 823, + 729 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "C.1 SOME PREPARATIONS ", + "text_level": 1, + "bbox": [ + 174, + 750, + 367, + 763 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Define the Bellman update error $\\zeta _ { h } ( s , a ) \\ : = \\ ( { \\mathcal T } _ { h } \\widehat V _ { h + 1 } ) ( s , a ) \\ - \\ - \\ - \\widetilde Q _ { h } ( s , a )$ and recall $\\widehat { \\pi } _ { h } ( s ) \\ =$ arg $\\operatorname* { m a x } _ { \\pi _ { h } } \\langle \\widehat { Q } _ { h } ( s , \\cdot ) , \\pi _ { h } ( \\cdot \\mid s ) \\rangle _ { \\mathcal { A } }$ , then by the direct application of Lemma H.8 ", + "bbox": [ + 173, + 773, + 826, + 809 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/f036dd51f54ff83e1fbc2531865d476277df7179bdb966b7d011606d688224d9.jpg", + "text": "$$\nV _ { 1 } ^ { \\pi } ( s ) - V _ { 1 } ^ { \\widehat { \\pi } } ( s ) \\leq \\sum _ { h = 1 } ^ { H } \\mathbb { E } _ { \\pi } \\left[ \\zeta _ { h } ( s _ { h } , a _ { h } ) \\ | \\ s _ { 1 } = s \\right] - \\sum _ { h = 1 } ^ { H } \\mathbb { E } _ { \\widehat { \\pi } } \\left[ \\zeta _ { h } ( s _ { h } , a _ { h } ) \\ | \\ s _ { 1 } = s \\right] .\n$$", + "text_format": "latex", + "bbox": [ + 235, + 818, + 763, + 862 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "The next lemma shows it is sufficient to bound the pessimistic penalty, which is the key in the proof. ", + "bbox": [ + 174, + 869, + 825, + 886 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Lemma C.1. Suppose with probability $1 - \\delta$ , it holds for all $h , s , a \\in [ H ] \\times S \\times A$ that $| ( \\widehat { T _ { h } } \\widehat { V _ { h + 1 } } -$ $\\widehat { \\mathcal { T } } _ { h } \\widehat { V } _ { h + 1 } ) ( s , a ) | \\ \\leq \\ \\Gamma _ { h } ( s , a )$ , then it implies $\\forall s , a , h \\in \\mathcal { S } \\times \\mathcal { A } \\times [ H ]$ , $0 \\leq \\zeta _ { h } ( s , a ) \\leq 2 \\Gamma _ { h } ( s , a )$ . ", + "bbox": [ + 173, + 891, + 825, + 925 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Furthermore, it holds for any policy $\\pi$ simultaneously, with probability $1 - \\delta$ , ", + "bbox": [ + 171, + 102, + 679, + 119 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/18e91ffb4e85d3e8f561a4a1a041d7b87c759fc6d168a61eb7fe3f9db99507f4.jpg", + "text": "$$\nV _ { 1 } ^ { \\pi } ( s ) - V _ { 1 } ^ { \\widehat { \\pi } } ( s ) \\leq \\sum _ { h = 1 } ^ { H } 2 \\cdot \\mathbb { E } _ { \\pi } \\left[ \\Gamma _ { h } ( s _ { h } , a _ { h } ) \\mid s _ { 1 } = s \\right] .\n$$", + "text_format": "latex", + "bbox": [ + 325, + 131, + 671, + 175 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Proof of Lemma C.1. We first show given $| ( \\mathcal { T } _ { h } \\widehat { V } _ { h + 1 } - \\widehat { \\mathcal { T } } _ { h } \\widehat { V } _ { h + 1 } ) ( s , a ) | \\ \\leq \\ \\Gamma _ { h } ( s , a )$ , then $0 ~ \\leq$ $\\zeta _ { h } ( s , a ) \\le 2 \\Gamma _ { h } ( s , a ) , \\forall s , a , h \\in \\mathcal { S } \\times \\bar { \\mathcal { A } } \\times [ H ]$ . ", + "bbox": [ + 173, + 212, + 825, + 244 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Step1: we first show $0 \\leq \\zeta _ { h } ( s , a ) , \\forall s , a , h \\in S \\times \\mathcal { A } \\times [ H ] .$ . ", + "bbox": [ + 171, + 250, + 568, + 266 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Indeed, if $\\bar { Q } _ { h } ( s , a ) ~ \\leq ~ 0$ , then by definition $\\widehat { Q } _ { h } ( s , a ) ~ = ~ 0$ and in this case $\\zeta _ { h } ( s , a ) : =$ $( { \\mathcal { T } } _ { h } { \\widehat { V } } _ { h + 1 } ) ( s , a ) - { \\widehat { Q } } _ { h } ( s , a ) = ( { \\mathcal { T } } _ { h } { \\widehat { V } } _ { h + 1 } ) ( s , a ) \\geq 0$ ; if $\\bar { Q } _ { h } ( s , a ) > 0$ , then $\\widehat { Q } _ { h } ( s , a ) \\ \\leq \\ \\bar { Q } _ { h } ( s , a )$ and ", + "bbox": [ + 173, + 272, + 826, + 321 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/93f06c32a975932a60fef58ba09d3a0a22cae47b483f130d31a147910274aa5d.jpg", + "text": "$$\n\\begin{array} { r l } & { \\zeta _ { h } ( s , a ) : = ( { \\mathcal { T } } _ { h } \\widehat { V } _ { h + 1 } ) ( s , a ) - \\widehat { Q } _ { h } ( s , a ) \\geq ( { \\mathcal { T } } _ { h } \\widehat { V } _ { h + 1 } ) ( s , a ) - \\bar { Q } _ { h } ( s , a ) } \\\\ & { \\qquad = ( { \\mathcal { T } } _ { h } \\widehat { V } _ { h + 1 } ) ( s , a ) - ( \\widehat { { \\mathcal { T } } _ { h } } \\widehat { V } _ { h + 1 } ) ( s , a ) + \\Gamma _ { h } ( s , a ) \\geq 0 . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 269, + 329, + 728, + 375 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Step2: next we show $\\zeta _ { h } ( s , a ) \\leq 2 \\Gamma _ { h } ( s , a ) , \\forall s , a , h \\in \\mathcal { S } \\times \\mathcal { A } \\times [ H ] .$ ", + "bbox": [ + 171, + 392, + 625, + 410 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Indeed, we have $\\widehat { Q } _ { h } ( s , a ) = \\operatorname* { m a x } ( \\bar { Q } _ { h } ( s , a ) , 0 )$ and this is because: $\\bar { Q } _ { h } ( x , a ) = ( \\widehat { T } _ { h } \\widehat { V } _ { h + 1 } ) ( x , a ) -$ $\\Gamma _ { h } ( x , a ) \\leq ( \\mathcal { T } _ { h } \\widehat { V } _ { h + 1 } ) ( x , a ) \\leq H - h + 1$ . Therefore, in this case we have: ", + "bbox": [ + 173, + 416, + 823, + 450 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/8e2fdb2d232ac6a3bd5aa3ef0deba8e414d90bb012bd648ffa2329bce7906a91.jpg", + "text": "$$\n\\begin{array} { r l } & { \\zeta _ { h } ( s , a ) : = ( { \\mathcal { T } } _ { h } { \\widehat { V } } _ { h + 1 } ) ( s , a ) - { \\widehat { Q } } _ { h } ( s , a ) \\leq ( { \\mathcal { T } } _ { h } { \\widehat { V } } _ { h + 1 } ) ( s , a ) - { \\bar { Q } } _ { h } ( s , a ) } \\\\ & { \\qquad = ( { \\mathcal { T } } _ { h } { \\widehat { V } } _ { h + 1 } ) ( s , a ) - ( { \\widehat { \\mathcal { T } } } _ { h } { \\widehat { V } } _ { h + 1 } ) ( s , a ) + \\Gamma _ { h } ( s , a ) \\leq 2 \\cdot \\Gamma _ { h } ( s , a ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 259, + 462, + 736, + 506 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "For the last statement, denote $\\mathfrak { F } : = \\{ 0 \\le \\zeta _ { h } ( s , a ) \\le 2 \\Gamma _ { h } ( s , a ) , \\forall s , a , h \\in \\mathcal { S } \\times \\mathcal { A } \\times [ H ]$ . Note conditional on $\\mathfrak { F }$ , then by equation 8, $\\begin{array} { r } { , V _ { 1 } ^ { \\pi } ( s ) - V _ { 1 } ^ { \\widehat { \\pi } } ( s ) \\leq \\sum _ { h = 1 } ^ { H } 2 \\cdot \\mathbb { E } _ { \\pi } [ \\Gamma _ { h } ( s _ { h } , a _ { h } ) \\mid s _ { 1 } = s ] } \\end{array}$ holds for any policy $\\pi$ almost surely. Therefore, ", + "bbox": [ + 173, + 523, + 826, + 571 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/08a12ee17209e642ceeb368aecea0c8b993be64f0c3e03b6f240f1c5548412be.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathbb { P } \\left[ \\forall \\pi , \\ V _ { 1 } ^ { \\pi } ( s ) - V _ { 1 } ^ { \\widehat { \\pi } } ( s ) \\leq \\displaystyle \\sum _ { h = 1 } ^ { H } 2 \\cdot \\mathbb { E } _ { \\pi } \\big [ \\Gamma _ { h } ( s _ { h } , a _ { h } ) \\mid s _ { 1 } = s \\big ] . \\right] } \\\\ & { = \\mathbb { P } \\Bigg [ \\forall \\pi , \\ V _ { 1 } ^ { \\pi } ( s ) - V _ { 1 } ^ { \\widehat { \\pi } } ( s ) \\leq \\displaystyle \\sum _ { h = 1 } ^ { H } 2 \\cdot \\mathbb { E } _ { \\pi } \\big [ \\Gamma _ { h } ( s _ { h } , a _ { h } ) \\mid s _ { 1 } = s \\bigg ] \\Bigg | \\mathfrak { F } \\Bigg ] \\cdot \\mathbb { P } [ \\mathfrak { F } ] } \\\\ & { + \\mathbb { P } \\left[ \\forall \\pi , \\ V _ { 1 } ^ { \\pi } ( s ) - V _ { 1 } ^ { \\widehat { \\pi } } ( s ) \\leq \\displaystyle \\sum _ { h = 1 } ^ { H } 2 \\cdot \\mathbb { R } _ { \\pi } \\big [ \\Gamma _ { h } ( s _ { h } , a _ { h } ) \\mid s _ { 1 } = s \\big ] \\Bigg | \\mathfrak { F } ^ { \\overline { { \\kappa } } } \\right] \\cdot \\mathbb { P } [ \\mathfrak { F } ^ { c } ] } \\\\ & { \\geq \\mathbb { P } \\Bigg [ \\forall \\pi , \\ V _ { 1 } ^ { \\pi } ( s ) - V _ { 1 } ^ { \\widehat { \\pi } } ( s ) \\leq \\displaystyle \\sum _ { h = 1 } ^ { H } 2 \\cdot \\mathbb { E } _ { \\pi } \\big [ \\Gamma _ { h } ( s _ { h } , a _ { h } ) \\mid s _ { 1 } = s \\big ] \\Bigg | \\mathfrak { F } \\Bigg ] \\cdot \\mathbb { P } [ \\mathfrak { F } ] \\geq 1 \\cdot \\mathbb { P } [ \\mathfrak { F } ] \\geq 1 - \\delta , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 194, + 583, + 802, + 763 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "which finishes the proof. ", + "bbox": [ + 173, + 792, + 338, + 808 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "C.2 BOUNDING $\\Big | ( { \\mathcal T } _ { h } \\widehat { V } _ { h + 1 } ) ( s , a ) - ( \\widehat { { \\mathcal T } } _ { h } \\widehat { V } _ { h + 1 } ) ( s , a ) \\Big | .$ ", + "text_level": 1, + "bbox": [ + 173, + 851, + 537, + 876 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "By Lemma C.1, it remains to bound $| ( { \\mathcal T } _ { h } \\widehat { V } _ { h + 1 } ) ( s , a ) - ( \\widehat { \\mathcal T } _ { h } \\widehat { V } _ { h + 1 } ) ( s , a ) |$ . Suppose $w _ { h }$ is the coefficient corresponding to the $\\mathcal { T } _ { h } \\widehat { V } _ { h + 1 }$ (such $w _ { h }$ exists by Lemma H.9), i.e. $\\mathcal { T } _ { h } \\widehat { V } _ { h + 1 } = \\phi ^ { \\top } w _ { h }$ , and recall ", + "bbox": [ + 173, + 890, + 823, + 926 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "$( \\widehat { T } _ { h } \\widehat { V } _ { h + 1 } ) ( s , a ) = \\phi ( s , a ) ^ { \\top } \\widehat { w } _ { h }$ , then: ", + "bbox": [ + 173, + 101, + 415, + 119 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/474425d8f3846016016b019bd799c067bde88f60244a4b9b14f5dce2ae45fa1d.jpg", + "text": "$$\n\\begin{array} { r l } & { \\quad \\left( \\mathcal { T } _ { h } \\widehat { V } _ { h + 1 } \\right) ( s , a ) - \\left( \\widehat { \\mathcal { T } } _ { h } \\widehat { V } _ { h + 1 } \\right) ( s , a ) = \\phi ( s , a ) ^ { \\top } ( w _ { h } - \\widehat { w } _ { h } ) } \\\\ & { = \\phi ( s , a ) ^ { \\top } w _ { h } - \\phi ( s , a ) ^ { \\top } \\widehat { \\Lambda } _ { h } ^ { - 1 } \\left( \\displaystyle \\sum _ { \\tau = 1 } ^ { K } \\phi \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\cdot \\left( r _ { h } ^ { \\tau } + \\widehat { V } _ { h + 1 } \\left( s _ { h + 1 } ^ { \\tau } \\right) \\right) / \\widehat { \\sigma } _ { h } ^ { 2 } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) \\right) } \\\\ & { = \\phi ( s , a ) ^ { \\top } w _ { h } - \\phi ( s , a ) ^ { \\top } \\widehat { \\Lambda } _ { h } ^ { - 1 } \\left( \\displaystyle \\sum _ { \\tau = 1 } ^ { K } \\phi \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\cdot \\left( \\mathcal { T } _ { h } \\widehat { V } _ { h + 1 } \\right) \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) / \\widehat { \\sigma } _ { h } ^ { 2 } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) \\right) } \\\\ & { { \\overset { { , } } { , } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 168, + 131, + 725, + 248 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/d985a0834b83e526509a28b18b216835fd3ac31b93c0da8d50e3d8bd3485b0ef.jpg", + "text": "$$\n+ \\phi ( s , a ) ^ { \\top } \\widehat { \\Lambda } _ { h } ^ { - 1 } \\left( \\sum _ { \\tau = 1 } ^ { K } \\phi \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\cdot \\left( r _ { h } ^ { \\tau } + \\widehat { V } _ { h + 1 } \\left( s _ { h + 1 } ^ { \\tau } \\right) - \\left( \\tau _ { h } \\widehat { V } _ { h + 1 } \\right) \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\right) / \\widehat { \\sigma } _ { h } ^ { 2 } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 217, + 268, + 833, + 310 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "The term (i) is dealt by the following lemma. ", + "bbox": [ + 174, + 348, + 470, + 364 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Lemma C.2. Recall $\\kappa$ in Assumption 2.2. Suppose $\\begin{array} { r } { K \\ge \\operatorname* { m a x } \\left\\{ 5 1 2 H ^ { 4 } / \\kappa ^ { 2 } \\log \\left( \\frac { 2 d } { \\delta } \\right) , 4 \\lambda H ^ { 2 } / \\kappa \\right\\} } \\end{array}$ , then with probability $1 - \\delta _ { : }$ , for all $s , a , h \\in \\mathcal S \\times \\mathcal A \\times [ H ]$ ", + "bbox": [ + 173, + 369, + 826, + 404 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/5df864ea3b54fac8520c9eb7b7537a004c4de3dc7587c274078d3cac9f38c97d.jpg", + "text": "$$\n\\phi ( s , a ) ^ { \\top } w _ { h } - \\phi ( s , a ) ^ { \\top } \\widehat { \\Lambda } _ { h } ^ { - 1 } \\left( \\sum _ { \\tau = 1 } ^ { K } \\phi \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\cdot \\left( \\mathcal { T } _ { h } \\widehat { V } _ { h + 1 } \\right) \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) / \\widehat { \\sigma } ^ { 2 } \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\right) \\Bigg | \\leq \\frac { 2 \\lambda H ^ { 3 } \\sqrt { d } / \\kappa } { K } .\n$$", + "text_format": "latex", + "bbox": [ + 181, + 415, + 823, + 458 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Proof. Recall $\\mathcal { T } _ { h } \\widehat { V } _ { h + 1 } = \\phi ^ { \\top } w _ { h }$ and apply Lemma H.6, we obtain with probability $1 - \\delta$ , for all $s , a , h \\in \\mathcal S \\times \\mathcal A \\times [ H ]$ , ", + "bbox": [ + 174, + 493, + 825, + 526 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/a4063bd519014fa410d64e777145eabac33280af056ea974c2adfb3b9665671b.jpg", + "text": "$$\n\\begin{array} { r l } & { \\phi ( s , a ) ^ { \\top } w _ { h } - \\phi ( s , a ) ^ { \\top } \\widehat \\Lambda _ { h } ^ { - 1 } \\left( \\displaystyle \\sum _ { \\tau = 1 } ^ { K } \\phi \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\cdot \\left( \\mathcal { T } _ { h } \\widehat { V } _ { h + 1 } \\right) \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) / \\widehat \\sigma ^ { 2 } \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\right) } \\\\ & { = \\phi ( s , a ) ^ { \\top } w _ { h } - \\phi ( s , a ) ^ { \\top } \\widehat \\Lambda _ { h } ^ { - 1 } \\left( \\displaystyle \\sum _ { \\tau = 1 } ^ { K } \\phi \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\cdot \\phi ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) ^ { \\top } w _ { h } / \\widehat \\sigma ^ { 2 } \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\right) } \\\\ & { = \\phi ( s , a ) ^ { \\top } w _ { h } - \\phi ( s , a ) ^ { \\top } \\widehat \\Lambda _ { h } ^ { - 1 } \\left( \\widehat { \\Lambda } _ { h } - \\lambda I \\right) w _ { h } = \\lambda \\cdot \\phi ( s , a ) ^ { \\top } \\widehat \\Lambda _ { h } ^ { - 1 } w _ { h } } \\\\ & { \\leq \\lambda \\| \\phi ( s , a ) \\| _ { \\widehat \\Lambda _ { h } ^ { - 1 } } \\cdot \\| w _ { h } \\| _ { \\widehat \\Lambda _ { h } ^ { - 1 } } \\leq \\displaystyle \\frac { \\lambda } { K } \\left\\| \\phi ( s , a ) \\right\\| _ { \\{ \\widehat \\Lambda _ { h } ^ { p } } - 1 } \\cdot \\left\\| w _ { h } \\right\\| _ { ( \\widehat { \\Lambda } _ { h } ^ { p } ) ^ { - 1 } } \\\\ & { \\leq \\displaystyle \\frac { \\lambda } { K } 1 \\cdot \\sqrt { \\left\\| ( \\widehat { \\Lambda } _ { h } ^ { p } ) ^ { - 1 } \\right\\| } \\cdot 2 H \\sqrt { d } \\cdot \\sqrt { \\left\\| ( \\widehat { \\Lambda } _ { h } ^ { p } ) ^ { - 1 } \\right\\| } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 227, + 537, + 769, + 722 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "where $\\tilde { \\Lambda } _ { h } ^ { p } : = \\mathbb { E } _ { \\mu , h } \\left[ \\widehat { \\sigma } _ { h } ( s , a ) ^ { - 2 } \\phi ( s , a ) \\phi ( s , a ) ^ { \\top } \\right]$ and the second inequality is by Lemma H.6 (with $\\phi ^ { \\prime } ~ = ~ \\phi / \\widehat { \\sigma } _ { h }$ and $\\| \\phi / { \\widehat { \\sigma } } _ { h } \\| \\ \\leq \\ \\| \\phi \\| \\ \\leq \\ 1 \\ : = \\ C )$ and the third inequality uses ${ \\sqrt { a ^ { \\top } \\cdot A \\cdot a } } \\leq$ $\\sqrt { \\| a \\| _ { 2 } \\| A \\| _ { 2 } \\| a \\| _ { 2 } } = \\| a \\| _ { 2 } \\sqrt { \\| A \\| _ { 2 } }$ with $a$ to be either $\\phi$ or $w _ { h }$ . Moreover, $\\lambda _ { \\operatorname* { m i n } } ( \\tilde { \\Lambda } _ { h } ^ { p } ) \\geq$ $\\kappa / \\operatorname* { m a x } _ { h , s , a } \\widehat { \\sigma } _ { h } ( s , a ) ^ { 2 } \\geq \\kappa / H ^ { 2 }$ implies $\\left\\| ( \\tilde { \\Lambda } _ { h } ^ { p } ) ^ { - 1 } \\right\\| \\leq H ^ { 2 } / \\kappa$ , therefore for all $s , a , h \\in \\mathcal S \\times \\mathcal A \\times [ H ]$ , with probability $1 - \\delta$ ", + "bbox": [ + 171, + 751, + 826, + 842 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/4f857aef8d40f7cd5fc09a28a495f6be61ddc3fd5b4196fdf1560d7a20c5c4f0.jpg", + "text": "$$\n\\phi ( s , a ) ^ { \\top } w _ { h } - \\phi ( s , a ) ^ { \\top } \\widehat { \\Lambda } _ { h } ^ { - 1 } \\left( \\sum _ { \\tau = 1 } ^ { K } \\phi \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\cdot \\left( \\mathcal { T } _ { h } \\widehat { V } _ { h + 1 } \\right) \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) / \\widehat { \\sigma } ^ { 2 } \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\right) \\Bigg | \\leq \\frac { 2 \\lambda H ^ { 3 } \\sqrt { d } / \\kappa } { K } .\n$$", + "text_format": "latex", + "bbox": [ + 181, + 853, + 825, + 897 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "For term (ii), denote: xτ = $\\begin{array} { r } { x _ { \\tau } = \\frac { \\phi ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) } { \\widehat { \\sigma } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) } , \\quad \\eta _ { \\tau } = \\left( r _ { h } ^ { \\tau } + { \\widehat V } _ { h + 1 } \\left( s _ { h + 1 } ^ { \\tau } \\right) - \\left( { \\mathcal T } _ { h } { \\widehat V } _ { h + 1 } \\right) \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\right) / { \\widehat { \\sigma } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) } , } \\end{array}$ , bthen by Cauchy inequality it follows ", + "bbox": [ + 173, + 99, + 833, + 140 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/2c2033056f9f694d9fef6cf473b0fde71c3ac8c8e0454e443c264a78bf904ca7.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle \\left| \\phi ( s , a ) ^ { \\top } \\widehat { \\Lambda } _ { h } ^ { - 1 } \\left( \\displaystyle \\sum _ { \\tau = 1 } ^ { K } \\phi \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\cdot \\left( r _ { h } ^ { \\tau } + \\widehat { V } _ { h + 1 } \\left( s _ { h + 1 } ^ { \\tau } \\right) - \\left( \\mathcal { T } _ { h } \\widehat { V } _ { h + 1 } \\right) \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\right) / \\widehat { \\sigma } _ { h } ^ { 2 } \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\right) \\right| } \\\\ { \\leq \\sqrt { \\phi ( s , a ) ^ { \\top } \\widehat { \\Lambda } _ { h } ^ { - 1 } \\phi ( s , a ) } \\cdot \\displaystyle \\| \\displaystyle \\sum _ { \\tau = 1 } ^ { K } x _ { \\tau } \\eta _ { \\tau } \\big \\| _ { \\widehat { \\Lambda } _ { h } ^ { - 1 } } } \\\\ { \\leq \\sqrt { \\phi ( s , a ) ^ { \\top } \\widehat { \\Lambda } _ { h } ^ { - 1 } \\phi ( s , a ) } \\cdot \\displaystyle \\| \\displaystyle \\sum _ { \\tau = 1 } ^ { K } x _ { \\tau } \\eta _ { \\tau } \\big \\| _ { \\widehat { \\Lambda } _ { h } ^ { - 1 } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 191, + 157, + 808, + 246 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "C.2.1 ANALYZING THE TERM $\\sqrt { \\phi ( s , a ) \\widehat { \\Lambda } _ { h } ^ { - 1 } \\phi ( s , a ) }$ ", + "bbox": [ + 173, + 285, + 535, + 309 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Recall (in Theorem 3.2) the estimated $\\begin{array} { r } { \\widehat { \\Lambda } _ { h } = \\sum _ { \\tau = 1 } ^ { K } \\phi \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\phi \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) ^ { \\top } / \\widehat { \\sigma } ^ { 2 } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) + \\lambda \\cdot I } \\end{array}$ and $\\begin{array} { r } { \\Lambda _ { h } = \\sum _ { \\tau = 1 } ^ { K } \\phi ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) ^ { \\top } \\phi ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) / \\sigma _ { \\widehat { V } _ { h + 1 } } ^ { 2 } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) + \\lambda I } \\end{array}$ h h h h b h h . Then we have the following lemma to control the term $\\sqrt { \\phi ( s , a ) \\widehat { \\Lambda } _ { h } ^ { - 1 } \\phi ( s , a ) }$ . ", + "bbox": [ + 171, + 324, + 826, + 388 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Lemma C.3. Denote the quantities $C _ { 1 } \\ = \\ \\operatorname* { m a x } \\{ 2 \\lambda , 1 2 8 \\log ( 2 d / \\delta ) , 1 2 8 H ^ { 4 } \\log ( 2 d / \\delta ) / \\kappa ^ { 2 } \\}$ and $\\begin{array} { r } { C _ { 2 } = \\operatorname* { m a x } \\{ \\frac { \\lambda ^ { 2 } } { \\kappa \\log ( ( \\lambda + K ) H / \\lambda \\delta ) } , 9 6 ^ { 2 } H ^ { 1 2 } d \\log ( ( \\lambda + K ) H / \\lambda \\delta ) / \\kappa ^ { 5 } \\} } \\end{array}$ . Suppose the number of episode $K$ satisfies $K > \\operatorname* { m a x } \\{ C _ { 1 } , C _ { 2 } \\}$ , then with probability $1 - \\delta$ , ", + "bbox": [ + 173, + 398, + 826, + 452 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/72da16f29276fb22dbc52b9a54d7ac4808c736b5a525b13c9405ef837e5870a2.jpg", + "text": "$$\n\\sqrt { \\phi ( s , a ) \\widehat \\Lambda _ { h } ^ { - 1 } \\phi ( s , a ) } \\le 2 \\sqrt { \\phi ( s , a ) \\Lambda _ { h } ^ { - 1 } \\phi ( s , a ) } , \\quad \\forall s , a \\in \\mathcal { S } \\times \\mathcal { A } .\n$$", + "text_format": "latex", + "bbox": [ + 282, + 469, + 714, + 497 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Proof of Lemma C.3. By definition $\\sqrt { \\phi ( s , a ) \\widehat { \\Lambda } _ { h } ^ { - 1 } \\phi ( s , a ) } = \\| \\phi ( s , a ) \\| _ { \\widehat { \\Lambda } _ { h } ^ { - 1 } }$ . Then denote ", + "bbox": [ + 171, + 553, + 748, + 579 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/7315700146a1a0ac0f660a8939b706a62a638ec0fcfad1e700ceef66f404a05c.jpg", + "text": "$$\n\\widehat { \\Lambda } _ { h } ^ { \\prime } = \\frac { 1 } { K } \\widehat { \\Lambda } _ { h } , \\quad \\Lambda _ { h } ^ { \\prime } = \\frac { 1 } { K } \\Lambda _ { h } ,\n$$", + "text_format": "latex", + "bbox": [ + 401, + 597, + 593, + 627 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "where $\\begin{array} { r } { \\Lambda _ { h } \\ = \\ \\sum _ { \\tau = 1 } ^ { K } \\phi ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) ^ { \\top } \\phi ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) / \\sigma _ { \\widehat { V } _ { h + 1 } } ^ { 2 } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) + \\lambda I . } \\end{array}$ . Under the condition of $K$ , by Lemma C.7, with probability $1 - \\delta$ ", + "bbox": [ + 174, + 646, + 825, + 683 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/0b55496ca8b38f5a50b3a1b2e1c6c4587a1d6adf5eedf5c4809544b52531d8c4.jpg", + "text": "$$\n\\begin{array} { r l } & { \\left\\| \\widehat { \\Lambda } _ { h } ^ { \\prime } - \\Lambda _ { h } ^ { \\prime } \\right\\| \\leq \\underset { s , a } { \\operatorname* { s u p } } \\left\\| \\frac { \\phi ( s , a ) \\phi ( s , a ) ^ { \\top } } { \\widehat { \\sigma } _ { h } ^ { 2 } ( s , a ) } - \\frac { \\phi ( s , a ) \\phi ( s , a ) ^ { \\top } } { \\widehat { \\sigma } _ { \\widehat { V } _ { h + 1 } } ^ { 2 } ( s , a ) } \\right\\| } \\\\ & { \\leq \\underset { s , a } { \\operatorname* { s u p } } \\left| \\frac { \\widehat { \\sigma } _ { h } ^ { 2 } ( s , a ) - \\sigma _ { \\widehat { V } _ { h + 1 } } ^ { 2 } ( s , a ) } { \\widehat { \\sigma } _ { h } ^ { 2 } ( s , a ) \\sigma _ { \\widehat { V } _ { h + 1 } } ^ { 2 } ( s , a ) } \\right| \\cdot \\left\\| \\phi ( s , a ) \\right\\| ^ { 2 } \\leq \\underset { s , a } { \\operatorname* { s u p } } \\left| \\frac { \\widehat { \\sigma } _ { h } ^ { 2 } ( s , a ) - \\sigma _ { \\widehat { V } _ { h + 1 } } ^ { 2 } ( s , a ) } { 1 } \\right| \\cdot 1 } \\\\ & { \\leq 1 2 \\sqrt { \\frac { H ^ { 4 } d } { \\kappa K } \\log \\left( \\frac { \\left( \\lambda + K \\right) H } { \\lambda \\delta } \\right) } + 1 2 \\lambda \\frac { H ^ { 2 } \\sqrt { d } } { \\kappa K } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 235, + 700, + 764, + 834 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Next by Lemma H.5 (with $\\phi$ to be $\\phi / \\sigma _ { \\widehat { V } _ { h + 1 } }$ and $C = 1$ ), it holds with probability $1 - \\delta$ , ", + "bbox": [ + 169, + 851, + 751, + 871 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/930944bad7cbc297d0ab3d035137d3c0317083086eee697073e9e0f0c6c2a0de.jpg", + "text": "$$\n\\left\\| \\Lambda _ { h } ^ { \\prime } - \\left( \\mathbb { E } _ { \\mu , h } [ \\phi ( s , a ) \\phi ( s , a ) ^ { \\top } / \\sigma _ { \\hat { V } _ { h + 1 } } ^ { 2 } ( s , a ) ] + \\frac { \\lambda } { K } I _ { d } \\right) \\right\\| \\leq \\frac { 4 \\sqrt { 2 } } { \\sqrt { K } } \\left( \\log \\frac { 2 d } { \\delta } \\right) ^ { 1 / 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 238, + 891, + 758, + 929 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Therefore by Weyl’s spectrum theorem and the condition $K$ > $\\operatorname* { m a x } \\{ 2 \\lambda , 1 2 8 \\log ( 2 d / \\delta ) , 1 2 8 H ^ { 4 } \\log ( 2 d / \\delta ) / \\kappa ^ { 2 } \\} .$ , the above implies ", + "bbox": [ + 171, + 102, + 823, + 133 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/0eaef0a745df068251121ccb4815b6c111cb574fd4d9b188402e6b893f998f80.jpg", + "text": "$$\n\\begin{array} { r l } & { \\| \\Lambda _ { t } ^ { \\prime } \\| = \\lambda _ { \\operatorname* { m a x } } ( N _ { h } ) \\leq \\lambda _ { \\operatorname* { m a x } } \\Big ( \\mathrm { E } _ { \\mu h } \\big [ \\phi ( s , a ) \\phi ( s , a ) \\big ] \\Big ) + \\frac { \\lambda } { K } + \\frac { 4 \\sqrt { 2 } } { \\sqrt { K } } \\Big ( \\log \\frac { 2 d } { \\delta } \\Big ) ^ { 1 / 2 } } \\\\ & { \\qquad - \\Big \\| \\mathrm { E } _ { \\mu h } [ \\delta ( s , a ) \\phi ( s , a ) ^ { \\top } / \\sigma _ { b \\mathrm { t } \\mathrm { \\scriptsize \\ ' \\ ' } } ^ { 2 } ( s , a ) \\mathbb { S } , a ] \\Big \\| _ { 2 } + \\frac { \\lambda } { K } + \\frac { 4 \\sqrt { 2 } } { \\sqrt { K } } \\Big ( \\log \\frac { 2 d } { \\delta } \\Big ) ^ { 1 / 2 } } \\\\ & { \\qquad \\leq \\| \\phi ( s , a ) \\| ^ { 2 } + \\frac { \\lambda } { K } + \\frac { 4 \\sqrt { 2 } } { \\sqrt { K } } \\Big ( \\log \\frac { 2 d } { \\delta } \\Big ) ^ { 1 / 2 } \\leq 1 + \\frac { \\lambda } { K } + \\frac { 4 \\sqrt { 2 } } { \\sqrt { K } } \\Big ( \\log \\frac { 2 d } { \\delta } \\Big ) ^ { 1 / 2 } \\leq 2 , } \\\\ & { \\qquad \\quad \\times _ { \\operatorname* { m i n } } \\big ( \\Delta _ { h } ^ { \\prime } \\big ) \\leq \\lambda _ { \\operatorname* { m i n } } \\Big ( \\mathrm { E } _ { \\mu h } \\big ) \\big ( \\tilde { \\phi } ( s , a ) \\phi ( s , a ) ^ { \\top } / \\sigma _ { b \\mathrm { t } \\mathrm { \\scriptsize \\ ' } + 1 } ^ { 2 } ( s , a ) \\big ] \\Big ) + \\frac { \\lambda } { K } - \\frac { 4 \\sqrt { 2 } } { \\sqrt { K } } \\Big ( \\log \\frac { 2 d } { \\delta } \\Big ) ^ { 1 / 2 } } \\\\ & { \\qquad \\quad \\geq \\lambda _ { \\operatorname* { m i n } } \\Big ( \\mathrm { E } _ { \\mu h } \\big [ \\phi ( s , a ) \\phi ( s , a ) ^ { \\top } / \\sigma _ { b \\mathrm { t } \\mathrm { \\scriptsize \\ ' } + 1 } ^ { 2 } ( s , a ) \\big ] \\Big ) - \\frac { 4 \\sqrt { 2 } } { \\sqrt { K } } \\Big ( \\log \\frac { 2 d } { \\delta } \\Big ) ^ { 1 / 2 } } \\\\ & \\qquad \\quad \\geq \\frac { \\kappa } { H ^ { 2 } } - \\frac { 4 \\sqrt { 2 } } { \\sqrt { K } } \\Big ( \\log \\frac \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 184, + 143, + 821, + 376 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Hence with probability $1 - \\delta , \\| \\Lambda _ { h } ^ { \\prime } \\| \\leq 2$ and $\\left\\| \\Lambda _ { h } ^ { \\prime - 1 } \\right\\| = 1 / \\lambda _ { \\operatorname* { m i n } } ( \\Lambda _ { h } ^ { \\prime } ) \\le 2 H ^ { 2 } / \\kappa$ . Similarly, one can show $\\left\\| \\widehat { \\Lambda } _ { h } ^ { \\prime - 1 } \\right\\| \\leq 2 H ^ { 2 } / \\kappa$ with high probability. ", + "bbox": [ + 173, + 383, + 826, + 428 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Now apply Lemma H.4 to $\\widehat { \\Lambda } _ { h } ^ { \\prime }$ and $\\Lambda _ { h } ^ { \\prime }$ and a union bound, we obtain with probability $1 - \\delta$ , for all $s , a$ ", + "bbox": [ + 171, + 433, + 823, + 452 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/d5967e38fba8c718d18ae360a13bbf55703eb7ad38d01b91497cb4c7f686fce2.jpg", + "text": "$$\n\\begin{array} { r l } { \\left\\| \\phi ( s , a ) \\right\\| _ { \\widetilde { \\Lambda } _ { h } ^ { - 1 } } \\leq \\left[ 1 + \\sqrt { \\left\\| \\Lambda _ { h } ^ { \\prime - 1 } \\right\\| \\left\\| \\Lambda _ { h } ^ { \\prime } \\right\\| } \\cdot \\left\\| \\hat { \\Lambda } _ { h } ^ { \\prime - 1 } \\right\\| \\cdot \\left\\| \\hat { \\Lambda } _ { h } ^ { \\prime } - \\Lambda _ { h } ^ { \\prime } \\right\\| \\right] \\cdot \\left\\| \\phi ( s , a ) \\right\\| _ { \\Lambda _ { h } ^ { \\prime - 1 } } } & { } \\\\ { \\leq \\left[ 1 + \\sqrt { \\frac { 2 H ^ { 2 } } { \\kappa } } \\cdot 1 \\cdot \\frac { 2 H ^ { 2 } } { \\kappa } \\cdot \\left\\| \\hat { \\Lambda } _ { h } ^ { \\prime } - \\Lambda _ { h } ^ { \\prime } \\right\\| \\right] \\cdot \\left\\| \\phi ( s , a ) \\right\\| _ { \\Lambda _ { h } ^ { \\prime - 1 } } } & { } \\\\ { \\leq \\left[ 1 + \\sqrt { \\frac { 4 \\Delta H ^ { 4 } } { \\kappa ^ { 2 } } \\left( \\sqrt { \\frac { H ^ { 4 } d } { \\kappa K } } \\log \\left( \\frac { ( \\lambda + K ) H } { \\lambda \\delta } \\right) + \\lambda \\frac { H ^ { 2 } \\sqrt { d } } { \\kappa K } \\right) } \\right] \\cdot \\left\\| \\phi ( s , a ) \\right\\| _ { \\Lambda _ { h } ^ { \\prime - 1 } } } & { } \\\\ { \\leq \\left[ 1 + \\sqrt { \\frac { 9 6 H ^ { 4 } } { \\kappa ^ { 2 } } \\sqrt { \\frac { H ^ { 4 } d } { \\kappa K } } \\log \\left( \\frac { ( \\lambda + K ) H } { \\lambda \\delta } \\right) } \\right] \\cdot \\left\\| \\phi ( s , a ) \\right\\| _ { \\Lambda _ { h } ^ { \\prime - 1 } } \\leq 2 \\left\\| \\phi ( s , a ) \\right\\| _ { \\Lambda _ { h } ^ { \\prime - 1 } } } & { } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 186, + 460, + 821, + 664 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "where the third inequality uses equation 11 and the last and the second last inequality use $\\begin{array} { r } { K > \\operatorname* { m a x } \\lbrace \\frac { \\lambda ^ { 2 } } { \\kappa \\log ( ( \\lambda + K ) H / \\lambda \\delta ) } , 9 6 ^ { 2 } H ^ { 1 2 } d \\log ( ( \\lambda + K ) H / \\lambda \\delta ) / \\kappa ^ { 5 } \\rbrace . } \\end{array}$ . Note the above is equivalent to $\\sqrt { \\phi ( s , a ) \\widehat \\Lambda _ { h } ^ { - 1 } \\phi ( s , a ) } \\le 2 \\sqrt { \\phi ( s , a ) \\Lambda _ { h } ^ { - 1 } \\phi ( s , a ) }$ by multiplying $1 / \\sqrt { K }$ on both sides. ", + "bbox": [ + 173, + 689, + 826, + 750 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "C.2.2 ANALYZING THE TERM $\\begin{array} { r } { | | \\sum _ { \\tau = 1 } ^ { K } x _ { \\tau } \\eta _ { \\tau } | | _ { \\widehat { \\Lambda } ^ { - 1 } } } \\end{array}$ ", + "bbox": [ + 173, + 795, + 516, + 814 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Lemma C.4. Recall $\\begin{array} { r } { x _ { \\tau } = \\frac { \\phi ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) } { \\widehat { \\sigma } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) } } \\end{array}$ $\\begin{array} { r } { \\frac { a _ { h } ^ { \\tau } } { a _ { h } ^ { \\tau } ) } a n d \\eta _ { \\tau } = \\left( r _ { h } ^ { \\tau } + \\widehat { V } _ { h + 1 } \\left( s _ { h + 1 } ^ { \\tau } \\right) - \\left( \\mathcal { T } _ { h } \\widehat { V } _ { h + 1 } \\right) ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) \\right) / \\widehat { \\sigma } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) . } \\end{array}$ . Let $\\begin{array} { r } { C _ { H , d , \\kappa , K } : = 3 6 \\sqrt { \\frac { H ^ { 4 } d ^ { 3 } } { \\kappa } \\log \\left( \\frac { ( \\lambda + K ) 2 K d H ^ { 2 } } { \\lambda \\delta } \\right) } + 1 2 \\lambda \\frac { H ^ { 2 } \\sqrt { d } } { \\kappa } } \\end{array}$ and denote ", + "bbox": [ + 173, + 823, + 836, + 880 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/f46644a87b6e47aba6cd17b7a42b936a0b8de966101822058b87a01b8e6a98f3.jpg", + "text": "$$\n\\xi : = \\operatorname* { s u p } _ { V \\in \\left[ 0 , H \\right] , s ^ { \\prime } \\sim P _ { h } \\left( s , a \\right) , h \\in \\left[ H \\right] } \\left| \\frac { r _ { h } + V \\left( s ^ { \\prime } \\right) - \\left( \\mathcal T _ { h } V \\right) \\left( s , a \\right) } { \\sigma _ { V } \\left( s , a \\right) } \\right| .\n$$", + "text_format": "latex", + "bbox": [ + 295, + 891, + 699, + 928 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "If $K \\ge 4 C _ { H , d , \\kappa , K } ^ { 2 }$ and $K \\ge \\widetilde { \\cal O } ( H ^ { 6 } d / \\kappa ) .$ , then with probability $1 - \\delta$ , ", + "bbox": [ + 171, + 101, + 629, + 121 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/87a788e78e4a4624516bff7ea43a860cc5631671d1b210be0cd2e6b2fa86694d.jpg", + "text": "$$\n\\left\\| \\sum _ { \\tau = 1 } ^ { K } x _ { \\tau } \\eta _ { \\tau } \\right\\| _ { \\widehat { \\Lambda } ^ { - 1 } } \\leq 1 6 \\sqrt { d \\log \\left( 1 + \\frac { K } { \\lambda d } \\right) } \\cdot \\log \\left( \\frac { 4 K ^ { 2 } } { \\delta } \\right) + 4 \\xi \\log \\left( \\frac { 4 K ^ { 2 } } { \\delta } \\right) \\leq \\widetilde { O } \\operatorname* { m a x } \\left\\{ \\sqrt { d } , \\xi \\right\\} ,\n$$", + "text_format": "latex", + "bbox": [ + 187, + 130, + 805, + 174 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "where $\\widetilde O$ absorbs the constants and Polylog terms. ", + "bbox": [ + 173, + 185, + 503, + 202 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Proof of Lemma C.4. By construction, we have $\\| x _ { \\tau } \\| \\le \\| \\phi / \\widehat { \\sigma } \\| \\le 1$ and by Lemma C.7, with probability $1 - \\delta / 3$ , ", + "bbox": [ + 174, + 223, + 825, + 256 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/1a05682e6bf307b8706bc1a9a1b48122d0ebfe3bee7380fbfaf98bbd7a772305.jpg", + "text": "$$\n\\left\\| \\sigma _ { \\widehat { V } _ { h + 1 } } - \\widehat { \\sigma } _ { h } \\right\\| _ { \\infty } = \\operatorname* { s u p } _ { s , a } \\left| \\frac { \\sigma _ { \\widehat { V } _ { h + 1 } } ^ { 2 } ( s , a ) - \\widehat { \\sigma } _ { h } ^ { 2 } ( s , a ) } { \\sigma _ { \\widehat { V } _ { h + 1 } } ( s , a ) + \\widehat { \\sigma } _ { h } ( s , a ) } \\right| \\leq \\frac { 1 } { 2 } \\left\\| \\sigma _ { \\widehat { V } _ { h + 1 } } ^ { 2 } - \\widehat { \\sigma } _ { h } ^ { 2 } \\right\\| _ { \\infty } \\leq C _ { H , d , \\kappa , K } \\sqrt { \\frac { 1 } { K } }\n$$", + "text_format": "latex", + "bbox": [ + 200, + 262, + 795, + 318 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Therefore, when $K \\geq 4 C _ { H , d , \\kappa , K } ^ { 2 } , C _ { H , d , \\kappa , K } \\sqrt { \\frac { 1 } { K } } \\leq 1 / 2 \\leq \\sigma _ { \\widehat { V } _ { h + 1 } } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) / 2$ and hence ", + "bbox": [ + 169, + 327, + 743, + 352 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/649ba0c516232a49c240d91990639d87ff0b65dea268745a3f6f1f097c46ea3d.jpg", + "text": "$$\n\\begin{array} { r l } & { \\eta _ { \\tau } | \\leq \\left| \\frac { r _ { h } ^ { \\tau } + \\widehat { V } _ { h + 1 } \\left( s _ { h + 1 } ^ { \\tau } \\right) - \\left( \\mathcal { T } _ { h } \\widehat { V } _ { h + 1 } \\right) \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) } { \\sigma _ { \\widehat { V } _ { h + 1 } } \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) - \\frac { C _ { H , d , \\kappa , K } } { K ^ { 1 / 2 } } } \\right| \\leq 2 \\left| \\frac { r _ { h } ^ { \\tau } + \\widehat { V } _ { h + 1 } \\left( s _ { h + 1 } ^ { \\tau } \\right) - \\left( \\mathcal { T } _ { h } \\widehat { V } _ { h + 1 } \\right) \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) } { \\sigma _ { \\widehat { V } _ { h + 1 } } \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) } \\right| } \\\\ & { \\quad \\leq 2 \\underset { V \\in [ 0 , H ] , s ^ { \\prime } \\sim P _ { h } ( s , a ) } { \\operatorname* { s u p } } \\left| \\frac { r + V \\left( s ^ { \\prime } \\right) - \\left( \\mathcal { T } _ { h } V \\right) \\left( s , a \\right) } { \\sigma _ { V } \\left( s , a \\right) } \\right| : = \\xi . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 361, + 825, + 450 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Next, for a fixed function $V$ , we define the Bellman error as $B _ { h } ( V ) ( s , a ) = r _ { h } + V ( s ^ { \\prime } ) - ( T _ { h } V ) ( s , a )$ , then ", + "bbox": [ + 173, + 477, + 826, + 507 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/7647db2ac043d0446feb6e873c5a35be596580529f0e35ed83643835f0b5eb85.jpg", + "text": "$$\n\\begin{array} { r l } & { - \\frac { \\nabla \\cdot \\mathbf { S } _ { \\perp + } [ \\hat { \\mathbf { x } } _ { \\perp } + \\hat { \\mathbf { x } } _ { \\perp } - ( \\hat { \\mathbf { x } } _ { \\perp } - [ ( \\hat { \\mathbf { x } } _ { \\perp } \\hat { \\mathbf { e } } _ { \\perp + } ^ { - \\mathbf { A } } ) ] ) \\cdot \\mathbf { \\bar { x } } _ { \\perp } \\hat { \\mathbf { e } } _ { \\perp } ^ { - \\mathbf { A } } ] \\cdot [ \\hat { \\mathbf { x } } _ { \\perp - } ] } { \\hat { \\mathcal { Q } } _ { \\perp + } [ \\hat { \\mathbf { x } } _ { \\perp } + \\hat { \\mathbf { x } } _ { \\perp } ] } } \\\\ & { - \\frac { \\nabla \\cdot \\mathbf { S } _ { \\perp + } [ \\hat { \\mathbf { x } } _ { \\perp } + \\hat { \\mathbf { x } } _ { \\perp } - \\hat { \\mathbf { x } } _ { \\perp } ] } { 2 } \\cdot \\frac { \\partial \\hat { \\mathbf { x } } _ { \\perp + } [ \\hat { \\mathbf { x } } _ { \\perp } + \\hat { \\mathbf { x } } _ { \\perp } ] \\cdot ( \\hat { \\mathbf { x } } _ { \\perp } - \\hat { \\mathbf { x } } _ { \\perp } ) } { \\partial \\hat { \\mathbf { x } } _ { \\perp } [ \\hat { \\mathbf { x } } _ { \\perp } + \\hat { \\mathbf { x } } _ { \\perp } ] } [ \\hat { \\mathbf { x } } _ { \\perp } \\hat { \\mathbf { e } } _ { \\perp + } ^ { \\mathbf { A } } ] \\hat { \\mathcal { Q } } _ { \\perp - } \\Bigg ] } \\\\ & \\phantom { = } \\frac { \\nabla \\cdot \\mathbf { S } _ { \\perp + } [ \\hat { \\mathbf { x } } _ { \\perp + } ^ { \\mathbf { B } } \\cdot \\mathbf { \\bar { x } } _ { \\perp } \\hat { \\mathbf { e } } _ { \\perp + } ^ { \\mathbf { B } } ] \\cdot [ \\hat { \\mathbf { x } } _ { \\perp } \\hat { \\mathbf { e } } _ { \\perp + } ^ { \\mathbf { B } } ] \\cdot [ \\hat { \\mathbf { x } } _ { \\perp - } \\mathbf { \\bar { x } } _ { \\perp } \\hat { \\mathbf { x } } _ { \\perp } \\hat { \\mathbf { x } } _ { \\perp } ] \\hat { \\mathcal { Q } } _ { \\perp - } } \\\\ & \\phantom { = } \\frac \\nabla \\cdot \\mathbf { S } _ { \\perp + } [ \\hat { \\mathbf { x } } _ \\ \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 264, + 516, + 839, + 832 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "where the first inequality is by Lemma H.11, the second inequality is by $\\mathcal { T } _ { h }$ is non-expansive, the third inequality is by Lemma C.8, the next equality is by Markovian property, and the fourth inequality is by Lemma C.7 and Lemma C.10. The fifth inequality uses definition $\\sigma _ { h , V } ( s , a ) ^ { 2 } : =$ $\\operatorname* { m a x } \\{ 1 , \\operatorname { V a r } _ { P _ { h } } ( V ) ( s , a ) \\}$ and the last one is by condition $K \\ge \\widetilde { O } ( H ^ { 6 } d / \\kappa )$ and $\\sigma _ { h , V ^ { \\star } } ( s , a ) ^ { 2 } : =$ $\\operatorname* { n a x } \\{ 1 , \\operatorname { V a r } _ { P _ { h } } ( V ^ { \\star } ) ( s , a ) \\} \\ \\geq \\ 1$ . Thus, by Bernstein inequality for self-normalized martingale ", + "bbox": [ + 173, + 849, + 826, + 925 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "(Lemma H.3),9 with probability $1 - \\delta$ , ", + "bbox": [ + 173, + 102, + 429, + 118 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/c035e458bda461595ff07b2ae3671e844ae1bf072f841fda7aa562c98230b031.jpg", + "text": "$$\n\\left\\| \\sum _ { \\tau = 1 } ^ { K } x _ { \\tau } \\eta _ { \\tau } \\right\\| _ { \\widehat { \\Lambda } ^ { - 1 } } \\leq \\widetilde { \\mathcal { O } } \\left( \\sqrt { d \\log \\left( 1 + \\frac { K } { \\lambda d } \\right) } \\cdot \\log \\left( \\frac { 4 K ^ { 2 } } { \\delta } \\right) \\right) + 4 \\xi \\log \\left( \\frac { 4 K ^ { 2 } } { \\delta } \\right) \\leq \\widetilde { \\mathcal { O } } \\operatorname* { m a x } \\left\\{ \\sqrt { d } , \\xi \\right\\}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 126, + 818, + 171 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "where $\\widetilde O$ absorbs the constants and Polylog terms. ", + "bbox": [ + 173, + 181, + 501, + 198 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Recall $\\mathcal { M } _ { 1 } , \\mathcal { M } _ { 2 } , \\mathcal { M } _ { 3 } , \\mathcal { M } _ { 4 }$ in List A. Based on the above results, we have the following key lemma: ", + "bbox": [ + 173, + 218, + 825, + 234 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Lemma C.5. Assume $K > \\operatorname* { m a x } \\{ \\mathcal { M } _ { 1 } , \\mathcal { M } _ { 2 } , \\mathcal { M } _ { 3 } , \\mathcal { M } _ { 4 } \\}$ , for any $0 < \\lambda < \\kappa ,$ , suppose $\\sqrt { d } > \\xi$ , $\\xi : = \\operatorname* { s u p } _ { V \\in [ 0 , H ] , \\ s ^ { \\prime } \\sim P _ { h } ( s , a ) , \\ h \\in [ H ] } \\left| \\frac { r _ { h } + V \\big ( s ^ { \\prime } \\big ) - ( T _ { h } V ) ( s , a ) } { \\sigma _ { V } ( s , a ) } \\right| .$ Then with probability $1 - \\delta _ { : }$ , for all $h , s , a \\in [ H ] \\times \\mathcal { S } \\times \\mathcal { A } ,$ , ", + "bbox": [ + 173, + 239, + 826, + 303 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/5ca4ed36e8c44d91c97ff962c4ad7623fd3c949f127a55a8acddea465251d4ab.jpg", + "text": "$$\n\\left| ( { \\cal T } _ { h } \\widehat V _ { h + 1 } - \\widehat T _ { h } \\widehat V _ { h + 1 } ) ( s , a ) \\right| \\leq \\widetilde O \\left( \\sqrt { d } \\sqrt { \\phi ( s , a ) \\Lambda _ { h } ^ { - 1 } \\phi ( s , a ) } \\right) + \\frac { 2 H ^ { 3 } \\sqrt { d } } { K } ,\n$$", + "text_format": "latex", + "bbox": [ + 253, + 310, + 743, + 345 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "where $\\begin{array} { r } { \\Lambda _ { h } = \\sum _ { \\tau = 1 } ^ { K } \\phi ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) ^ { \\top } \\phi ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) / \\sigma _ { \\widehat { V } _ { h + 1 } } ^ { 2 } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) + \\lambda I } \\end{array}$ and $\\widetilde O$ absorbs the universal constants and Polylog terms. ", + "bbox": [ + 173, + 353, + 823, + 391 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Proof of Lemma C.5. Combing equation 9, Lemma C.2, equation 10, Lemma C.3 and C.4 and a union bound to finish the proof. □ ", + "bbox": [ + 174, + 410, + 825, + 440 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "C.3 PROOF OF THE FIRST PART OF THEOREM 3.2 ", + "text_level": 1, + "bbox": [ + 173, + 458, + 526, + 474 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Theorem C.6 (First part of Theorem 3.2). Let $K$ be the number of episodes. Suppose $\\sqrt { d } > \\xi$ , where $\\begin{array} { r } { \\xi : = \\operatorname* { s u p } _ { V \\in [ 0 , H ] , \\ s ^ { \\prime } \\sim P _ { h } ( s , a ) , \\ h \\in [ H ] } \\left| \\frac { r _ { h } + V \\left( s ^ { \\prime } \\right) - ( \\mathcal T _ { h } V ) ( s , a ) } { \\sigma _ { V } ( s , a ) } \\right| } \\end{array}$ and $K > \\operatorname* { m a x } \\{ \\mathcal { M } _ { 1 } , \\mathcal { M } _ { 2 } , \\mathcal { M } _ { 3 } , \\mathcal { M } _ { 4 } \\} ^ { 1 0 }$ . Then for any $0 < \\lambda < \\kappa$ , with probability $1 - \\delta$ , for all policy $\\pi$ simultaneously, the output $\\widehat { \\pi }$ of Algorithm $^ { l }$ satisfies ", + "bbox": [ + 173, + 483, + 826, + 559 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/4163ed88fbfb0446b698a10f32143c31548b24a7dda8bc846707cd7f2e1e8390.jpg", + "text": "$$\nv ^ { \\pi } - v ^ { \\widehat { \\pi } } \\leq \\widetilde { O } \\left( \\sqrt { d } \\cdot \\sum _ { h = 1 } ^ { H } \\mathbb { E } _ { \\pi } \\left[ \\left( \\phi ( \\cdot , \\cdot ) ^ { \\top } \\Lambda _ { h } ^ { - 1 } \\phi ( \\cdot , \\cdot ) \\right) ^ { 1 / 2 } \\right] \\right) + \\frac { 2 H ^ { 4 } \\sqrt { d } } { K }\n$$", + "text_format": "latex", + "bbox": [ + 274, + 568, + 723, + 611 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "where $\\begin{array} { r } { \\Lambda _ { h } = \\sum _ { \\tau = 1 } ^ { K } \\frac { \\phi ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) \\cdot \\phi ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) ^ { \\top } } { \\sigma _ { \\hat { V } _ { h + 1 } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) } ^ { 2 } } + \\lambda I _ { d } } \\end{array}$ 1 φ(sτh,aτh)·φ(sτh,aτh)>2 + λId and O absorbs the universal constants and the Polylog terms. ", + "bbox": [ + 174, + 619, + 826, + 660 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Proof of Theorem C.6. Combing Lemma C.1 and Lemma C.5, we directly have with probability $1 - \\delta$ , for all policy $\\pi$ simultaneously, ", + "bbox": [ + 171, + 680, + 823, + 710 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/a0a189412cb0543c770e196f77cccb7d9349335e0a73429fc8248c1912cd7bed.jpg", + "text": "$$\nV _ { 1 } ^ { \\pi } ( s ) - V _ { 1 } ^ { \\widehat { \\pi } } ( s ) \\leq \\widetilde { O } \\left( \\sqrt { d } \\cdot \\sum _ { h = 1 } ^ { H } \\mathbb { E } _ { \\pi } \\left[ \\left( \\phi ( \\cdot , \\cdot ) ^ { \\top } \\Lambda _ { h } ^ { - 1 } \\phi ( \\cdot , \\cdot ) \\right) ^ { 1 / 2 } \\middle | s _ { 1 } = s \\right] \\right) + \\frac { 2 H ^ { 4 } \\sqrt { d } } { K } ,\n$$", + "text_format": "latex", + "bbox": [ + 209, + 718, + 761, + 762 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "now take the initial distribution $d _ { 1 }$ on both sides to get the stated result. ", + "bbox": [ + 174, + 770, + 642, + 785 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "C.4 TWO INTERMEDIATE RESULTS ", + "text_level": 1, + "bbox": [ + 176, + 803, + 428, + 818 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "The next two lemmas provide intermediate results in finishing the whole proofs. ", + "bbox": [ + 176, + 829, + 696, + 845 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "C.4.1 BOUNDING THE VARIANCE ", + "text_level": 1, + "bbox": [ + 174, + 103, + 419, + 118 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Lemma C.7. Recall the definition $\\widehat { \\sigma } _ { h } ( \\cdot , \\cdot ) ^ { 2 } = \\operatorname* { m a x } \\{ 1 , \\widehat { \\mathrm { V a r } } _ { P _ { h } } \\widehat { V } _ { h + 1 } ( \\cdot , \\cdot ) \\} + 1$ and $\\sigma _ { \\widehat { V } _ { h + 1 } } ( \\cdot , \\cdot ) ^ { 2 } : =$ $\\operatorname * { m a x } \\{ 1 , \\mathrm { V a r } _ { P _ { h } } \\widehat { V } _ { h + 1 } ( \\cdot , \\cdot ) \\} \\ + \\ 1 .$ Moreover, $\\begin{array} { r l r } { \\left[ \\widehat { \\mathrm { V a r } } _ { h } \\widehat { V } _ { h + 1 } \\right] ( \\cdot , \\cdot ) } & { { } = } & { \\big \\langle \\phi ( \\cdot , \\cdot ) , \\bar { \\beta } _ { h } \\big \\rangle _ { \\left[ 0 , ( H - h + 1 ) ^ { 2 } \\right] } - } \\end{array}$ $\\left[ \\left. \\phi ( \\cdot , \\cdot ) , \\bar { \\theta } _ { h } \\right. _ { [ 0 , H - h + 1 ] } \\right] ^ { 2 }$ (where $\\bar { \\beta } _ { h }$ and ${ \\bar { \\theta } } _ { h }$ are defined in Algorithm 1). Let $K \\_ { \\mathbf { \\alpha } } \\geq$ max $\\textstyle \\left\\{ 5 1 2 ( 1 / \\kappa ) ^ { 2 } \\log \\left( { \\frac { 4 H d } { \\delta } } \\right) , 4 \\lambda / \\kappa \\right\\}$ , then with probability $1 - \\delta$ , ", + "bbox": [ + 171, + 125, + 826, + 205 + ], + "page_idx": 22 + }, + { + "type": "equation", + "img_path": "images/4959c8b6eeb0f2e8c592f6e868a69cc424c54159b8eb9bf92754632b30b8f55f.jpg", + "text": "$$\n\\operatorname* { s u p } _ { h } \\lvert | \\widehat { \\sigma } _ { h } ^ { 2 } - \\sigma _ { \\widehat { V } _ { h + 1 } } ^ { 2 } | | _ { \\infty } \\leq 3 6 \\sqrt { \\frac { H ^ { 4 } d ^ { 3 } } { \\kappa K } \\log \\left( \\frac { ( \\lambda + K ) 2 K d H ^ { 2 } } { \\lambda \\delta } \\right) } + 1 2 \\lambda \\frac { H ^ { 2 } \\sqrt { d } } { \\kappa K } .\n$$", + "text_format": "latex", + "bbox": [ + 250, + 208, + 746, + 251 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Proof. Step1: we first show for all $h , s , a \\in [ H ] \\times \\mathcal { S } \\times \\mathcal { A }$ , with probability $1 - \\delta$ ", + "bbox": [ + 173, + 262, + 710, + 280 + ], + "page_idx": 22 + }, + { + "type": "equation", + "img_path": "images/76e06e7e5a40551abe4d1ea2c771c1c951fd803010230e4eaec573f0929b67f4.jpg", + "text": "$$\n\\langle \\phi ( s , a ) , \\bar { \\beta } _ { h } \\rangle _ { [ 0 , ( H - h + 1 ) ^ { 2 } ] } - \\mathbb { P } _ { h } ( \\widehat { V } _ { h + 1 } ) ^ { 2 } ( s , a ) \\bigg | \\leq 1 2 \\sqrt { \\frac { H ^ { 4 } d ^ { 3 } } { \\kappa K } \\log \\left( \\frac { ( \\lambda + K ) 2 K d H ^ { 2 } } { \\lambda \\delta } \\right) } + 4 \\lambda \\frac { H ^ { 2 } \\sqrt { d } } { \\kappa K } .\n$$", + "text_format": "latex", + "bbox": [ + 181, + 284, + 826, + 325 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Proof of Step1. Note ", + "text_level": 1, + "bbox": [ + 173, + 335, + 318, + 351 + ], + "page_idx": 22 + }, + { + "type": "equation", + "img_path": "images/d4ae04fbf2d87610ece95c037c58b17692bfea472e6b047855949db1679bac79.jpg", + "text": "$$\n\\begin{array} { r l } & { | \\phi ( s , a ) , \\widetilde { \\beta } _ { h } [ 0 , ( J - h + 1 ) ^ { 2 } ] - \\widetilde { \\Gamma } _ { h } ( \\widehat { V } _ { h + 1 } ) ^ { 2 } ( s , a ) | \\leq | \\phi ( s , a ) , \\widetilde { \\beta } _ { h } - \\widetilde { \\Gamma } _ { h } ( \\widehat { V } _ { h + 1 } ) ^ { 2 } ( s , a ) | } \\\\ & { = | \\phi ( s , a ) ^ { \\top } \\widetilde { \\Sigma } _ { h } ^ { - 1 } \\displaystyle \\sum _ { \\tau = 1 } ^ { K } \\phi ( \\widehat { s } _ { h } ^ { \\top } , \\widetilde { a } _ { h } ^ { \\top } ) \\cdot \\widehat { V } _ { h + 1 } ( \\widehat { s } _ { h + 1 } ^ { \\top } ) ^ { 2 } - \\widetilde { \\Gamma } _ { h } ( \\widehat { V } _ { h + 1 } ) ^ { 2 } ( s , a ) | } \\\\ & { = | \\phi ( s , a ) ^ { \\top } \\widetilde { \\Sigma } _ { h } ^ { - 1 } \\displaystyle \\sum _ { \\tau = 1 } ^ { K } \\phi ( \\widehat { s } _ { h } ^ { \\top } , \\widetilde { a } _ { h } ^ { \\top } ) \\cdot \\widehat { V } _ { h + 1 } ( \\widehat { s } _ { h + 1 } ^ { \\top } ) ^ { 2 } - \\phi ( s , a ) ^ { \\top } \\int _ { \\gamma } ( \\widehat { V } _ { h + 1 } ) ^ { 2 } ( s ^ { \\top } ) d r _ { h } ( s ^ { \\top } ) | } \\\\ & { = | \\phi ( s , a ) ^ { \\top } \\widetilde { \\Sigma } _ { h } ^ { - 1 } \\displaystyle \\sum _ { \\tau = 1 } ^ { K } \\phi ( \\widehat { s } _ { h } ^ { \\top } , a ) ^ { \\top } \\cdot \\widehat { V } _ { h + 1 } ( \\widehat { s } _ { h + 1 } ^ { \\top } ) ^ { 2 } - \\phi ( s , a ) ^ { \\top } \\displaystyle \\sum _ { \\tau = 1 } ^ { K } \\phi ( \\widehat { s } _ { h } ^ { \\top } , a _ { h } ^ { \\top } ) ( \\phi ( \\widehat { s } _ { h } ^ { \\top } , a ) ^ { \\top } ) ^ { \\top } + \\lambda I ) \\int _ { \\mathcal { O } } ( \\widehat { V } _ { h + 1 } ) ^ { 2 } ( s ^ { \\top } ) d r _ { h } } \\\\ & \\ : \\sum _ { \\tau = 1 } ^ { K } | \\phi ( s , a ) ^ { \\top } \\widetilde { \\Sigma } _ h \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 354, + 839, + 560 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "For $\\textcircled{2}$ , since $\\begin{array} { r } { K \\ge \\operatorname* { m a x } \\left\\{ 5 1 2 ( 1 / \\kappa ) ^ { 2 } \\log \\left( \\frac { 4 H d } { \\delta } \\right) , 4 \\lambda / \\kappa \\right\\} } \\end{array}$ , by Lemma H.6 and a union bound over $h \\in [ H ]$ , with probability $1 - \\delta$ for all $h , s , a \\in [ H ] \\times \\mathcal { S } \\times \\mathcal { A }$ , ", + "bbox": [ + 173, + 573, + 823, + 606 + ], + "page_idx": 22 + }, + { + "type": "equation", + "img_path": "images/23376ab2f63991b35ca9ec8e192ec4b0189c2fa854baf0a5f5969b62bbfb1aba.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle \\mathfrak { D } \\le \\lambda \\| \\phi ( s , a ) \\| _ { \\bar { \\Sigma } _ { h } ^ { - 1 } } \\left\\| \\int _ { S } ( \\widehat { V } _ { h + 1 } ) ^ { 2 } ( s ^ { \\prime } ) d \\nu _ { h } ( s ^ { \\prime } ) \\right\\| _ { \\bar { \\Sigma } _ { h } ^ { - 1 } } } \\\\ { \\displaystyle \\quad \\le \\lambda \\frac { 2 } { \\sqrt { K } } \\| \\phi ( s , a ) \\| _ { ( \\Sigma _ { h } ^ { p } ) ^ { - 1 } } \\frac { 2 } { \\sqrt { K } } \\left\\| \\int _ { S } ( \\widehat { V } _ { h + 1 } ) ^ { 2 } ( s ^ { \\prime } ) d \\nu _ { h } ( s ^ { \\prime } ) \\right\\| _ { ( \\Sigma _ { h } ^ { p } ) ^ { - 1 } } \\le 4 \\lambda \\left\\| ( \\Sigma _ { h } ^ { p } ) ^ { - 1 } \\right\\| \\frac { H ^ { 2 } \\sqrt { d } } { K } \\le 4 \\lambda \\frac { H ^ { 2 } \\vee \\lambda } { \\kappa K } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 609, + 839, + 689 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "For $\\textcircled{1}$ , we have ", + "bbox": [ + 173, + 705, + 281, + 720 + ], + "page_idx": 22 + }, + { + "type": "equation", + "img_path": "images/9b8bbc961017fefa852e03bb274a44859133d2cfb6014ec838728bfaac9f8d25.jpg", + "text": "$$\n\\mathbb { O } \\leq \\| \\phi ( s , a ) \\| _ { \\bar { \\Sigma } _ { h } ^ { - 1 } } \\left\\| \\sum _ { \\tau = 1 } ^ { K } \\phi ( \\bar { s } _ { h } ^ { \\tau } , \\bar { a } _ { h } ^ { \\tau } ) \\cdot \\Big ( \\widehat { V } _ { h + 1 } ( \\bar { s } _ { h + 1 } ^ { \\tau } ) ^ { 2 } - \\mathbb { P } _ { h } ( \\widehat { V } _ { h + 1 } ) ^ { 2 } ( \\bar { s } _ { h } ^ { \\tau } , \\bar { a } _ { h } ^ { \\tau } ) \\Big ) \\right\\| _ { \\bar { \\Sigma } _ { h } ^ { - 1 } }\n$$", + "text_format": "latex", + "bbox": [ + 235, + 724, + 764, + 768 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Bounding using covering. Note for any fix $V _ { h + 1 }$ , we can define $x _ { \\tau } = \\phi ( \\bar { s } _ { h } ^ { \\tau } , \\bar { a } _ { h } ^ { \\tau } )$ $( \\| \\phi \\| _ { 2 } \\leq 1 )$ and $\\eta _ { \\tau } = V _ { h + 1 } ( \\bar { s } _ { h + 1 } ^ { \\tau } ) ^ { 2 } - \\mathbb { P } _ { h } ( V _ { h + 1 } ) ^ { 2 } ( \\bar { s } _ { h } ^ { \\tau } , \\bar { a } _ { h } ^ { \\tau } )$ is $H ^ { 2 }$ -subgaussian, by Lemma H.2 (where $t = K$ and $L = 1$ ) with probability $1 - \\delta$ , ", + "bbox": [ + 173, + 773, + 823, + 819 + ], + "page_idx": 22 + }, + { + "type": "equation", + "img_path": "images/8eabf166d25b6b7a883df2c8e47d32bb173413ae33e8c7d4be815ef403a7ff28.jpg", + "text": "$$\n\\left\\| \\sum _ { \\tau = 1 } ^ { K } \\phi ( \\bar { s } _ { h } ^ { \\tau } , \\bar { a } _ { h } ^ { \\tau } ) \\cdot \\left( V _ { h + 1 } \\big ( \\bar { s } _ { h + 1 } ^ { \\tau } \\big ) ^ { 2 } - \\mathbb { P } _ { h } \\big ( V _ { h + 1 } \\big ) ^ { 2 } \\big ( \\bar { s } _ { h } ^ { \\tau } , \\bar { a } _ { h } ^ { \\tau } \\big ) \\right) \\right\\| _ { \\bar { \\Sigma } _ { h } ^ { - 1 } } \\leq \\sqrt { 8 H ^ { 4 } \\cdot \\frac d 2 \\log \\left( \\frac { \\lambda + K } { \\lambda \\delta } \\right) }\n$$", + "text_format": "latex", + "bbox": [ + 223, + 823, + 772, + 866 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "let $\\mathcal { N } _ { h } ( \\epsilon )$ be the minimal $\\epsilon$ -cover (with respect the supremum norm) of $\\mathcal { V } _ { h } : = \\{ V _ { h } : V _ { h } ( \\cdot ) =$ $\\begin{array} { r } { \\operatorname* { m a x } _ { a \\in { \\cal A } } \\{ \\operatorname* { m i n } \\{ \\phi ( s , a ) ^ { \\top } \\theta - C _ { 1 } \\sqrt { d \\cdot \\phi ( \\cdot , \\cdot ) ^ { \\top } \\widehat \\Lambda _ { h } ^ { - 1 } \\phi ( \\cdot , \\cdot ) } - C _ { 2 } , H - h + 1 \\} ^ { + } \\} \\} . } \\end{array}$ . That is, for any ", + "bbox": [ + 173, + 880, + 826, + 929 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "$V \\in \\mathcal { V } _ { h }$ , there exists a value function $V ^ { \\prime } \\in \\mathcal { N } _ { h } ( \\epsilon )$ such that $\\begin{array} { r } { \\operatorname* { s u p } _ { s \\in \\mathcal { S } } | V ( s ) - V ^ { \\prime } ( s ) | < \\epsilon . } \\end{array}$ . Now by a union bound, we obtain with probability $1 - \\delta$ ", + "bbox": [ + 173, + 102, + 826, + 131 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/3806d4eaa019bad60be45633e388e1d2392273c77c301aeea4192dbc48c2e2ae.jpg", + "text": "$$\n\\operatorname* { s u p } _ { \\tilde { h } + 1 \\in \\mathcal { N } _ { h + 1 } ( \\epsilon ) } \\left\\| \\sum _ { \\tau = 1 } ^ { K } \\phi ( \\bar { s } _ { h } ^ { \\tau } , \\bar { a } _ { h } ^ { \\tau } ) \\cdot \\left( V _ { h + 1 } ( \\bar { s } _ { h + 1 } ^ { \\tau } ) ^ { 2 } - \\mathbb { P } _ { h } ( V _ { h + 1 } ) ^ { 2 } ( \\bar { s } _ { h } ^ { \\tau } , \\bar { a } _ { h } ^ { \\tau } ) \\right) \\right\\| _ { \\bar { \\Sigma } _ { h } ^ { - 1 } } \\leq \\sqrt { 8 H ^ { 4 } \\cdot \\frac { d } { 2 } \\log \\left( \\frac { \\lambda + K } { \\lambda \\delta } | \\mathcal { N } _ { h + 1 } ( 6 , 7 ) | \\bar { \\mu } _ { h } ( 7 ) | \\right) }\n$$", + "text_format": "latex", + "bbox": [ + 181, + 136, + 839, + 179 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "which implies ", + "bbox": [ + 173, + 184, + 267, + 199 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/719fdc0fafcc2b68173fffa41a3576c9a6010c3c65875232d2fbb6fa86bcdec4.jpg", + "text": "$$\n\\begin{array} { r l } & { \\displaystyle \\left\\| \\displaystyle \\sum _ { \\tau = 1 } ^ { K } \\phi ( \\bar { s } _ { h } ^ { \\tau } , \\bar { a } _ { h } ^ { \\tau } ) \\cdot \\Big ( \\widehat { V } _ { h + 1 } ( \\bar { s } _ { h + 1 } ^ { \\tau } ) ^ { 2 } - \\mathbb { P } _ { h } ( \\widehat { V } _ { h + 1 } ) ^ { 2 } ( \\bar { s } _ { h } ^ { \\tau } , \\bar { a } _ { h } ^ { \\tau } ) \\Big ) \\right\\| _ { \\bar { \\Sigma } _ { h } ^ { - 1 } } } \\\\ & { \\displaystyle \\leq \\sqrt { 8 H ^ { 4 } \\cdot \\frac { d } { 2 } \\log \\Big ( \\frac { \\lambda + K } { \\lambda \\delta } | \\mathcal { N } _ { h + 1 } ( \\epsilon ) | \\Big ) } + 4 H ^ { 2 } \\sqrt { \\epsilon ^ { 2 } K ^ { 2 } / \\lambda } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 292, + 204, + 707, + 295 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "choosing $\\epsilon = d \\sqrt { \\lambda } / K$ , applying Lemma B.3 of Jin et al. (2021b)11 to the covering number $\\mathcal { N } _ { h + 1 } ( \\epsilon )$ w.r.t. $\\nu _ { h + 1 }$ , we can further bound above by ", + "bbox": [ + 173, + 300, + 826, + 330 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/fc2c575702cd3610d423bbb6d294737b34f8270a6637ef57e2f7fd3dc379aaaf.jpg", + "text": "$$\n\\leq \\sqrt { 8 H ^ { 4 } \\cdot \\frac { d ^ { 3 } } { 2 } \\log \\left( \\frac { \\lambda + K } { \\lambda \\delta } 2 d H K \\right) } + 4 H ^ { 2 } \\sqrt { d ^ { 2 } } \\leq 6 \\sqrt { H ^ { 4 } \\cdot d ^ { 3 } \\log \\left( \\frac { \\lambda + K } { \\lambda \\delta } 2 d H K \\right) }\n$$", + "text_format": "latex", + "bbox": [ + 217, + 334, + 781, + 377 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Apply a union bound for $h \\in [ H ]$ , we have with probability $1 - \\delta$ , for all $h \\in [ H ]$ , ", + "bbox": [ + 173, + 388, + 714, + 406 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/f705aea47bb52b7d6aea94b2fa2828d3be33531ee0be36f4f90e9ad89d3dd218.jpg", + "text": "$$\n\\left| \\sum _ { \\tau = 1 } ^ { K } \\phi ( \\bar { s } _ { h } ^ { \\tau } , \\bar { a } _ { h } ^ { \\tau } ) \\cdot \\left( \\widehat { V } _ { h + 1 } ( \\bar { s } _ { h + 1 } ^ { \\tau } ) ^ { 2 } - \\mathbb { P } _ { h } ( \\widehat { V } _ { h + 1 } ) ^ { 2 } ( \\bar { s } _ { h } ^ { \\tau } , \\bar { a } _ { h } ^ { \\tau } ) \\right) \\right| _ { \\bar { \\Sigma } _ { h } ^ { - 1 } } \\leq 6 \\sqrt { H ^ { 4 } d ^ { 3 } \\log \\left( \\frac { ( \\lambda + K ) 2 K d H ^ { 2 } } { \\lambda \\delta } \\right) }\n$$", + "text_format": "latex", + "bbox": [ + 181, + 410, + 828, + 458 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "and similar to $\\textcircled{2}$ , with probability $1 - \\delta$ for all $h , s , a \\in [ H ] \\times \\mathcal { S } \\times \\mathcal { A }$ , ", + "bbox": [ + 173, + 467, + 637, + 484 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/85fbd2fd64ba5f536e7c4cb62dcd3c083b8df47cc3300324d11c0a6bf19910af.jpg", + "text": "$$\n\\left. \\phi ( s , a ) \\right. _ { \\bar { \\Sigma } _ { h } ^ { - 1 } } \\leq \\frac { 2 \\left. ( \\Sigma _ { h } ^ { p } ) ^ { - 1 } \\right. ^ { 1 / 2 } } { \\sqrt { K } } \\leq \\frac { 2 } { \\sqrt { \\kappa K } } .\n$$", + "text_format": "latex", + "bbox": [ + 357, + 488, + 643, + 529 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Combing equation 13, equation 14, equation 15 and equation 16 we obtain with probability $1 - \\delta$ for all $h , s , a \\in [ H ] \\times \\mathcal { S } \\times \\mathcal { A }$ , ", + "bbox": [ + 173, + 532, + 823, + 563 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/412ac7e7c817e310b1900749746da9ab1dff50880e790abb89b290c5114fe35d.jpg", + "text": "$$\n\\langle \\phi ( s , a ) , \\bar { \\beta } _ { h } \\rangle _ { [ 0 , ( H - h + 1 ) ^ { 2 } ] } - \\mathbb { P } _ { h } ( \\widehat { V } _ { h + 1 } ) ^ { 2 } ( s , a ) \\bigg | \\leq 1 2 \\sqrt { \\frac { H ^ { 4 } d ^ { 3 } } { \\kappa K } \\log \\left( \\frac { ( \\lambda + K ) 2 K d H ^ { 2 } } { \\lambda \\delta } \\right) } + 4 \\lambda \\frac { H ^ { 2 } \\sqrt { d } } { \\kappa K } .\n$$", + "text_format": "latex", + "bbox": [ + 181, + 566, + 826, + 609 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Step2: we show for all $h , s , a \\in [ H ] \\times \\mathcal { S } \\times \\mathcal { A }$ , with probability $1 - \\delta$ ", + "bbox": [ + 173, + 619, + 632, + 637 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/160c57c3069119f3c5936d92d696dd0d6d1fe9a95e671cceec47f5f5d2e65577.jpg", + "text": "$$\n\\left| \\langle \\phi ( s , a ) , \\bar { \\theta } _ { h } \\rangle _ { [ 0 , H - h + 1 ] } - \\mathbb { P } _ { h } ( \\widehat { V } _ { h + 1 } ) ( s , a ) \\right| \\leq 1 2 \\sqrt { \\frac { H ^ { 2 } d ^ { 3 } } { \\kappa K } \\log \\left( \\frac { ( \\lambda + K ) 2 K d H ^ { 2 } } { \\lambda \\delta } \\right) } + 4 \\lambda \\frac { H \\sqrt { d } } { \\kappa K } .\n$$", + "text_format": "latex", + "bbox": [ + 184, + 642, + 813, + 684 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "The proof of Step2 follows nearly the identical way as Step1 except $\\widehat { V } _ { h } ^ { 2 }$ is replaced by $\\widehat { V } _ { h }$ . ", + "bbox": [ + 174, + 702, + 763, + 720 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/4f6970fb828b8f1d188f23adcb9be59219d003fd94ae7ee45da55f9a53f0e9c5.jpg", + "text": "$$\n\\begin{array} { r } { \\mathbf { S t e p 3 : W e ~ p r o v e ~ s u p } _ { h } \\vert \\vert \\widehat \\sigma _ { h } ^ { 2 } - \\sigma _ { \\widehat { V } _ { h } } ^ { 2 } \\vert \\vert _ { \\infty } \\leq 3 6 \\sqrt { \\frac { H ^ { 4 } d ^ { 3 } } { \\kappa K } \\log \\left( \\frac { ( \\lambda + K ) 2 K d H ^ { 2 } } { \\lambda \\delta } \\right) } + 1 2 \\lambda \\frac { H ^ { 2 } \\sqrt { d } } { \\kappa K } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 171, + 726, + 732, + 761 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Proof of Step3. By equation 17, ", + "bbox": [ + 173, + 765, + 388, + 780 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/0e384c5d91264a0a26a3ddf234eeb18e3d63e6b2aec171b7b4df4d87f7954227.jpg", + "text": "$$\n\\begin{array} { r l } & { \\quad \\left| \\left[ \\big < \\phi ( \\cdot , \\cdot ) , \\bar { \\theta } _ { h } \\big > _ { [ 0 , H - h + 1 ] } \\right] ^ { 2 } - \\left[ \\mathbb { P } _ { h } ( \\widehat { V } _ { h + 1 } ) ( s , a ) \\right] ^ { 2 } \\right| } \\\\ & { = \\left| \\big < \\phi ( s , a ) , \\bar { \\theta } _ { h } \\big > _ { [ 0 , H - h + 1 ] } + \\mathbb { P } _ { h } ( \\widehat { V } _ { h + 1 } ) ( s , a ) \\right| \\cdot \\left| \\big < \\phi ( s , a ) , \\bar { \\theta } _ { h } \\big > _ { [ 0 , H - h + 1 ] } - \\mathbb { P } _ { h } ( \\widehat { V } _ { h + 1 } ) ( s , a ) \\right| } \\\\ & { \\leq 2 H \\cdot \\left| \\big < \\phi ( s , a ) , \\bar { \\theta } _ { h } \\big > _ { [ 0 , H - h + 1 ] } - \\mathbb { P } _ { h } ( \\widehat { V } _ { h + 1 } ) ( s , a ) \\right| \\leq 2 4 \\sqrt { \\frac { H ^ { 4 } d ^ { 3 } } { \\kappa K } \\log \\left( \\frac { \\big ( \\lambda + K \\big ) 2 K d H ^ { 2 } } { \\lambda \\delta } \\right) } + 8 \\lambda \\frac { H ^ { 2 } \\sqrt { \\lambda + 1 } } { \\kappa K } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 784, + 839, + 885 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Combining this with Step1 we receive $\\forall h , s , a \\in [ H ] \\times S \\times A$ , with probability $1 - \\delta$ ", + "bbox": [ + 173, + 102, + 740, + 119 + ], + "page_idx": 24 + }, + { + "type": "equation", + "img_path": "images/f4b951b2e5e1765085db5698e6c04160a1f223f35745f5b15263dcf3463ae348.jpg", + "text": "$$\n\\left| \\widehat { \\mathrm { V a r } } _ { h } \\widehat { V } _ { h + 1 } ( s , a ) - \\mathrm { V a r } _ { P _ { h } } \\widehat { V } _ { h + 1 } ( s , a ) \\right| \\leq 3 6 \\sqrt { \\frac { H ^ { 4 } d ^ { 3 } } { \\kappa K } \\log \\left( \\frac { ( \\lambda + K ) 2 K d H ^ { 2 } } { \\lambda \\delta } \\right) } + 1 2 \\lambda \\frac { H ^ { 2 } \\sqrt { d } } { \\kappa K } .\n$$", + "text_format": "latex", + "bbox": [ + 191, + 126, + 805, + 169 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Finally, by the non-expansiveness of operator $\\operatorname* { m a x } \\{ 1 , \\cdot \\}$ , we have the stated result. ", + "bbox": [ + 171, + 175, + 714, + 191 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "C.4.2 A CRUDE BOUND ON $\\operatorname* { s u p } _ { h } \\lvert | V _ { h } ^ { \\star } - \\widehat { V } _ { h } \\rvert | _ { \\infty }$ . ", + "bbox": [ + 173, + 227, + 509, + 246 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Lemma C.8. Define $\\begin{array} { r l r } { \\widehat { \\sigma } _ { h } ( s , a ) } & { { } = } & { \\sqrt { \\operatorname* { m a x } \\left\\{ 1 , \\widehat { \\mathrm { V a r } } _ { P _ { h } } \\widehat { V } _ { h + 1 } ( s , a ) \\right\\} + 1 } , \\quad i f \\quad K \\quad \\geq } \\end{array}$ $\\operatorname* { m a x } \\{ \\mathcal { M } _ { 1 } , \\mathcal { M } _ { 2 } , \\mathcal { M } _ { 3 } , \\mathcal { M } _ { 4 } \\}$ and $K > C \\cdot H ^ { 4 } \\kappa ^ { 2 }$ , then with probability at least $1 - \\delta$ , ", + "bbox": [ + 173, + 256, + 826, + 304 + ], + "page_idx": 24 + }, + { + "type": "equation", + "img_path": "images/3c7cadfe349706768342616197eb287aa483058a78bdf4afd5994dc2d3682e27.jpg", + "text": "$$\n\\operatorname* { s u p } _ { h } \\left\\| V _ { h } ^ { \\star } - \\widehat { V } _ { h } \\right\\| _ { \\infty } \\leq \\widetilde { O } \\left( \\frac { H ^ { 2 } \\sqrt { d } } { \\sqrt { \\kappa K } } \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 379, + 310, + 619, + 352 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Proof. Step1: We show with probability at least $1 - \\delta$ $\\begin{array} { r } { \\delta , \\operatorname* { s u p } _ { h } \\left\\| V _ { h } ^ { \\star } - V _ { h } ^ { \\widehat { \\pi } } \\right\\| _ { \\infty } \\leq \\widetilde { O } \\left( \\frac { H ^ { 2 } \\sqrt { d } } { \\sqrt { \\kappa K } } \\right) . } \\end{array}$ ", + "bbox": [ + 168, + 369, + 769, + 392 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Indeed, combing Lemma C.1 and Lemma C.5, similar to the proof of Theorem C.6, we directly have with probability $1 - \\delta$ , for all policy $\\pi$ simultaneously, and for all $s \\in S$ , $h \\in [ H ]$ ", + "bbox": [ + 173, + 398, + 828, + 429 + ], + "page_idx": 24 + }, + { + "type": "equation", + "img_path": "images/ba549b28f81a91fe5e15a2e227d956572975a13f5ec929ef06c7c77f6138186a.jpg", + "text": "$$\nV _ { h } ^ { \\pi } ( s ) - V _ { h } ^ { \\widehat { \\pi } } ( s ) \\leq \\widetilde { O } \\left( \\sqrt { d } \\cdot \\sum _ { t = h } ^ { H } \\mathbb { E } _ { \\pi } \\left[ \\left( \\phi ( \\cdot , \\cdot ) ^ { \\top } \\Lambda _ { t } ^ { - 1 } \\phi ( \\cdot , \\cdot ) \\right) ^ { 1 / 2 } \\middle | s _ { h } = s \\right] \\right) + \\frac { 2 H ^ { 4 } \\sqrt { d } } { K } ,\n$$", + "text_format": "latex", + "bbox": [ + 209, + 435, + 761, + 479 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Next, since $\\begin{array} { r } { K \\ge \\operatorname* { m a x } \\left\\{ 5 1 2 ( 1 / \\kappa ) ^ { 2 } \\log \\left( \\frac { 4 H d } { \\delta } \\right) , 4 \\lambda / \\kappa \\right\\} } \\end{array}$ , by Lemma H.6 and a union bound over $h \\in [ H ]$ , with probability $1 - \\delta$ ", + "bbox": [ + 173, + 486, + 825, + 518 + ], + "page_idx": 24 + }, + { + "type": "equation", + "img_path": "images/f502bdf0db8af076015aed19a6047ebf175f0cabd3fd5e1d6d786c32d898dbf9.jpg", + "text": "$$\n\\operatorname* { s u p } _ { s , a } \\| \\phi ( s , a ) \\| _ { \\hat { \\Lambda } _ { h } ^ { - 1 } } \\leq \\frac { 2 } { \\sqrt { K } } \\operatorname* { s u p } _ { s , a } \\| \\phi ( s , a ) \\| _ { \\Lambda _ { h } ^ { p - 1 } } \\leq \\frac { 2 H } { \\sqrt { \\kappa K } } , \\forall h \\in [ H ] .\n$$", + "text_format": "latex", + "bbox": [ + 276, + 525, + 720, + 559 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Lastly, taking $\\pi = \\pi ^ { \\star }$ in equation 18 to obtain ", + "bbox": [ + 173, + 566, + 478, + 582 + ], + "page_idx": 24 + }, + { + "type": "equation", + "img_path": "images/ab74c525c6a278049b9876c8be6f74a9eb95027342b1013941c6ecd34eadc214.jpg", + "text": "$$\n\\begin{array} { c l } { \\displaystyle 0 \\leq { V _ { h } ^ { \\pi ^ { * } } ( s ) - V _ { h } ^ { \\widehat { \\pi } } ( s ) \\leq \\widetilde O \\left( \\sqrt { d } \\cdot \\displaystyle \\sum _ { t = h } ^ { H } \\mathbb { E } _ { \\pi ^ { * } } \\left[ \\left( \\phi ( \\cdot , \\cdot ) ^ { \\top } \\Lambda _ { t } ^ { - 1 } \\phi ( \\cdot , \\cdot ) \\right) ^ { 1 / 2 } \\Big | s _ { h } = s \\right] \\right) + \\frac { 2 H ^ { 4 } \\sqrt { d } } { K } } } \\\\ { \\displaystyle \\leq \\widetilde O \\left( \\frac { H ^ { 2 } \\sqrt { d } } { \\sqrt { \\kappa K } } \\right) + \\frac { 2 H ^ { 4 } \\sqrt { d } } { K } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 191, + 589, + 781, + 676 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "This implies by using the condition $K > C \\cdot H ^ { 4 } \\kappa ^ { 2 }$ , we finish the proof of Step1. ", + "bbox": [ + 173, + 681, + 702, + 699 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Step2: We show with probability $\\begin{array} { r } { 1 - \\delta , \\operatorname* { s u p } _ { h } { \\left\\| { \\widehat { V } } _ { h } - V _ { h } ^ { \\widehat { \\pi } } \\right\\| _ { \\infty } } \\leq \\widetilde { O } \\left( \\frac { H ^ { 2 } \\sqrt { d } } { \\sqrt { \\kappa K } } \\right) . } \\end{array}$ ", + "bbox": [ + 173, + 704, + 671, + 731 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Indeed, applying Extended Value Difference Lemma H.7 for $\\pi = \\pi ^ { \\prime } = \\widehat { \\pi }$ , then with probability $1 - \\delta$ , for all $s , h$ ", + "bbox": [ + 174, + 736, + 828, + 765 + ], + "page_idx": 24 + }, + { + "type": "equation", + "img_path": "images/424f3fc11ee126d5be85b3f4c29be9f2eaf8103d3bae63b44fca45864c1dc6d7.jpg", + "text": "$$\n\\begin{array} { r l } & { ~ \\displaystyle | \\widehat { V } _ { h } ( s ) - V _ { h } ^ { \\widehat { \\pi } } ( s ) | = \\displaystyle | \\sum _ { t = h } ^ { H } \\mathbb { E } _ { \\widehat { \\pi } } [ \\widehat { Q } _ { h } ( s _ { h } , a _ { h } ) - ( \\mathcal { T } _ { h } \\widehat { V } _ { h + 1 } ) ( s _ { h } , a _ { h } ) | s _ { h } = s ] | } \\\\ & { \\leq \\displaystyle \\sum _ { t = h } ^ { H } \\| ( \\widehat { \\mathcal { T } } _ { h } \\widehat { V } _ { h + 1 } - \\mathcal { T } _ { h } \\widehat { V } _ { h + 1 } ) ( s , a ) \\| + \\| \\Gamma _ { h } ( s , a ) \\| } \\\\ & { { \\leq } \\widetilde { O } ( H \\sqrt { d } \\| \\sqrt { \\phi ( s , a ) \\Lambda _ { h } ^ { - 1 } \\phi ( s , a ) } \\| ) + \\frac { 4 H ^ { 4 } \\sqrt { d } } { K } \\leq \\widetilde { O } ( \\frac { H ^ { 2 } \\sqrt { d } } { \\sqrt { \\kappa K } } ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 246, + 772, + 750, + 904 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "where the second inequality uses Lemma ${ \\mathrm { C } } . 5 ^ { 1 2 }$ and the last inequality follows the same procedure as Step1. ", + "bbox": [ + 173, + 102, + 825, + 133 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Step3: Combine Step1 and Step2, by triangular inequality and a union bound we finish the proof of the lemma. ", + "bbox": [ + 171, + 138, + 825, + 167 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Remark C.9. Note as an intermediate calculation, equation $^ { 1 9 }$ ensures a learning bound with order $\\widetilde { \\cal O } \\big ( \\frac { H ^ { 2 } \\sqrt { d } } { \\sqrt { \\kappa K } } \\big )$ . Here, the convergence rate is the standard statistical rate $\\textstyle { \\frac { 1 } { \\sqrt { K } } }$ and the $H ^ { 2 }$ dependence is loose. However, the feature dependence $\\sqrt { d / \\kappa }$ is roughly tight, since, in the well-explored case (Assumption 2 of Wang et al. (2021a)), √ $\\kappa = 1 / d$ and the $\\sqrt { d / \\kappa } = \\sqrt { d ^ { 2 } }$ recovers the optimal feature dependence $d H \\sqrt { T }$ in the online setting (Zhou et al., 2021a). If $\\kappa \\ll 1 / d ,$ , then doing offline learning requires sample size proportional to $d / \\kappa ,$ , which reveals offline $R L$ is harder when the exploration of behavior policy is insufficient. When $\\kappa = 0$ , learning the optimal policy accurately cannot be guaranteed even if the sample/episode size $K \\infty$ . ", + "bbox": [ + 173, + 198, + 826, + 327 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "C.5 PROOF OF THE SECOND PART OF THEOREM 3.2 ", + "bbox": [ + 174, + 343, + 544, + 359 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Lemma C.10. Recall $\\widehat { \\sigma } _ { h } = \\sqrt { \\operatorname* { m a x } \\left\\{ 1 , \\widehat { \\mathrm { V a r } } _ { P _ { h } } \\widehat { V } _ { h + 1 } \\right\\} + 1 }$ and $\\sigma _ { h } ^ { \\star } = \\sqrt { \\operatorname* { m a x } \\left\\{ 1 , \\mathrm { V a r } _ { P _ { h } } V _ { h + 1 } ^ { \\star } \\right\\} + 1 } .$ . Let $\\begin{array} { r } { K \\ge \\operatorname* { m a x } \\left\\{ 5 1 2 ( 1 / \\kappa ) ^ { 2 } \\log \\left( \\frac { 4 H d } { \\delta } \\right) , 4 \\lambda / \\kappa \\right\\} } \\end{array}$ and $K \\ge \\operatorname* { m a x } \\{ \\mathcal { M } _ { 1 } , \\mathcal { M } _ { 2 } , \\mathcal { M } _ { 3 } , \\mathcal { M } _ { 4 } \\}$ , then with probability $1 - \\delta$ , ", + "bbox": [ + 173, + 367, + 826, + 429 + ], + "page_idx": 25 + }, + { + "type": "equation", + "img_path": "images/ec14809aea10c1a5dbf9381ecb8244a76d3a73b1d50835ed0d928ab437f01e6f.jpg", + "text": "$$\n\\operatorname* { s u p } _ { h } \\vert \\vert \\widehat { \\sigma } _ { h } ^ { 2 } - \\sigma _ { h } ^ { \\star 2 } \\vert \\vert _ { \\infty } \\leq \\widetilde { O } \\left( \\frac { H ^ { 3 } \\sqrt { d } } { \\sqrt { \\kappa K } } \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 379, + 428, + 619, + 470 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Proof. By definition and the non-expansiveness of $\\operatorname* { m a x } \\{ 1 , \\cdot \\} + 1$ , we have ", + "bbox": [ + 174, + 486, + 668, + 502 + ], + "page_idx": 25 + }, + { + "type": "equation", + "img_path": "images/22eb4cd699d7db194269f8a2277efcf636aaccf41f32ef2b4e65064d1f788669.jpg", + "text": "$$\n\\begin{array} { r l } & { \\quad \\left\\| \\sigma _ { \\widehat { V } _ { h + 1 } } ^ { 2 } - \\sigma _ { h } ^ { \\star 2 } \\right\\| _ { \\infty } \\leq \\left\\| \\operatorname { V a r } \\widehat { V } _ { h + 1 } - \\operatorname { V a r } V _ { h + 1 } ^ { \\star } \\right\\| _ { \\infty } } \\\\ & { \\leq \\left\\| \\mathbb { P } _ { h } \\left( \\widehat { V } _ { h + 1 } ^ { 2 } - V _ { h + 1 } ^ { \\star 2 } \\right) \\right\\| _ { \\infty } + \\left\\| ( \\mathbb { P } _ { h } \\widehat { V } _ { h + 1 } ) ^ { 2 } - ( \\mathbb { P } _ { h } V _ { h + 1 } ^ { \\star } ) ^ { 2 } \\right\\| _ { \\infty } } \\\\ & { \\leq \\left\\| \\widehat { V } _ { h + 1 } ^ { 2 } - V _ { h + 1 } ^ { \\star 2 } \\right\\| _ { \\infty } + \\left\\| ( \\mathbb { P } _ { h } \\widehat { V } _ { h + 1 } + \\mathbb { P } _ { h } V _ { h + 1 } ^ { \\star } ) ( \\mathbb { P } _ { h } \\widehat { V } _ { h + 1 } - \\mathbb { P } _ { h } V _ { h + 1 } ^ { \\star } ) \\right\\| _ { \\infty } } \\\\ & { \\leq 2 H \\left\\| \\widehat { V } _ { h + 1 } - V _ { h + 1 } ^ { \\star } \\right\\| _ { \\infty } + 2 H \\left\\| \\mathbb { P } _ { h } \\widehat { V } _ { h + 1 } - \\mathbb { P } _ { h } V _ { h + 1 } ^ { \\star } \\right\\| _ { \\infty } \\leq \\widetilde { \\mathcal { O } } \\left( \\frac { H ^ { 3 } \\sqrt { d } } { \\sqrt { \\kappa K } } \\right) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 254, + 508, + 743, + 637 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "with probability $1 - \\delta$ for all $h \\in [ H ]$ , where the last inequality comes from Lemma C.8. Combining this with Lemma C.7, we have the stated result. ", + "bbox": [ + 171, + 643, + 825, + 672 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Lemma C.11. Denote the quantities $C _ { 1 } = \\operatorname* { m a x } \\{ 2 \\lambda , 1 2 8 \\log ( 2 d / \\delta ) , 1 2 8 H ^ { 4 } \\log ( 2 d / \\delta ) / \\kappa ^ { 2 } \\}$ and $\\begin{array} { r } { C _ { 2 } = \\operatorname* { m a x } \\{ \\frac { \\lambda ^ { 2 } } { \\kappa \\log ( ( \\lambda + K ) H / \\lambda \\delta ) } , 9 6 ^ { 2 } H ^ { 1 2 } d \\log ( ( \\lambda + K ) H / \\lambda \\delta ) / \\kappa ^ { 5 } \\} } \\end{array}$ . Suppose the number of episode $K$ satisfies $K > \\operatorname* { m a x } \\{ C _ { 1 } , C _ { 2 } \\}$ , then with probability $1 - \\delta$ , ", + "bbox": [ + 173, + 681, + 825, + 733 + ], + "page_idx": 25 + }, + { + "type": "equation", + "img_path": "images/0afe9a095521196b73798b4aadea8daeb7346d645eaaa22cd1a14bbb516186f7.jpg", + "text": "$$\n\\sqrt { \\phi ( s , a ) \\Lambda _ { h } ^ { - 1 } \\phi ( s , a ) } \\le 2 \\sqrt { \\phi ( s , a ) \\Lambda _ { h } ^ { \\star - 1 } \\phi ( s , a ) } , \\quad \\forall s , a \\in \\mathcal { S } \\times \\mathcal { A } ,\n$$", + "text_format": "latex", + "bbox": [ + 279, + 739, + 717, + 767 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Proof of Lemma C.11. By definition $\\sqrt { \\phi ( s , a ) \\Lambda _ { h } ^ { - 1 } \\phi ( s , a ) } = \\| \\phi ( s , a ) \\| _ { \\Lambda _ { h } ^ { - 1 } } .$ . Then denote ", + "bbox": [ + 176, + 785, + 753, + 809 + ], + "page_idx": 25 + }, + { + "type": "equation", + "img_path": "images/4cf2251ae3e307003067d53ebd4cbe5b6a64e97ebcbe36ca70502c914b7f81ed.jpg", + "text": "$$\n\\Lambda _ { h } ^ { \\prime } = \\frac { 1 } { K } \\Lambda _ { h } , \\quad { \\Lambda _ { h } ^ { \\star } } ^ { \\prime } = \\frac { 1 } { K } \\Lambda _ { h } ^ { \\star } ,\n$$", + "text_format": "latex", + "bbox": [ + 400, + 816, + 596, + 847 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "where $\\begin{array} { r } { \\Lambda _ { h } \\ = \\ \\sum _ { \\tau = 1 } ^ { K } \\phi ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) ^ { \\top } \\phi ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) / \\sigma _ { V _ { h + 1 } ^ { \\star } } ^ { 2 } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) + \\lambda I . } \\end{array}$ . Under the condition of $K$ , by Lemma C.10, with probability $1 - \\delta$ ", + "bbox": [ + 173, + 99, + 825, + 136 + ], + "page_idx": 26 + }, + { + "type": "equation", + "img_path": "images/526db28629d9a68511d8dba74ba4673be24b340046629ae03743fb39351a09f9.jpg", + "text": "$$\n\\begin{array} { r l } & { \\left. \\Lambda _ { h } ^ { \\star ^ { \\prime } } - \\Lambda _ { h } ^ { \\prime } \\right. \\leq \\underset { s , a } { \\operatorname* { s u p } } \\left. \\frac { \\phi ( s , a ) \\phi ( s , a ) ^ { \\top } } { \\sigma _ { h } ^ { \\star 2 } ( s , a ) } - \\frac { \\phi ( s , a ) \\phi ( s , a ) ^ { \\top } } { \\sigma _ { \\hat { V } _ { h + 1 } } ^ { 2 } ( s , a ) } \\right. } \\\\ & { \\leq \\underset { s , a } { \\operatorname* { s u p } } \\left. \\frac { \\sigma _ { h } ^ { \\star 2 } ( s , a ) - \\sigma _ { \\hat { V } _ { h + 1 } } ^ { 2 } ( s , a ) } { \\sigma _ { h } ^ { \\star 2 } ( s , a ) \\sigma _ { \\hat { V } _ { h + 1 } } ^ { 2 } ( s , a ) } \\right. \\cdot \\left. \\phi ( s , a ) \\right. ^ { 2 } \\leq \\underset { s , a } { \\operatorname* { s u p } } \\left. \\frac { \\sigma _ { h } ^ { \\star 2 } ( s , a ) - \\sigma _ { \\hat { V } _ { h + 1 } } ^ { 2 } ( s , a ) } { 1 } \\right. \\cdot 1 } \\\\ & { \\leq \\tilde { O } \\left( \\frac { H ^ { 3 } \\sqrt { d } } { \\sqrt { \\kappa K } } \\right) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 215, + 140, + 754, + 270 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Next by Lemma H.5 (with $\\phi$ to be $\\phi / \\sigma _ { V _ { h + 1 } ^ { \\star } }$ and $C = 1$ ), it holds with probability $1 - \\delta$ , ", + "bbox": [ + 169, + 275, + 751, + 294 + ], + "page_idx": 26 + }, + { + "type": "equation", + "img_path": "images/652fa363c9f42c0535cfb6248feb0df6df83e64beb364647e1bf65a4883a66ef.jpg", + "text": "$$\n\\left\\| \\Lambda _ { h } ^ { \\star ^ { \\prime } } - \\left( \\mathbb { E } _ { \\mu , h } [ \\phi ( s , a ) \\phi ( s , a ) ^ { \\top } / \\sigma _ { V _ { h + 1 } ^ { \\star } } ^ { 2 } ( s , a ) ] + \\frac { \\lambda } { K } I _ { d } \\right) \\right\\| \\leq \\frac { 4 \\sqrt { 2 } } { \\sqrt { K } } \\left( \\log \\frac { 2 d } { \\delta } \\right) ^ { 1 / 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 235, + 299, + 759, + 338 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Therefore by Weyl’s spectrum theorem and the condition $K$ > $\\operatorname* { m a x } \\{ 2 \\lambda , 1 2 8 \\log ( 2 d / \\delta ) , 1 2 8 H ^ { 4 } \\log ( 2 d / \\delta ) / \\kappa ^ { 2 } \\}$ , the above implies ", + "bbox": [ + 176, + 340, + 821, + 371 + ], + "page_idx": 26 + }, + { + "type": "equation", + "img_path": "images/dcf54322bd8ce9562fd57dbefd2e43c8878238c74f9ab8763019dab9c5972d88.jpg", + "text": "$$\n\\begin{array} { r l } { \\left\\| { { { \\Lambda } _ { k } ^ { \\prime } } } \\right\\| = \\lambda { \\operatorname* { m s u p } _ { i } } ( \\Lambda _ { k } ^ { \\prime } ) \\le \\lambda { \\operatorname* { m s u p } _ { i } } \\left( { \\operatorname* { \\sum } _ { n , h } } [ \\phi ( s , a ) \\phi ( s , a ) ] ^ { \\top } \\rho _ { \\mathcal { N } _ { k + 1 } ^ { \\prime } } ^ { \\sigma } ( s , a ) \\right) + \\frac { \\lambda } { K } + \\frac { 4 \\sqrt { 2 } } { \\sqrt { K } } \\left( \\log \\frac { 2 d } { \\delta } \\right) ^ { 1 / 2 } } & \\\\ { \\le \\left\\| { { \\mathbb { E } _ { \\rho , h } } [ \\phi ( s , a ) \\phi ( s , a ) ] ^ { \\top } } \\rho _ { \\mathcal { N } _ { k + 1 } ^ { \\prime } } ^ { \\sigma } ( s , a ) \\right\\| + \\frac { \\lambda } { K } + \\frac { 4 \\sqrt { 2 } } { \\sqrt { K } } \\left( \\log \\frac { 2 d } { \\delta } \\right) ^ { 1 / 2 } } & \\\\ { \\le \\| \\phi ( s , a ) \\| ^ { 2 } + \\frac { \\lambda } { K } + \\frac { 4 \\sqrt { 2 } } { \\sqrt { K } } \\left( \\log \\frac { 2 d } { \\delta } \\right) ^ { 1 / 2 } \\le 1 + \\frac { \\lambda } { K } + \\frac { 4 \\sqrt { 2 } } { \\sqrt { K } } \\left( \\log \\frac { 2 d } { \\delta } \\right) ^ { 1 / 2 } \\le 2 , } & \\\\ { \\operatorname* { m i n } ( \\Lambda _ { k } ^ { \\prime } ) \\le \\lambda { \\operatorname* { m i n } } \\left( { \\mathbb { E } _ { \\rho , h } } [ \\phi ( s , a ) \\phi ( s , a ) ] ^ { \\top } \\rho _ { \\mathcal { N } _ { k + 1 } ^ { \\prime } ( s , a ) } ^ { \\sigma } | \\right) + \\frac { \\lambda } { K } - \\frac { 4 \\sqrt { 2 } } { \\sqrt { K } } \\left( \\log \\frac { 2 d } { \\delta } \\right) ^ { 1 / 2 } } & \\\\ { \\le \\lambda { \\operatorname* { m i n } } \\left( { \\mathbb { E } _ { \\rho , h } } [ \\phi ( s , a ) \\phi ( s , a ) ] ^ { \\top } \\rho _ { \\mathcal { N } _ { k + 1 } ^ { \\prime } ( s , a ) } ^ { \\sigma } | \\right) - \\frac { 4 \\sqrt { 2 } } { \\sqrt { K } } \\left( \\log \\frac { 2 d } { \\delta } \\right) ^ { 1 / 2 } } & \\\\ \\ge \\frac { \\kappa } { L } - \\frac { 4 \\sqrt { 2 } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 183, + 376, + 825, + 607 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Hence with probability $1 - \\delta$ , $\\left\\| \\Lambda _ { h } ^ { \\star ^ { \\prime } } \\right\\| \\leq 2$ and $\\left\\| { \\boldsymbol { \\Lambda } } _ { h } ^ { \\star ^ { \\prime } - 1 } \\right\\| = 1 / { \\lambda _ { \\operatorname* { m i n } } ( \\boldsymbol { \\Lambda } _ { h } ^ { \\star ^ { \\prime } } ) } \\le 2 H ^ { 2 } / \\kappa$ . Similarly, $\\left\\| \\Lambda _ { h } ^ { ' - 1 } \\right\\| \\le 2 H ^ { 2 } / \\kappa$ with high probability. ", + "bbox": [ + 173, + 609, + 826, + 659 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Now apply Lemma H.4 to $\\Lambda _ { h } ^ { \\star ^ { \\prime } }$ and $\\Lambda _ { h } ^ { \\prime }$ and a union bound, we obtain with probability $1 - \\delta$ , for all $s , a$ ", + "bbox": [ + 165, + 666, + 825, + 698 + ], + "page_idx": 26 + }, + { + "type": "equation", + "img_path": "images/27fb18c65eff53dc82250ad63a633e3cb2aaf15e407851db0caeaac205d94b02.jpg", + "text": "$$\n\\begin{array} { r l } { \\| \\phi ( s , a ) \\| _ { \\Lambda _ { h } ^ { \\prime - 1 } } \\leq \\left[ 1 + \\sqrt { \\left\\| \\Lambda _ { h } ^ { \\star ^ { \\prime } - 1 } \\right\\| \\left\\| \\Lambda _ { h } ^ { \\star ^ { \\prime } } \\right\\| \\cdot \\left\\| \\Lambda _ { h } ^ { \\prime - 1 } \\right\\| \\cdot \\left\\| \\Lambda _ { h } ^ { \\star ^ { \\prime } } - \\Lambda _ { h } ^ { \\prime } \\right\\| } \\right] \\cdot \\| \\phi ( s , a ) \\| _ { \\Lambda _ { h } ^ { \\star ^ { \\prime } - 1 } } } & { } \\\\ { \\leq \\left[ 1 + \\sqrt { \\frac { 2 H ^ { 2 } } { \\kappa } \\cdot 1 \\cdot \\frac { 2 H ^ { 2 } } { \\kappa } \\cdot \\left\\| \\Lambda _ { h } ^ { \\star ^ { \\prime } } - \\Lambda _ { h } ^ { \\prime } \\right\\| } \\right] \\cdot \\| \\phi ( s , a ) \\| _ { \\Lambda _ { h } ^ { \\star ^ { \\prime } - 1 } } } & { } \\\\ { \\leq \\left[ 1 + \\sqrt { \\frac { H ^ { 4 } } { \\kappa ^ { 2 } } \\left[ \\tilde { \\mathcal { O } } \\left( \\frac { H ^ { 3 } \\sqrt { d } } { \\sqrt { \\kappa K } } \\right) \\right] } \\right] \\cdot \\| \\phi ( s , a ) \\| _ { \\Lambda _ { h } ^ { \\star ^ { \\prime } - 1 } } \\leq 2 \\left\\| \\phi ( s , a ) \\right\\| _ { \\Lambda _ { h } ^ { \\star ^ { \\prime } - 1 } } } & { } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 217, + 699, + 779, + 832 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "where the third inequality uses equation 20 and the last inequality uses $K \\_ { > }$ $\\begin{array} { r } { \\operatorname* { m a x } \\{ \\frac { \\lambda ^ { 2 } } { \\kappa \\log ( ( \\lambda + K ) H / \\lambda \\delta ) } , \\overset { } { 9 } 6 ^ { 2 } \\dot { H } ^ { 1 2 } d \\log ( ( \\lambda + K ) H / \\lambda \\delta ) / \\kappa ^ { 5 } \\} } \\end{array}$ . The claimed result follows straightforwardly by multiplying $1 / \\sqrt { K }$ on both sides of the above. ", + "bbox": [ + 173, + 851, + 826, + 904 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Proof of Theorem 3.2. The first part of the theorem has been shown in Theorem C.6. For the second part, apply Theorem C.6 with $\\pi = \\pi ^ { \\star }$ , then with probability $1 - \\delta$ , ", + "bbox": [ + 169, + 103, + 825, + 132 + ], + "page_idx": 27 + }, + { + "type": "equation", + "img_path": "images/6fa7eacbb016f34f419b8c971fc787a07f9badeea298ee2ab92255c68893d28f.jpg", + "text": "$$\nv ^ { \\pi ^ { \\star } } - v ^ { \\widehat { \\pi } } \\leq \\widetilde O \\left( \\sqrt { d } \\cdot \\sum _ { h = 1 } ^ { H } \\mathbb E _ { \\pi ^ { \\star } } \\left[ \\left( \\phi ( \\cdot , \\cdot ) ^ { \\top } \\Lambda _ { h } ^ { - 1 } \\phi ( \\cdot , \\cdot ) \\right) ^ { 1 / 2 } \\right] \\right) + \\frac { 2 H ^ { 4 } \\sqrt { d } } { K } ,\n$$", + "text_format": "latex", + "bbox": [ + 264, + 135, + 732, + 179 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "Now apply Lemma C.11 and a union bound, with probability $1 - \\delta$ , ", + "bbox": [ + 173, + 180, + 619, + 195 + ], + "page_idx": 27 + }, + { + "type": "equation", + "img_path": "images/f267a949e8c56d1b709e3869b520b7e1a83149d97bc5bed955e0d89e65dd1ed0.jpg", + "text": "$$\n0 \\leq v ^ { \\star } - v ^ { \\widehat { \\pi } } \\leq \\widetilde { O } \\left( \\sqrt { d } \\cdot \\sum _ { h = 1 } ^ { H } \\mathbb { E } _ { \\pi ^ { \\star } } \\left[ \\left( \\phi ( \\cdot , \\cdot ) ^ { \\top } \\Lambda _ { h } ^ { \\star - 1 } \\phi ( \\cdot , \\cdot ) \\right) ^ { 1 / 2 } \\right] \\right) + \\frac { 2 H ^ { 4 } \\sqrt { d } } { K } .\n$$", + "text_format": "latex", + "bbox": [ + 251, + 198, + 746, + 242 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "D PROOF OF THEOREM 3.3 ", + "text_level": 1, + "bbox": [ + 174, + 285, + 416, + 301 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "First of all, we show the following lemma. ", + "bbox": [ + 174, + 315, + 452, + 330 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "Lemma D.1. Suppose $K > \\operatorname* { m a x } \\{ \\mathcal { M } _ { 1 } , \\mathcal { M } _ { 2 } , \\mathcal { M } _ { 3 } , \\mathcal { M } _ { 4 } \\}$ . Plug ", + "bbox": [ + 173, + 333, + 581, + 349 + ], + "page_idx": 27 + }, + { + "type": "equation", + "img_path": "images/146979c426a5db4602cc39be48033f2f69fbfe2197072e8f93e047713e381914.jpg", + "text": "$$\n\\mathbf { \\Sigma } _ { h } ^ { \\scriptscriptstyle { T } } ( s , a ) \\gets \\phi ( s , a ) ^ { \\scriptscriptstyle { T } } \\left| \\widehat { \\Lambda } _ { h } ^ { - 1 } \\sum _ { \\tau = 1 } ^ { K } \\frac { \\phi \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\cdot \\left( r _ { h } ^ { \\tau } + \\widehat { V } _ { h + 1 } \\left( s _ { h + 1 } ^ { \\tau } \\right) - \\left( \\widehat { T } _ { h } \\widehat { V } _ { h + 1 } \\right) \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\right) } { \\widehat { \\sigma } _ { h } ^ { 2 } \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) } \\right| + \\widetilde { O } ( \\frac { H ^ { 3 } d / \\kappa } { K } )\n$$", + "text_format": "latex", + "bbox": [ + 181, + 352, + 839, + 402 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "in Algorithm $I$ and let $\\mathcal { T } _ { h }$ be the Bellman operator and $\\widehat { \\mathcal { T } } _ { h }$ be the approximated Bellman operator. Then we have with probability $1 - \\delta$ : ", + "bbox": [ + 174, + 406, + 826, + 436 + ], + "page_idx": 27 + }, + { + "type": "equation", + "img_path": "images/9f7ff5e2c1095e1bfd32eea1092ce4ba940c47d796bad2abfbf8de4105f980ff.jpg", + "text": "$$\n\\vert ( \\mathcal T _ { h } \\widehat V _ { h + 1 } - \\widehat T _ { h } \\widehat V _ { h + 1 } ) ( s , a ) \\vert \\le \\Gamma _ { h } ^ { I } ( s , a ) , \\quad \\forall s , a \\in \\mathcal S \\times \\mathcal A .\n$$", + "text_format": "latex", + "bbox": [ + 305, + 439, + 692, + 460 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "Proof of Lemma $D . I$ . Suppose $w _ { h }$ is the coefficient corresponding to the $\\mathcal { T } _ { h } \\widehat { V } _ { h + 1 }$ (such $w _ { h }$ exists by Lemma H.9), i.e. $\\mathcal { T } _ { h } \\widehat { V } _ { h + 1 } = \\phi ^ { \\top } w _ { h }$ , and recall $( \\widehat { T } _ { h } \\widehat { V } _ { h + 1 } ) ( s , a ) = \\phi ( s , a ) ^ { \\top } \\widehat { w } _ { h }$ , then: ", + "bbox": [ + 173, + 474, + 825, + 508 + ], + "page_idx": 27 + }, + { + "type": "equation", + "img_path": "images/78ae013359ab7a4ee09d98b7722f0ef2502118ababfbf0597ff5967168f7f27b.jpg", + "text": "$$\n\\begin{array} { r l } & { \\quad \\left( \\mathcal { T } _ { h } \\widehat { V } _ { h + 1 } \\right) ( s , a ) - \\left( \\widehat { \\mathcal { T } } _ { h } \\widehat { V } _ { h + 1 } \\right) ( s , a ) = \\phi ( s , a ) ^ { \\top } \\left( w _ { h } - \\widehat { w } _ { h } \\right) } \\\\ & { = \\phi ( s , a ) ^ { \\top } w _ { h } - \\phi ( s , a ) ^ { \\top } \\widehat { \\Lambda } _ { h } ^ { - 1 } \\left( \\displaystyle \\sum _ { \\tau = 1 } ^ { K } \\phi \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\cdot \\left( r _ { h } ^ { \\tau } + \\widehat { V } _ { h + 1 } \\left( s _ { h + 1 } ^ { \\tau } \\right) \\right) / \\widehat { \\sigma } _ { h } ^ { 2 } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) \\right) } \\\\ & { = \\phi ( s , a ) ^ { \\top } w _ { h } - \\phi ( s , a ) ^ { \\top } \\widehat { \\Lambda } _ { h } ^ { - 1 } \\left( \\displaystyle \\sum _ { \\tau = 1 } ^ { K } \\phi \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\cdot \\left( \\mathcal { T } _ { h } \\widehat { V } _ { h + 1 } \\right) \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) / \\widehat { \\sigma } _ { h } ^ { 2 } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) \\right) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 169, + 511, + 727, + 628 + ], + "page_idx": 27 + }, + { + "type": "equation", + "img_path": "images/491af15c9227ba63aa3ba4f1726761e8542c6afb392743b728f21bad30e276bf.jpg", + "text": "$$\n+ \\phi ( s , a ) ^ { \\top } \\widehat { \\Lambda } _ { h } ^ { - 1 } ( \\sum _ { \\tau = 1 } ^ { K } \\phi ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) \\cdot ( r _ { h } ^ { \\tau } + \\widehat { V } _ { h + 1 } ( s _ { h + 1 } ^ { \\tau } ) - ( \\widehat { T } _ { h } \\widehat { V } _ { h + 1 } ) ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) ) / \\widehat { \\sigma } _ { h } ^ { 2 } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) ) )\n$$", + "text_format": "latex", + "bbox": [ + 218, + 647, + 828, + 690 + ], + "page_idx": 27 + }, + { + "type": "equation", + "img_path": "images/669d24a364e9d8442ab1f0e01a511afccd0823c65fe15c598acca9df13d87cf9.jpg", + "text": "$$\n+ \\phi ( s , a ) ^ { \\top } \\widehat { \\Lambda } _ { h } ^ { - 1 } \\left( \\sum _ { \\tau = 1 } ^ { K } \\phi \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\cdot \\left( \\left( \\widehat { \\mathcal { T } } _ { h } \\widehat { V } _ { h + 1 } \\right) \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) - \\left( \\mathcal { T } _ { h } \\widehat { V } _ { h + 1 } \\right) \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\right) / \\widehat { \\sigma } _ { h } ^ { 2 } \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\right)\n$$", + "text_format": "latex", + "bbox": [ + 220, + 710, + 833, + 753 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "For term (i), by Lemma C.2 it is bounded by $\\frac { 2 \\lambda H ^ { 3 } \\sqrt { d } / \\kappa } { K }$ with probability $1 - \\delta / 2$ . 13 ", + "bbox": [ + 174, + 791, + 718, + 814 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "For term (ii), it is bounded by ", + "bbox": [ + 173, + 818, + 369, + 833 + ], + "page_idx": 27 + }, + { + "type": "equation", + "img_path": "images/741d25f0b7b8cea593f287434ecaf9b13dd87c696298b108548a77eeb21ec440.jpg", + "text": "$$\n\\phi ( s , a ) ^ { \\top } \\left| \\widehat { \\Lambda } _ { h } ^ { - 1 } \\sum _ { \\tau = 1 } ^ { K } \\frac { \\phi \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\cdot \\left( r _ { h } ^ { \\tau } + \\widehat { V } _ { h + 1 } \\left( s _ { h + 1 } ^ { \\tau } \\right) - \\left( \\widehat { \\mathcal { T } } _ { h } \\widehat { V } _ { h + 1 } \\right) \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\right) } { \\widehat { \\sigma } _ { h } ^ { 2 } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) } \\right| .\n$$", + "text_format": "latex", + "bbox": [ + 243, + 835, + 753, + 886 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "For term (iii), by Cauchy inequality ", + "bbox": [ + 174, + 103, + 410, + 119 + ], + "page_idx": 28 + }, + { + "type": "equation", + "img_path": "images/33929071e051bb8174417bc7c0901551d0f698899c850c868627005f4cc0e3b6.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle \\phi ( s , a ) ^ { \\top } \\widehat \\Lambda _ { h } ^ { - 1 } \\left( \\displaystyle \\sum _ { \\tau = 1 } ^ { K } \\phi \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\cdot \\left( \\left( \\widehat { \\gamma } _ { h } \\widehat V _ { h + 1 } \\right) \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) - \\left( \\widehat { \\gamma } _ { h } \\widehat V _ { h + 1 } \\right) \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\right) / \\widehat \\sigma _ { h } ^ { 2 } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) \\right) } \\\\ { \\displaystyle \\leq \\left\\| \\phi ( s , a ) \\right\\| _ { \\widehat \\Lambda _ { h } ^ { - 1 } } \\cdot \\displaystyle \\left\\| \\sum _ { \\tau = 1 } ^ { K } \\phi \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\cdot \\left( \\left( \\widehat { \\gamma } _ { h } \\widehat V _ { h + 1 } \\right) \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) - \\left( \\widehat { \\gamma } _ { h } \\widehat V _ { h + 1 } \\right) \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\right) / \\widehat \\sigma _ { h } ^ { 2 } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) \\right\\| _ { \\widehat \\Lambda _ { h } ^ { - 1 } } } \\\\ { \\displaystyle \\leq \\displaystyle \\frac { 2 H } { \\sqrt { \\kappa K } } \\cdot \\displaystyle \\left\\| \\sum _ { \\tau = 1 } ^ { K } \\phi \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\cdot \\left( \\left( \\widehat { \\gamma } _ { h } \\widehat V _ { h + 1 } \\right) \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) - \\left( \\widehat { \\gamma } _ { h } \\widehat V _ { h + 1 } \\right) \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\right) / \\widehat \\sigma _ { h } ^ { 2 } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) \\right\\| _ { \\widehat \\Lambda _ { h } ^ { - 1 } } } \\\\ \\displaystyle \\leq \\frac { 2 H } { \\sqrt { \\kappa K } } \\cdot \\displaystyle \\tilde { O } ( \\frac { H ^ { 2 } \\sqrt { d / \\kappa } } { \\sqrt { K } } ) \\cdot \\sqrt { d } = \\tilde { O } ( \\frac H ^ { 3 } d / \\kappa \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 178, + 122, + 815, + 303 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "where the first inequality is by Lemma H.6 (with $\\phi ^ { \\prime } = \\phi / \\widehat { \\sigma } _ { h }$ and $\\| \\phi / { \\widehat { \\sigma } } _ { h } \\| \\leq \\| \\phi \\| \\leq 1 : = C )$ and the third inequality uses ${ \\sqrt { a ^ { \\top } \\cdot A \\cdot a } } \\leq { \\sqrt { \\left\\| a \\right\\| _ { 2 } \\left\\| A \\right\\| _ { 2 } \\left\\| a \\right\\| _ { 2 } } } = \\left\\| a \\right\\| _ { 2 } { \\sqrt { \\left\\| A \\right\\| _ { 2 } } }$ with $a$ to be either $\\phi$ or $w _ { h }$ . Moreover, $\\begin{array} { r } { \\lambda _ { \\operatorname* { m i n } } ( \\widetilde { \\Lambda } _ { h } ^ { p } ) \\ge \\kappa / \\operatorname* { m a x } _ { h , s , a } \\widehat { \\sigma } _ { h } ( s , a ) ^ { 2 } \\ge \\kappa / H ^ { 2 } } \\end{array}$ implies $\\left\\| ( \\tilde { \\Lambda } _ { h } ^ { p } ) ^ { - 1 } \\right\\| \\leq H ^ { 2 } / \\kappa$ . ", + "bbox": [ + 173, + 301, + 826, + 358 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "The second inequality is true by denoting $x _ { \\tau } = \\phi ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) / \\widehat { \\sigma } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } )$ and ", + "bbox": [ + 174, + 364, + 651, + 381 + ], + "page_idx": 28 + }, + { + "type": "equation", + "img_path": "images/f2e7f2135ccad198c5784031582a838b4ac23cd04f0bb387f4b207e3db1409a3.jpg", + "text": "$$\n\\eta _ { \\tau } = \\left( \\left( \\widehat { T } _ { h } \\widehat { V } _ { h + 1 } \\right) ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) - \\left( T _ { h } \\widehat { V } _ { h + 1 } \\right) ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) \\right) / \\widehat { \\sigma } _ { h } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } )\n$$", + "text_format": "latex", + "bbox": [ + 289, + 383, + 707, + 411 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "and use Lemma H.10 as the condition for applying Lemma H.2. By collecting those three terms together we have the result. ", + "bbox": [ + 174, + 414, + 823, + 443 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "D.1 PROOF OF THEOREM 3.3 ", + "text_level": 1, + "bbox": [ + 174, + 458, + 390, + 473 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "Proof. Use Lemma D.1 as the condition for Lemma C.1 and average over initial distribution $d _ { 1 }$ , we obtain with probability $1 - \\delta$ , ", + "bbox": [ + 173, + 484, + 825, + 513 + ], + "page_idx": 28 + }, + { + "type": "equation", + "img_path": "images/3a0d45f810b808550b45aaab8a91ed8a37d01e24874f8dd71bd4d1f5e5db04cc.jpg", + "text": "$$\n\\begin{array} { l } { v ^ { \\pi } - v ^ { \\pi } \\leq } \\\\ { \\displaystyle \\sum _ { h = 1 } ^ { H } \\mathbb { E } _ { \\pi _ { h } } \\left[ \\phi ( s , a ) \\right] ^ { \\top } \\left| \\widehat { \\Lambda } _ { h } ^ { - 1 } \\displaystyle \\sum _ { \\tau = 1 } ^ { K } \\frac { \\phi \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\cdot \\left( r _ { h } ^ { \\tau } + \\widehat { V } _ { h + 1 } \\left( s _ { h + 1 } ^ { \\tau } \\right) - \\left( \\widehat { \\mathcal { T } } _ { h } \\widehat { V } _ { h + 1 } \\right) \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\right) } { \\widehat { \\sigma } _ { h } ^ { 2 } \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) } \\right| + \\widetilde { O } ( \\frac { H ^ { 4 } d / } { K } ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 171, + 522, + 816, + 589 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "Denote $\\begin{array} { r } { A _ { h } : = \\sum _ { \\tau = 1 } ^ { K } \\frac { \\phi ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) \\cdot \\left( r _ { h } ^ { \\tau } + \\widehat { V } _ { h + 1 } \\left( s _ { h + 1 } ^ { \\tau } \\right) - \\left( \\tau _ { h } \\widehat { V } _ { h + 1 } \\right) \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\right) } { \\widehat { \\sigma } _ { h } ^ { 2 } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) } } \\end{array}$ ", + "bbox": [ + 173, + 599, + 630, + 627 + ], + "page_idx": 28 + }, + { + "type": "equation", + "img_path": "images/1c1b5aa50ce8df94b8efcae8486f1d57b0c588a509c4625690cc765bc6fbc3b9.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathbb { E } _ { \\pi _ { h } } \\left[ \\phi ( s , a ) \\right] ^ { \\top } \\left| \\widehat { \\Lambda } _ { h } ^ { - 1 } \\displaystyle \\sum _ { \\tau = 1 } ^ { K } \\frac { \\phi \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\cdot \\left( r _ { h } ^ { \\tau } + \\widehat { V } _ { h + 1 } \\left( s _ { h + 1 } ^ { \\tau } \\right) - \\left( \\widehat { T } _ { h } \\widehat { V } _ { h + 1 } \\right) \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\right) } { \\widehat { \\sigma } _ { h } ^ { 2 } \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) } \\right| } \\\\ & { \\leq \\mathbb { E } _ { \\pi _ { h } } \\left[ \\phi \\right] ^ { \\top } \\cdot \\left| \\widehat { \\Lambda } _ { h } ^ { - 1 } A _ { h } \\right| + \\mathbb { E } _ { \\pi _ { h } } \\left[ \\phi \\right] ^ { \\top } \\left| \\widehat { \\Lambda } _ { h } ^ { - 1 } \\displaystyle \\sum _ { \\tau = 1 } ^ { K } \\frac { \\phi \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\cdot \\left( 7 _ { h } \\widehat { V } _ { h + 1 } \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) - \\widehat { T } _ { h } \\widehat { V } _ { h + 1 } \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\right) } { \\widehat { \\sigma } _ { h } ^ { 2 } \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) } \\right| } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 183, + 631, + 816, + 736 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "For the second term, it can be bounded similar to term (iii) in Lemma D.1 and for the first term we have the following: ", + "bbox": [ + 173, + 755, + 823, + 785 + ], + "page_idx": 28 + }, + { + "type": "equation", + "img_path": "images/51c2f29c3c26597a16833d6cb7a516babc4504603a4b0f58ce219e96b5ecd7c1.jpg", + "text": "$$\n\\begin{array} { r l } & { \\quad \\mathbb E _ { \\pi _ { h } } \\left[ \\phi \\right] ^ { \\top } \\cdot \\left| \\widehat \\Lambda _ { h } ^ { - 1 } A _ { h } \\right| = \\mathbb E _ { \\pi _ { h } } \\left[ \\phi \\right] ^ { \\top } \\cdot \\widehat \\Lambda _ { h } ^ { - 1 } \\cdot \\widehat \\Lambda _ { h } \\left| \\widehat \\Lambda _ { h } ^ { - 1 } A _ { h } \\right| \\leq \\| \\mathbb E _ { \\pi _ { h } } \\left[ \\phi \\right] \\| _ { \\widehat \\Lambda _ { h } ^ { - 1 } } \\cdot \\left\\| \\widehat \\Lambda _ { h } | \\widehat \\Lambda _ { h } ^ { - 1 } A _ { h } \\right| \\Big \\| _ { \\widehat \\Lambda _ { h } ^ { - 1 } } } \\\\ & { \\leq \\| \\mathbb E _ { \\pi _ { h } } [ \\phi ] \\| _ { \\widehat \\Lambda _ { h } ^ { - 1 } } \\cdot \\| | A _ { h } | \\| _ { \\widehat \\Lambda _ { h } ^ { - 1 } } \\leq \\widetilde O ( \\sqrt d ) \\| \\mathbb E _ { \\pi _ { h } } [ \\phi ] \\| _ { \\widehat \\Lambda _ { h } ^ { - 1 } } \\leq \\widetilde O ( \\sqrt d \\| \\mathbb E _ { \\pi _ { h } } [ \\phi ] \\| _ { \\Lambda _ { h } ^ { - 1 } } ) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 200, + 786, + 800, + 844 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "where the first inequality uses Cauchy’s inequality, the second inequality uses $\\widehat { \\Lambda } _ { h }$ is coordinate-wise positive (since we assume here $\\phi \\geq 0$ ), the third inequality is identical to the analysis in Section C.2.2 and the fourth inequality is identical to the analysis in Section C.2.1 with $\\phi$ replaced by $\\mathbb { E } [ \\phi ]$ . Plug this back to equation 22 we finish the proof for the first part. For the second part, converting $\\Lambda _ { h } ^ { - 1 }$ to $\\Lambda _ { h } ^ { \\star - 1 }$ is identical to Section C.5. This finishes the proof. □ ", + "bbox": [ + 173, + 849, + 826, + 925 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "E PROOF OF MINIMAX LOWER BOUND THEOREM 3.5 ", + "text_level": 1, + "bbox": [ + 173, + 102, + 633, + 118 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "The proof follows the lower bound proof of Zanette et al. (2021). For completeness, we provide all the details in below. ", + "bbox": [ + 174, + 132, + 823, + 161 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "E.1 CONSTRUCTION ", + "text_level": 1, + "bbox": [ + 174, + 178, + 330, + 193 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "Similar to the proof of [Zanette et al. (2021), Theorem 2], we construct a family of MDPs, each parameterized by a Boolean vector $u = ( u _ { 1 } , \\dotsc , u _ { H } )$ with each $u _ { h } \\in \\{ - 1 , + 1 \\} ^ { d - 2 }$ for $h \\in [ H ]$ . The MDPs share the same transition kernel and are only different in the reward observations. ", + "bbox": [ + 174, + 204, + 826, + 247 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "State space: At each time step $h$ , there are two states $S = \\{ + 1 , - 1 \\}$ . ", + "bbox": [ + 173, + 258, + 633, + 275 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "Action space: The action space $\\mathcal { A } = \\{ - 1 , 0 , + 1 \\} ^ { d - 2 }$ ", + "bbox": [ + 176, + 279, + 532, + 294 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "Feature map: The feature map $\\phi : \\mathcal { S } \\times \\mathcal { A } \\mapsto \\mathbb { R } ^ { d }$ is given by ", + "bbox": [ + 174, + 296, + 578, + 313 + ], + "page_idx": 29 + }, + { + "type": "equation", + "img_path": "images/e1dfeff32ff79a9bba1fe0656c47f727ed4ccbb2dceebd6cfa36c56acdd7a11a.jpg", + "text": "$$\n\\phi ( + 1 , a ) = \\left( \\frac { \\frac { a } { \\sqrt { 2 d } } } { \\frac { 1 } { \\sqrt { 2 } } } \\right) \\in \\mathbb R ^ { d } , \\qquad \\phi ( - 1 , a ) = \\left( \\begin{array} { c } { \\frac { a } { \\sqrt { 2 d } } } \\\\ { 0 } \\\\ { \\frac { 1 } { \\sqrt { 2 } } } \\end{array} \\right) \\in \\mathbb R ^ { d } .\n$$", + "text_format": "latex", + "bbox": [ + 326, + 318, + 728, + 371 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "The construction ensures the condition $\\| \\phi ( s , a ) \\| _ { 2 } \\leq 1$ for any $( s , a ) \\in S \\times A$ . ", + "bbox": [ + 235, + 376, + 748, + 392 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "Transition kernel: The transition probability $P _ { h } ( s ^ { \\prime } \\mid s , a )$ is independent of action $a$ . In other words, the Markov decision process reduces to a homogeneous Markov chain with transition matrix ", + "bbox": [ + 174, + 395, + 825, + 436 + ], + "page_idx": 29 + }, + { + "type": "equation", + "img_path": "images/9dad2d0043ca8e47f0b00dc96d3912a5b0e04e58bea25883de45fbe47c94cc26.jpg", + "text": "$$\n\\mathbf { P } = \\left( \\begin{array} { l l } { \\frac { 1 } { 2 } } & { \\frac { 1 } { 2 } } \\\\ { \\frac { 1 } { 2 } } & { \\frac { 1 } { 2 } } \\end{array} \\right) \\in \\mathbb { R } ^ { 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 455, + 440, + 599, + 477 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "By letting ", + "bbox": [ + 232, + 481, + 300, + 496 + ], + "page_idx": 29 + }, + { + "type": "equation", + "img_path": "images/d7d1a311b7efc080ab4bc8dae6e7dde33d0b8694fea5efd5991f12efc9118abe.jpg", + "text": "$$\n\\nu _ { h } ( + 1 ) = \\nu _ { h } ( - 1 ) = \\left( \\begin{array} { l } { \\mathbf { 0 } _ { d - 2 } } \\\\ { \\frac { 1 } { \\sqrt { 2 } } } \\\\ { \\frac { 1 } { \\sqrt { 2 } } } \\end{array} \\right) \\in \\mathbb { R } ^ { d } ,\n$$", + "text_format": "latex", + "bbox": [ + 401, + 502, + 653, + 555 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "we have $P _ { h } ( s ^ { \\prime } \\mid s , a ) = \\langle \\phi ( s , a ) , \\nu _ { h } ( s ^ { \\prime } ) \\rangle$ to be a valid probability transition. ", + "bbox": [ + 230, + 560, + 730, + 577 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "Reward observations: For any MDP $M _ { u }$ , at each times step $h$ , the reward follows a Gaussian distribution with ", + "bbox": [ + 176, + 579, + 821, + 608 + ], + "page_idx": 29 + }, + { + "type": "equation", + "img_path": "images/2fc126cd1b7a83e0305039588068d91021dbbd64b4b1f32b25e948df25672477.jpg", + "text": "$$\nR _ { u , h } ( s , a ) \\sim \\mathcal { N } \\bigg ( \\frac { s } { \\sqrt { 6 } } + \\frac { \\delta } { \\sqrt { 2 d } } \\langle a , u _ { h } \\rangle , 1 \\bigg ) ,\n$$", + "text_format": "latex", + "bbox": [ + 387, + 611, + 666, + 647 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "where $\\delta \\in \\left[ 0 , \\frac { 1 } { \\sqrt { 3 d } } \\right]$ determines to what extent the MDP models are different from each other. The mean reward function satisfies $r _ { u , h } ( s , a ) = \\langle \\phi ( s , a ) , \\theta _ { u , h } \\rangle$ with ", + "bbox": [ + 230, + 651, + 823, + 686 + ], + "page_idx": 29 + }, + { + "type": "equation", + "img_path": "images/387b539601f534a0926c91e4b507cc96d4d34cc1274e0146ca9cfacadda5d3e3.jpg", + "text": "$$\n\\theta _ { u , h } = \\left( \\begin{array} { c } { \\delta u _ { h } } \\\\ { \\frac { 1 } { \\sqrt { 3 } } } \\\\ { - \\frac { 1 } { \\sqrt { 3 } } } \\end{array} \\right) \\in \\mathbb { R } ^ { d } .\n$$", + "text_format": "latex", + "bbox": [ + 447, + 693, + 607, + 746 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "Offline data collection Scheme: The dataset D = {(sτh, aτh, rτh, sτh+1)}h∈[H]τ∈[K] consist of $K$ i.i.d. trajectories. All the trajectories initiate from uniform distribution. We take a behavior policy $\\textstyle \\mu ( \\cdot \\mid s )$ that is independent of state $s$ . Let $\\{ e _ { 1 } , e _ { 2 } , \\ldots , e _ { d - 2 } \\}$ be the canonical bases of $\\mathbb { R } ^ { d - 2 }$ and ${ \\bf 0 } _ { d - 2 } \\in \\mathbb { R } ^ { \\hat { d } - 2 }$ be the zero vector. The behavior policy $\\mu$ is set as ", + "bbox": [ + 173, + 755, + 826, + 818 + ], + "page_idx": 29 + }, + { + "type": "equation", + "img_path": "images/5ade7d5551401bacb16a2df6386c1dee7e5afc577aa295846707b3df45a7b9b2.jpg", + "text": "$$\n\\begin{array} { r } { \\mu ( e _ { j } \\mid s ) = \\frac { 1 } { d } \\quad \\mathrm { f o r ~ a n y ~ } j \\in [ d - 2 ] \\qquad \\mathrm { a n d } \\qquad \\mu ( \\mathbf { 0 } _ { d - 2 } \\mid s ) = \\frac { 2 } { d } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 308, + 821, + 746, + 842 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "E.2 OVERVIEW OF PROOF ", + "text_level": 1, + "bbox": [ + 174, + 856, + 366, + 869 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "The proof of the theorem is based on Assouad’s method, where we first reduce the problem to binary hypothesis tests (Lemmas E.1 and E.2) and then connect the testing error to the uncertainty quantity in the upper bound (Lemma E.3). ", + "bbox": [ + 174, + 881, + 823, + 924 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "Lemma E.1 (Reduction to testing). There exists a universal constant $c _ { 1 } > 0$ such thata ", + "bbox": [ + 173, + 103, + 750, + 119 + ], + "page_idx": 30 + }, + { + "type": "equation", + "img_path": "images/2df7381d6c3389c357588db2de8e89b9f36e13434fd9e80ac39e43fdb9b7e874.jpg", + "text": "$$\n\\operatorname* { i n f } _ { \\hat { \\pi } } \\operatorname* { m a x } _ { u \\in \\mathcal { U } } \\mathbb { E } _ { u } \\left[ V _ { u } ^ { \\star } - V _ { u } ^ { \\widehat { \\pi } } \\right] \\geq c _ { 1 } \\delta \\sqrt { d } H \\operatorname* { m i n } _ { u , u ^ { \\prime } \\in \\mathcal { U } : D _ { H } ( u ^ { \\prime } ; u ) = 1 } \\operatorname* { i n f } _ { \\psi } \\big [ \\mathbb { P } _ { u } ( \\psi \\neq u ) + \\mathbb { P } _ { u ^ { \\prime } } ( \\psi \\neq u ^ { \\prime } ) \\big ] ,\n$$", + "text_format": "latex", + "bbox": [ + 194, + 126, + 772, + 154 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "where $\\widehat { \\pi }$ denotes the output of any algorithm that maps from observations to an estimated policy. $\\psi$ is bany test function for parameter $u$ and $D _ { H }$ is the hamming distance. ", + "bbox": [ + 174, + 162, + 825, + 191 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "Lemma E.2. There exists a universal constant $c _ { 2 } > 0$ such that when taking $\\begin{array} { r } { \\delta : = \\frac { c _ { 2 } d } { \\sqrt { K } } } \\end{array}$ ", + "bbox": [ + 171, + 196, + 803, + 215 + ], + "page_idx": 30 + }, + { + "type": "equation", + "img_path": "images/1626d6434a77e1532534edfdb818bab2f7aa1e26a910bdd43b01a5bda53846b9.jpg", + "text": "$$\n\\operatorname* { m i n } _ { u , u ^ { \\prime } \\in \\mathcal { U } : D _ { H } ( u ^ { \\prime } ; u ) = 1 } \\operatorname* { i n f } _ { \\psi } \\left[ \\mathbb { P } _ { u } ( \\psi \\neq u ) + \\mathbb { P } _ { u ^ { \\prime } } ( \\psi \\neq u ^ { \\prime } ) \\right] \\geq \\frac { 1 } { 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 310, + 224, + 686, + 257 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "When $\\begin{array} { r } { K \\gtrsim d ^ { 3 } , \\delta : = \\frac { c _ { 2 } d } { \\sqrt { K } } } \\end{array}$ ensures that $\\delta \\leq 1 / \\sqrt { 3 d }$ . Combining Lemmas E.1 and E.2 yields a lower bound ", + "bbox": [ + 174, + 273, + 826, + 306 + ], + "page_idx": 30 + }, + { + "type": "equation", + "img_path": "images/0df3e157b981b28b9ecb2ed0f5831484d2eb41e45a32efe00e7c6f2e94bd2ac7.jpg", + "text": "$$\n\\operatorname* { i n f } _ { \\widehat { \\pi } } \\operatorname* { m a x } _ { u \\in \\mathcal { U } } \\mathbb { E } _ { u } \\left[ V _ { u } ^ { \\star } - V _ { u } ^ { \\widehat { \\pi } } \\right] \\geq c \\ \\frac { d \\sqrt { d } H } { \\sqrt { K } } ,\n$$", + "text_format": "latex", + "bbox": [ + 377, + 313, + 619, + 349 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "where lower $c > 0$ isto universal constan uncertainty term E.3 to connect the abovefor the chosen linear MDP $\\begin{array} { r } { \\sqrt { d } \\cdot \\sum _ { h = 1 } ^ { H } \\sqrt { \\mathbb { E } _ { \\pi ^ { \\star } } [ \\phi ] ^ { \\top } ( \\Lambda _ { h } ^ { \\star } ) ^ { - 1 } \\mathbb { E } _ { \\pi ^ { \\star } } [ \\phi ] } } \\end{array}$ $\\mathcal { M }$ ", + "bbox": [ + 173, + 356, + 825, + 401 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "Lemma E.3. There exists a universal constant $c _ { 3 } > 0$ such that for all $M \\in \\mathcal { M }$ , ", + "bbox": [ + 173, + 406, + 704, + 421 + ], + "page_idx": 30 + }, + { + "type": "equation", + "img_path": "images/84c24c590868f817e43aaaebdf6a7cd6f6fefa63544a96a29fd83ffe4f9f7227.jpg", + "text": "$$\n\\sum _ { h = 1 } ^ { H } \\sqrt { \\mathbb { E } _ { \\pi ^ { \\star } } [ \\phi ] ^ { \\top } ( \\Lambda _ { h } ^ { \\star } ) ^ { - 1 } \\mathbb { E } _ { \\pi ^ { \\star } } [ \\phi ] } \\le c _ { 3 } \\frac { d H } { \\sqrt { K } } .\n$$", + "text_format": "latex", + "bbox": [ + 357, + 429, + 637, + 473 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "Plugging inequality equation E.3 into the bound equation 25, we obtain the minimax lower bound equation 6 in the statement of theorem. ", + "bbox": [ + 171, + 487, + 826, + 517 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "E.3 REDUCTION TO TESTING VIA ASSOUAD’S METHOD ", + "text_level": 1, + "bbox": [ + 173, + 534, + 571, + 549 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "Proof of Lemma E.1. For any index vector $u = ( u _ { 1 } , \\ldots , u _ { H } ) \\in \\mathcal { U } = \\{ - 1 , + 1 \\} ^ { ( d - 2 ) \\times H }$ , the optimal policy for MDP instance $M _ { u }$ is simply ", + "bbox": [ + 173, + 560, + 823, + 590 + ], + "page_idx": 30 + }, + { + "type": "equation", + "img_path": "images/26b197a0e24a5468951bb74c2920ae4910de2848d5339e57c4f5a56ddc38fa0f.jpg", + "text": "$$\n\\pi _ { h } ^ { \\star } ( \\cdot ) = u _ { h } \\qquad { \\mathrm { f o r ~ } } h \\in [ H ] .\n$$", + "text_format": "latex", + "bbox": [ + 403, + 598, + 594, + 616 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "Similar to the proof of Lemma 9 in Zanette et al. (2021), we can show that the value suboptimality of policy $\\pi$ on MDP $M _ { u }$ is given by ", + "bbox": [ + 174, + 623, + 825, + 652 + ], + "page_idx": 30 + }, + { + "type": "equation", + "img_path": "images/2b43216e57f9fb9f0219e9019fe50774d8c02552f67ae9630f35f6686a0d63ba.jpg", + "text": "$$\nV _ { u } ^ { \\star } - V _ { u } ^ { \\pi } = \\frac { \\delta } { \\sqrt { 2 d } } \\sum _ { h = 1 } ^ { H } \\left\\| u _ { h } - \\mathbb { E } _ { \\pi } [ a _ { h } ] \\right\\| _ { 1 } .\n$$", + "text_format": "latex", + "bbox": [ + 366, + 660, + 632, + 704 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "Define $u ^ { \\pi } = ( u _ { 1 } ^ { \\pi } , \\ldots , u _ { H } ^ { \\pi } )$ with $u _ { h } ^ { \\pi } : = \\mathrm { s i g n } \\left( \\mathbb { E } _ { \\pi } [ a _ { h } ] \\right)$ , then the $\\ell _ { 1 }$ -norm is lower bounded as ", + "bbox": [ + 171, + 712, + 776, + 729 + ], + "page_idx": 30 + }, + { + "type": "equation", + "img_path": "images/dc43c845fe4a0b965b33999cb90b3e743d2a88445faa97c88fe263d1ee2497d5.jpg", + "text": "$$\n\\begin{array} { r } { \\left\\| u _ { h } - \\mathbb { E } _ { \\pi } [ a _ { h } ] \\right\\| _ { 1 } \\ge D _ { H } ( u _ { h } ^ { \\pi } ; u _ { h } ) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 387, + 737, + 607, + 758 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "where $D _ { H } ( \\cdot ; \\cdot )$ denotes the Hamming distance. It follows that ", + "bbox": [ + 173, + 765, + 581, + 781 + ], + "page_idx": 30 + }, + { + "type": "equation", + "img_path": "images/cd698ccbd7f0d8e7b946421711c5d8f45ced613dd48d328acb36233aae2c5eea.jpg", + "text": "$$\nV _ { u } ^ { \\star } - V _ { u } ^ { \\pi } \\geq \\frac { \\delta } { \\sqrt { 2 d } } D _ { H } ( u ^ { \\pi } ; u ) .\n$$", + "text_format": "latex", + "bbox": [ + 398, + 787, + 599, + 823 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "We then apply Assouad’s method (Lemma 2.12 in Sampson and Guttorp (1992)) and obtain that ", + "bbox": [ + 173, + 829, + 803, + 845 + ], + "page_idx": 30 + }, + { + "type": "equation", + "img_path": "images/0b0baeb06bc364e550d5dd6612451325e14c5c58e66d650444cfdb232b1a9819.jpg", + "text": "$$\n\\operatorname* { i n f } _ { \\substack { \\hat { u } \\in \\mathcal { U } } } \\operatorname* { m a x } _ { \\boldsymbol { u } \\in \\mathcal { U } } \\mathbb { E } _ { \\boldsymbol { u } } \\big [ D _ { \\boldsymbol { H } } ( \\hat { u } ; \\boldsymbol { u } ) \\big ] \\geq \\frac { ( d - 2 ) H } { 2 } \\operatorname* { m i n } _ { \\substack { \\boldsymbol { u } , \\boldsymbol { u } ^ { \\prime } \\in \\mathcal { U } : D _ { \\boldsymbol { H } } ( \\boldsymbol { u } ^ { \\prime } ; \\boldsymbol { u } ) = 1 } } \\operatorname* { i n f } _ { \\boldsymbol { \\psi } } \\big [ \\mathbb { P } _ { \\boldsymbol { u } } ( \\boldsymbol { \\psi } \\neq \\boldsymbol { u } ) + \\mathbb { P } _ { \\boldsymbol { u } ^ { \\prime } } ( \\boldsymbol { \\psi } \\neq \\boldsymbol { u } ^ { \\prime } ) \\big ] ,\n$$", + "text_format": "latex", + "bbox": [ + 189, + 852, + 779, + 886 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "where $\\psi$ is any test functions mapping from observations to $\\{ u , u ^ { \\prime } \\}$ . Combining inequalities equation 27 and equation 28, we finish the proof. □ ", + "bbox": [ + 174, + 895, + 828, + 924 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "E.4 LOWER BOUND ON THE TESTING ERROR ", + "text_level": 1, + "bbox": [ + 173, + 103, + 495, + 118 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "Proof of Lemma $E . 2$ . The proof of Lemma E.2 is similar to that of Lemma 10 in Zanette et al. (2021). We first apply Theorem 2.12 in Sampson and Guttorp (1992) to lower bound the testing error using Kullback–Leibler divergence and obtain ", + "bbox": [ + 173, + 128, + 828, + 172 + ], + "page_idx": 31 + }, + { + "type": "equation", + "img_path": "images/03bba56ee7da373f751e758361460cde4c0c408ae5e8736b415f1aa5096762cd.jpg", + "text": "$$\n\\operatorname* { m i n } _ { \\substack { \\tau , u ^ { \\prime } \\in \\mathcal { U } : D _ { H } ( u ^ { \\prime } ; u ) = 1 } } \\operatorname* { i n f } _ { \\psi } \\left[ \\mathbb { P } _ { u } ( \\psi \\neq u ) + \\mathbb { P } _ { u ^ { \\prime } } ( \\psi \\neq u ^ { \\prime } ) \\right] \\geq 1 - \\left( \\frac { 1 } { 2 } \\operatorname* { m a x } _ { \\substack { u , u ^ { \\prime } \\in \\mathcal { U } : D _ { H } ( u ^ { \\prime } ; u ) = 1 } } D _ { \\mathrm { K L } } ( Q _ { u } \\| Q _ { u ^ { \\prime } } ) \\right) ^ { 1 / 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 181, + 179, + 838, + 210 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "It only remains to estimate $D _ { \\mathrm { K L } } ( \\mathcal { Q } _ { u } \\Vert \\mathcal { Q } _ { u ^ { \\prime } } )$ . ", + "bbox": [ + 174, + 232, + 459, + 248 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "The probability density $\\mathcal { Q } _ { u }$ takes the form ", + "bbox": [ + 174, + 253, + 452, + 270 + ], + "page_idx": 31 + }, + { + "type": "equation", + "img_path": "images/f704540d4daec5b3af6eb98b0c2f3750e751b757ccda1952ab9801a4e09dd934.jpg", + "text": "$$\n{ \\mathcal { Q } } _ { u } ( { \\mathcal { D } } ) = \\prod _ { k = 1 } ^ { K } \\xi _ { 1 } ( s _ { 1 } ^ { k } ) \\prod _ { h = 1 } ^ { H } \\mu ( a _ { h } ^ { k } \\mid s _ { h } ^ { k } ) \\left[ R _ { u , h } ( s _ { h } ^ { k } , a _ { h } ^ { k } ) \\right] ( r _ { h } ^ { k } ) \\mathbb { P } _ { h } ( s _ { h + 1 } ^ { k } \\mid s _ { h } ^ { k } , a _ { h } ^ { k } )\n$$", + "text_format": "latex", + "bbox": [ + 250, + 275, + 746, + 319 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "where $\\xi _ { 1 } = \\left[ \\textstyle { \\frac { 1 } { 2 } } , \\frac { 1 } { 2 } \\right]$ is the initial distribution. It follows that ", + "bbox": [ + 173, + 324, + 558, + 343 + ], + "page_idx": 31 + }, + { + "type": "equation", + "img_path": "images/c361c204140da4db2570694ae47c94812bcd3651f50d11ddca9f361bf632e2ef.jpg", + "text": "$$\n\\begin{array} { l } { { \\displaystyle D _ { K L } \\big ( \\mathcal { Q } _ { u } \\big \\| \\mathcal { Q } _ { u ^ { \\prime } } \\big ) = \\mathbb { E } _ { u } \\big [ \\log \\big ( \\mathcal { Q } _ { u } / \\mathcal { Q } _ { u ^ { \\prime } } \\big ) \\big ] } \\ ~ } \\\\ { { \\displaystyle = K \\cdot \\sum _ { h = 1 } ^ { H } \\mathbb { E } _ { u } \\Big [ \\log \\big ( \\big [ R _ { u , h } ( s _ { h } ^ { 1 } , a _ { h } ^ { 1 } ) \\big ] ( r _ { h } ^ { 1 } ) \\big / \\big [ R _ { u ^ { \\prime } , h } ( s _ { h } ^ { 1 } , a _ { h } ^ { 1 } ) \\big ] ( r _ { h } ^ { 1 } ) \\big ) \\Big ] } \\ ~ } \\\\ { { \\displaystyle = \\frac { K } { d } \\sum _ { j = 1 } ^ { d - 2 } D _ { \\mathrm { K L } } \\Big ( \\mathcal { N } \\big ( \\frac { \\delta } { \\sqrt { 2 d } } \\langle e _ { j } , u _ { h } \\rangle , 1 \\big ) \\ \\Big \\| \\mathcal { N } \\big ( \\frac { \\delta } { \\sqrt { 2 d } } \\langle e _ { j } , u _ { h } ^ { \\prime } \\rangle , 1 \\big ) \\Big ) . } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 285, + 364, + 712, + 478 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "If we take $\\begin{array} { r } { \\delta = \\frac { c _ { 2 } d } { \\sqrt { K } } } \\end{array}$ , then inequality equation 29 ensures inequality equation 24, as claimed in the statement of the lemma. ", + "bbox": [ + 176, + 491, + 825, + 523 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "E.5 CONNECTION TO THE UNCERTAINTY TERM ", + "text_level": 1, + "bbox": [ + 173, + 561, + 514, + 577 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "Proof of Lemma E.3. We first calculate the explicit form of the inverse of variance-rescaled covariance matrix $\\Lambda _ { h } ^ { \\star , p }$ . For each time step $h \\in [ H ]$ , the value function $V _ { u , h + 1 } ^ { \\star }$ takes the form ", + "bbox": [ + 173, + 587, + 826, + 618 + ], + "page_idx": 31 + }, + { + "type": "equation", + "img_path": "images/95bbc5a37d479b48d714767c7391c37e8272e002bfb4f79df31475f1ffabe892.jpg", + "text": "$$\n\\begin{array} { r } { V _ { u , h + 1 } ^ { \\star } = \\mathbb { E } _ { \\pi ^ { \\star } } r _ { u , h + 1 } + \\bigl ( \\mathbb { P } _ { h + 1 } ^ { \\pi ^ { \\star } } V _ { u , h + 2 } ^ { \\star } \\bigr ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 367, + 625, + 629, + 646 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "Sinc $\\begin{array} { r } { \\colon \\bigl ( \\mathbb { P } _ { h + 1 } V _ { u , h + 2 } ^ { \\star } \\bigr ) ( + 1 ) = \\bigl ( \\mathbb { P } _ { h + 1 } V _ { u , h + 2 } ^ { \\star } \\bigr ) ( - 1 ) \\mathrm { ~ a n d ~ } r _ { u , h + 1 } ( + 1 , a ) - r _ { u , h + 1 } ( - 1 , a ) = 2 / \\sqrt { 6 } } \\end{array}$ , we have ", + "bbox": [ + 169, + 651, + 825, + 684 + ], + "page_idx": 31 + }, + { + "type": "equation", + "img_path": "images/ea89f73cc1527b0d313fe1e05447089a4c60dbf4c109b679b23131f4b411c331.jpg", + "text": "$$\n\\operatorname { V a r } _ { P _ { h } } ( V _ { u , h + 1 } ^ { \\star } ) ( + 1 , a ) = \\operatorname { V a r } _ { P _ { h } } ( \\mathbb { E } _ { \\pi ^ { \\star } } r _ { u , h + 1 } ) ( + 1 , a ) = \\frac 1 6 .\n$$", + "text_format": "latex", + "bbox": [ + 303, + 686, + 694, + 717 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "Similarly, ", + "bbox": [ + 173, + 720, + 240, + 736 + ], + "page_idx": 31 + }, + { + "type": "equation", + "img_path": "images/d0f76ad4e54054ccc30d18ab5c112b329e34afc7f3ec2141429969ee8180bb24.jpg", + "text": "$$\n\\operatorname { V a r } _ { P _ { h } } ( V _ { u , h + 1 } ^ { \\star } ) ( - 1 , a ) = \\operatorname { V a r } _ { P _ { h } } ( V _ { u , h + 1 } ^ { \\star } ) ( + 1 , a ) = \\frac { 1 } { 6 } .\n$$", + "text_format": "latex", + "bbox": [ + 316, + 742, + 681, + 772 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "By routine calculation, we find that the population-level rescaled covariance matrix takes the form ", + "bbox": [ + 169, + 776, + 816, + 791 + ], + "page_idx": 31 + }, + { + "type": "equation", + "img_path": "images/368c47d0536fb454399f7b2e6439e98ceb7915b6eaa9e0fb3599c0336a37f507.jpg", + "text": "$$\n\\Lambda _ { h } ^ { \\star , p } = \\frac { 3 K } { 2 } \\left( \\begin{array} { c c } { \\frac { 2 } { d ^ { 2 } } \\mathbf { I } _ { d - 2 } } & { \\frac { 1 } { d \\sqrt { d } } \\mathbf { 1 } _ { ( d - 2 ) \\times 2 } } \\\\ { \\frac { 1 } { d \\sqrt { d } } \\mathbf { 1 } _ { 2 \\times ( d - 2 ) } } & { \\mathbf { I } _ { 2 } } \\end{array} \\right) \\in \\mathbb { R } ^ { d \\times d }\n$$", + "text_format": "latex", + "bbox": [ + 316, + 796, + 681, + 840 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "for any $h \\in [ H ]$ . Applying Gaussian elimination on $\\Lambda _ { h } ^ { \\star , p }$ , we have ", + "bbox": [ + 173, + 844, + 609, + 862 + ], + "page_idx": 31 + }, + { + "type": "equation", + "img_path": "images/845a75076e13f05d301bc2194eeb2a28df7efbbcc1eecd00034442f949a0c4ce.jpg", + "text": "$$\n( \\Lambda _ { h } ^ { \\star , p } ) ^ { - 1 } = \\frac { 2 } { 3 K } \\left( \\begin{array} { c c } { { \\frac { d ^ { 2 } } { 2 } \\bigl \\{ { \\bf I } _ { d - 2 } + \\frac { 1 } { d - 2 } { \\bf 1 } _ { ( d - 2 ) \\times ( d - 2 ) } \\bigr \\} } } & { { - \\frac { d \\sqrt { d } } { 2 ( d - 2 ) } { \\bf 1 } _ { ( d - 2 ) \\times 2 } } } \\\\ { { - \\frac { d \\sqrt { d } } { 2 ( d - 2 ) } { \\bf 1 } _ { 2 \\times ( d - 2 ) } } } & \\frac { 1 } { d - 2 } \\left( \\begin{array} { c c } { { d - 1 } } & { { 1 } } \\\\ { { 1 } } & { { d - 1 } } \\end{array} \\right) \\right) . \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 243, + 867, + 756, + 922 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "For each time step $h \\in [ H ]$ , we have (by Jensen’s inequality) ", + "bbox": [ + 173, + 102, + 575, + 119 + ], + "page_idx": 32 + }, + { + "type": "equation", + "img_path": "images/8456631e02a79f03e5db48b39901100235e634bcbd9267c83299588f2b3afa2f.jpg", + "text": "$$\n\\sqrt { \\mathbb { E } _ { \\pi ^ { \\star } } [ \\phi ] ^ { \\top } ( \\Lambda _ { h } ^ { \\star } ) ^ { - 1 } \\mathbb { E } _ { \\pi ^ { \\star } } [ \\phi ] } \\le \\frac { 1 } { 2 } \\bigl \\| \\phi ( + 1 , u _ { h } ) \\bigr \\| _ { ( \\Lambda _ { h } ^ { \\star , p } ) ^ { - 1 } } + \\frac { 1 } { 2 } \\bigl \\| \\phi ( - 1 , u _ { h } ) \\bigr \\| _ { ( \\Lambda _ { h } ^ { \\star , p } ) ^ { - 1 } } .\n$$", + "text_format": "latex", + "bbox": [ + 240, + 125, + 758, + 155 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "Recall that by our construction, ", + "bbox": [ + 173, + 160, + 382, + 174 + ], + "page_idx": 32 + }, + { + "type": "equation", + "img_path": "images/db59ea7733aa787868ea4c5dcedf88127148ef39af6419159dbc363dfe50c2c5.jpg", + "text": "$$\n\\phi ( + 1 , u _ { h } ) = \\left( \\begin{array} { c c } { \\frac { u _ { h } } { \\sqrt { 2 d } } } \\\\ { \\frac { 1 } { \\sqrt { 2 } } } \\\\ { 0 } \\end{array} \\right) \\in \\mathbb R ^ { d } , \\qquad \\phi ( - 1 , u _ { h } ) = \\left( \\begin{array} { c c } { \\frac { u _ { h } } { \\sqrt { 2 d } } } \\\\ { 0 } \\\\ { \\frac { 1 } { \\sqrt { 2 } } } \\end{array} \\right) \\in \\mathbb R ^ { d } .\n$$", + "text_format": "latex", + "bbox": [ + 289, + 181, + 709, + 233 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "It follows that ", + "bbox": [ + 173, + 238, + 267, + 252 + ], + "page_idx": 32 + }, + { + "type": "equation", + "img_path": "images/cfda210747c2f9cc1b67b07c4ac07e229826f2380b33d89bd1d329b80929a9d4.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle \\left\\| \\phi ( + 1 , u _ { h } ) \\right\\| _ { ( \\Lambda _ { h } ^ { \\star , p } ) ^ { - 1 } } ^ { 2 } = \\left\\| \\phi ( - 1 , u _ { h } ) \\right\\| _ { ( \\Lambda _ { h } ^ { \\star , p } ) ^ { - 1 } } ^ { 2 } } \\\\ { \\displaystyle = \\frac 2 { 3 K } \\biggl \\{ \\frac d 4 u _ { h } ^ { \\top } \\{ \\mathbf I _ { d - 2 } + \\frac 1 { d - 2 } \\mathbf 1 _ { ( d - 2 ) \\times ( d - 2 ) } \\} u _ { h } - \\frac d { 2 ( d - 2 ) } \\mathbf 1 _ { d - 2 } ^ { \\top } u _ { h } + \\frac d { 2 ( d - 2 ) } \\biggr \\} } \\\\ { \\displaystyle = \\frac 2 { 3 K } \\biggl \\{ \\frac { d ^ { 2 } } + \\frac { d } { 4 ( d - 2 ) } \\bigl ( 1 - \\mathbf 1 _ { d - 2 } ^ { \\top } u _ { h } \\bigr ) ^ { 2 } + \\frac 1 4 \\biggr \\} } \\\\ { \\displaystyle \\le \\frac 2 { 3 K } \\biggl \\{ \\frac { d ^ { 2 } } 4 + \\frac { d ( d - 1 ) ^ { 2 } } { 4 ( d - 2 ) } + \\frac 1 4 \\biggr \\} = \\frac 2 { 3 K } \\biggl \\{ \\frac { d ^ { 2 } } 2 + \\frac { d - 1 } { 2 ( d - 2 ) } \\biggr \\} \\lesssim d ^ { 2 } / K . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 235, + 256, + 761, + 390 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "Therefore, ", + "bbox": [ + 173, + 393, + 245, + 409 + ], + "page_idx": 32 + }, + { + "type": "equation", + "img_path": "images/abe82989f98b8a57411622561bbd1ec91eff4ac37b1d297c7f3fb979b3dbf170.jpg", + "text": "$$\n\\sqrt { \\mathbb { E } _ { \\pi ^ { \\star } } [ \\phi ] ^ { \\top } ( \\Lambda _ { h } ^ { \\star } ) ^ { - 1 } \\mathbb { E } _ { \\pi ^ { \\star } } [ \\phi ] } \\lesssim d / \\sqrt { K } .\n$$", + "text_format": "latex", + "bbox": [ + 377, + 415, + 620, + 441 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "Taking the summation over $h \\in [ H ]$ , we obtain the bound equation 26 as claimed in the lemma statement. ", + "bbox": [ + 173, + 446, + 826, + 477 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "F A NUMERICAL SIMULATION ", + "text_level": 1, + "bbox": [ + 174, + 517, + 442, + 534 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "F.1 A LINEAR MDP CONSTRUCTION ", + "text_level": 1, + "bbox": [ + 174, + 547, + 441, + 564 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "We consider a synthetic linear MDP example that is similar to Min et al. (2021) but with some modifications for the offline learning task. The MDP instance we use consists of $| S | = 2$ states and $| { \\mathcal { A } } | = 1 0 0$ actions, and feature dimension $d = 1 0 0$ . We set ${ \\mathcal { S } } = \\{ 0 , 1 \\}$ and $\\mathcal { A } = \\{ 0 , 1 , \\ldots , 9 9 \\}$ respectively. For each action $a \\ \\in \\ \\{ 0 , 1 , \\ldots , 9 9 \\}$ , we use binary encoding to obtain a vector $\\textbf { a } \\in \\ \\bar { \\mathbb { R } } ^ { 8 }$ using its binary representation (i.e. each coordinate is either 0 or 1). we interchangebly use $a$ and and its vector representation a for the ease of explanation. We first define $\\delta ( s , a ) = { \\left\\{ \\begin{array} { l l } { 1 } & { { \\mathrm { ~ i f ~ } } \\mathbf { 1 } \\{ s = 0 \\} = \\mathbf { 1 } \\{ a = 0 \\} } \\\\ { 0 } & { { \\mathrm { ~ o t h e r w i s e } } } \\end{array} \\right. }$ , then the non-stationary linear MDP is specified by the following configuration ", + "bbox": [ + 173, + 574, + 826, + 705 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "• Feature mapping: ", + "bbox": [ + 218, + 718, + 349, + 733 + ], + "page_idx": 32 + }, + { + "type": "equation", + "img_path": "images/b36c814460d8351217b297043f305d5cafab9180121e0bceb455c6032878d5a3.jpg", + "text": "$$\n\\phi ( s , a ) = \\left( \\mathbf { a } ^ { \\top } , \\delta ( s , a ) , 1 - \\delta ( s , a ) \\right) ^ { \\top } \\in \\mathbb { R } ^ { 1 0 }\n$$", + "text_format": "latex", + "bbox": [ + 379, + 731, + 676, + 753 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "• The true measure $\\nu _ { h }$ ", + "bbox": [ + 218, + 758, + 367, + 772 + ], + "page_idx": 32 + }, + { + "type": "equation", + "img_path": "images/777daf5c7ba850994e43e8a969674ff9d5da0bb32892f33162de262151a89821.jpg", + "text": "$$\n\\pmb { \\nu } _ { h } ( s ) = ( 0 , \\dots , 0 , ( 1 - s ) \\oplus \\alpha _ { h } , s \\oplus \\alpha _ { h } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 385, + 779, + 668, + 796 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "where $\\{ \\alpha _ { h } \\} _ { h \\in [ H ] }$ is a sequence of integers taking values 0 or 1 and $\\oplus$ is the standard XOR operator. We define ", + "bbox": [ + 230, + 803, + 823, + 829 + ], + "page_idx": 32 + }, + { + "type": "equation", + "img_path": "images/4f1adc907f64bca7eef416606994fbfbaf0a4b0b9d33b02357fbe47700777a5d.jpg", + "text": "$$\n\\theta _ { h } \\equiv ( 0 , \\ldots , 0 , r , 1 - r ) \\in \\mathbb { R } ^ { 1 0 }\n$$", + "text_format": "latex", + "bbox": [ + 423, + 828, + 632, + 847 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "with the choice of $r = 0 . 9$ . The transition follows $P _ { h } ( s ^ { \\prime } | s , a ) = \\langle \\phi ( s , a ) , \\nu _ { h } ( s ^ { \\prime } ) \\rangle$ and the mean reward function $r _ { h } ( s , a ) = \\langle \\phi ( s , a ) , \\theta _ { h } \\rangle$ . ", + "bbox": [ + 230, + 848, + 823, + 878 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "• Behavior policy: always choose action $a = 0$ with probability $p$ , and other actions uniformly with probability $( 1 - p ) / 9 9$ . The initial distribution chooses $s = 0$ and $s = 1$ with equal probability $1 / 2$ . We use $p = 0 . 6$ . ", + "bbox": [ + 220, + 882, + 823, + 924 + ], + "page_idx": 32 + }, + { + "type": "image", + "img_path": "images/34259f6dd9b65de1f148aa38cd9fca4ea9b5c62a39196fb77cf679acac11f205.jpg", + "image_caption": [ + "Figure 1: Comparison between PEVI and VAPVI in the non-stationary linear MDP instance described above. In each figure, y-axis denotes suboptimality gap $v ^ { \\star } - v ^ { \\widehat { \\pi } }$ , $\\mathbf { X }$ -axis denotes number of episodes $K$ . The problem horizons are fixed to be $H = 2 0 , 3 0 , 5 0$ . The solid line denotes the average suboptimality gap over 50 trials and the error bar area is the corresponding standard deviation. The range of $K$ is from 5 to 1000. " + ], + "image_footnote": [], + "bbox": [ + 199, + 109, + 799, + 239 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "F.2 EMPIRICAL COMPARISON BETWEEN PEVI AND VAPVI ON THE CONSTRUCTED LINEAR MDP ", + "text_level": 1, + "bbox": [ + 176, + 347, + 816, + 375 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "We compare Pessimistic Value Iteration (PEVI) in Jin et al. (2021b) and our VAPVI Algorithm 1 in Figure 1, with horizon to be $H = 2 0$ , 30, 50. In addition, we add the non-pessimistic version for both algorithms, i.e. least-square value iteration (LSVI) and variance-aware value iteration (VAVI). The true optimal value $v ^ { \\star }$ is computed via value iteration using the underlying transition kernels. For the empirical validation of VAPVI, we do not split the data and, in particular, in all the methods we choose $\\lambda = 0 . 0 1$ (instead of $\\lambda = 1$ used in theory (Jin et al., 2021b) which causes over-regularization in the simulation). ", + "bbox": [ + 173, + 386, + 825, + 484 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "We can observe VAPVI outperforms PEVI and the gap becomes larger when horizon $H$ increases. One main reason for this to happen is due to the bonus used in PEVI (Jin et al., 2021b) ", + "bbox": [ + 174, + 489, + 825, + 517 + ], + "page_idx": 33 + }, + { + "type": "equation", + "img_path": "images/594c0bc419c0cda2a54dd4500e87cb0c48d58e5891ae7bc01b341d5e77c72610.jpg", + "text": "$$\nO \\left[ d H \\cdot \\left( \\phi ( \\cdot , \\cdot ) ^ { \\top } \\Sigma _ { h } ^ { - 1 } \\phi ( \\cdot , \\cdot ) \\right) ^ { 1 / 2 } \\right]\n$$", + "text_format": "latex", + "bbox": [ + 392, + 518, + 604, + 550 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "is overly pessimistic comparing to our ", + "bbox": [ + 178, + 551, + 431, + 565 + ], + "page_idx": 33 + }, + { + "type": "equation", + "img_path": "images/4c5c22846ddf533900a827f047cb7c45b860d7c8a8107b8c8484b90932c5abff.jpg", + "text": "$$\nO \\left[ \\sqrt { d } \\cdot \\left( \\phi ( \\cdot , \\cdot ) ^ { \\top } \\Lambda _ { h } ^ { - 1 } \\phi ( \\cdot , \\cdot ) \\right) ^ { 1 / 2 } \\right]\n$$", + "text_format": "latex", + "bbox": [ + 393, + 568, + 604, + 599 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "when $H$ becomes larger and this could potentially make the learning less accurate. In addition, both non-pessimistic algorithms exhibit similar accuracy, and this is partially owing to our truncation scheme $\\widehat { \\sigma } _ { h } ( \\cdot , \\cdot ) ^ { 2 } \\gets \\operatorname* { m a x } \\{ 1 , \\widehat { \\mathrm { V a r } } _ { P _ { h } } \\widehat { V } _ { h + 1 } ( \\cdot , \\cdot ) \\}$ so $\\widehat { \\sigma } _ { h } ( \\cdot , \\cdot ) ^ { 2 }$ will just be 1 when the estimated variance b bis small. Lastly, variance-aware pessimism eventually outperforms non-pessimism algorithms when sample size is large and this might come from that the pessimistic bonus is estimated more accurately when more samples are collected. ", + "bbox": [ + 173, + 601, + 826, + 688 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "G SOME MISSING DERIVATIONS AND DISCUSSIONS ", + "text_level": 1, + "bbox": [ + 173, + 707, + 611, + 723 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "G.1 REGARDING COVERAGE ASSUMPTION ", + "text_level": 1, + "bbox": [ + 176, + 737, + 480, + 751 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "Now we discuss the feature coverage assumption. Indeed, even if Assumption 2.2 is not satisfied, we can still learn in the effective subspan of $\\dot { \\Sigma } _ { h } ^ { p } : = \\mathbb { E } _ { \\boldsymbol { \\mu } , h } \\left[ \\boldsymbol { \\phi } ( \\boldsymbol { s } , \\boldsymbol { a } ) \\boldsymbol { \\phi } ( \\boldsymbol { s } , \\boldsymbol { a } ) ^ { \\top } \\right] ^ { - }$ . Concretely, since $\\Sigma _ { h } ^ { p }$ is symmetric, by orthogonal decomposition we have $\\Sigma _ { h } ^ { p } = Z _ { h } \\Lambda Z _ { h } ^ { \\top }$ , where $Z _ { h }$ (can be estimated using the samples for practical purpose) consists of orthogonal basis and $\\Lambda$ consists of eigenvalues of $\\Sigma _ { h } ^ { p }$ in the diagonal. Suppose we do not have a full coverage, i.e. ", + "bbox": [ + 173, + 762, + 826, + 835 + ], + "page_idx": 33 + }, + { + "type": "equation", + "img_path": "images/fe1e90df137357cb059b6bdbc0fed0be19c068893d866b2332233e5ee2a96157.jpg", + "text": "$$\n\\Lambda = \\mathrm { d i a g } [ \\lambda _ { 1 } , \\lambda _ { 2 } , . . . , \\lambda _ { d ^ { \\prime } } , 0 , . . . , 0 ] \\quad \\mathrm { w i t h } \\quad d ^ { \\prime } < d\n$$", + "text_format": "latex", + "bbox": [ + 331, + 838, + 658, + 854 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "then we can create transformed features $\\phi _ { h } ^ { \\prime } ( s , a ) = Z _ { h } \\cdot \\phi _ { h } ( s , a )$ , and then ", + "bbox": [ + 173, + 856, + 668, + 872 + ], + "page_idx": 33 + }, + { + "type": "equation", + "img_path": "images/f3741c4b2a0a4e1c0b8293b70723c3bfd849a91f12022d51d21ca81b5c178447.jpg", + "text": "$$\n\\mathbb { E } _ { \\mu , h } \\left[ \\phi _ { h } ^ { \\prime } ( s , a ) \\phi _ { h } ^ { \\prime } ( s , a ) ^ { \\top } \\right] = \\Lambda = \\mathrm { d i a g } [ \\lambda _ { 1 } , \\lambda _ { 2 } , . . . , \\lambda _ { d ^ { \\prime } } , 0 , . . . , 0 ] .\n$$", + "text_format": "latex", + "bbox": [ + 292, + 875, + 702, + 893 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "Then we can do learning w.r.t. the truncated features $\\phi _ { h } ^ { \\prime } | _ { 1 : d ^ { \\prime } }$ ’s instead of the original $\\phi$ . It reduces to the weaker notion of $\\begin{array} { r } { \\kappa : = \\operatorname* { m i n } _ { h \\in [ H ] } \\{ \\kappa _ { h } : s . t } \\end{array}$ . $\\kappa _ { h } =$ smallest positive eigenvalue at time $h \\}$ . ", + "bbox": [ + 169, + 895, + 828, + 925 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "When reducing Theorem 3.2,3.3 to the tabular case, set $\\phi ( s , a ) = \\mathbf { 1 } _ { s , a }$ , $d = S A$ , $\\lambda = 0$ , and recall by Assumption 3.4 (let’s assume $\\pi ^ { \\star }$ is a deterministic policy as it always exists in tabular MDP) $\\begin{array} { r } { C ^ { \\star } : = \\operatorname* { s u p } _ { h , s , a } { d _ { h } ^ { \\pi ^ { \\star } } ( s , a ) / d _ { h } ^ { \\mu } ( s , a ) } } \\end{array}$ , then for Theorem 3.2 ", + "bbox": [ + 173, + 127, + 826, + 171 + ], + "page_idx": 34 + }, + { + "type": "equation", + "img_path": "images/59f2ef5560f1d835ea32092803d87e5991fa88dec2ee5dad95457f4c6dd2a4a6.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathcal { U } _ { \\ge } \\displaystyle \\sum _ { u = 1 } ^ { N } \\frac { \\mu } { \\mu } \\left. \\nabla ^ { \\mu } u ^ { \\mu } ( x ) \\frac { \\mu } { \\mu } \\nabla ^ { \\nu } \\cdot \\nabla ^ { \\mu } u ^ { \\nu } \\right. _ { \\perp } - \\mathcal { U } _ { \\ge } \\displaystyle \\sum _ { u = 1 } ^ { N } \\frac { \\mu } { \\mu } \\sum _ { u = 1 } ^ { N } \\mathcal { U } _ { \\ge } \\nu ( u ) \\hat { \\nu } ^ { \\mu } \\mathbb { I } _ { \\ge } \\mathbb { I } _ { \\ge u } ^ { \\nu } } \\\\ & { - \\mathcal { U } _ { \\ge } \\displaystyle \\sum _ { u = 1 } ^ { N } \\frac { \\mu } { \\mu } \\sum _ { u = 1 } ^ { N } \\mathcal { U } _ { \\ge } \\mathbb { I } _ { \\ge u } ^ { \\nu } \\mathbb { I } _ { \\ge u } ^ { \\nu } \\mathbb { I } _ { \\ge u } ^ { \\nu } \\mathbb { I } _ { \\ge u } ^ { \\nu } } \\\\ & { = - \\mathcal { U } _ { \\ge } \\displaystyle \\sum _ { u = 1 } ^ { N } \\mathbb { I } _ { \\ge u \\ne 1 } ^ { N } \\mathbb { I } _ { \\ge u } ^ { \\nu } \\mathbb { I } _ { \\ge u } ^ { \\nu } \\mathbb { I } _ { \\ge u } ^ { \\nu } \\mathbb { I } _ { \\ge u } ^ { \\nu } } \\\\ & { \\le \\mathcal { U } _ { \\ge } \\displaystyle \\sum _ { u = 1 } ^ { N } \\mathbb { I } _ { \\ge u \\ne 1 } ^ { N } \\mathbb { I } _ { \\ge u \\ne 1 } ^ { N } \\mathbb { I } _ { \\ge u \\ne 1 } ^ { N } \\mathbb { I } _ { \\ge u \\ne 1 } ^ { N } \\mathbb { I } _ { \\ge u \\ne 1 } ^ { N } - \\mathbb { I } _ { \\ge } ^ { N } \\mathbb { I } _ { \\ge u \\ne 1 } ^ { N } \\mathbb { I } _ { \\ge u \\ne 1 } ^ { N } } \\\\ & { \\le \\mathcal { U } _ { \\ge } \\displaystyle \\sum _ { u = 1 } ^ { N } \\mathbb { I } _ { \\ge u \\ne 1 } ^ { N } \\mathbb { I } _ { \\ge u \\ne 1 } ^ { N } \\mathbb { I } _ { \\ge u \\ne 1 } ^ { N } \\mathbb { I } _ { \\ge u \\ne 1 } ^ { N } \\mathbb { I } _ { \\ge u \\ne 1 } ^ { N } \\mathbb { I } _ { \\ge u \\ne 1 } ^ { N } } \\\\ & - \\mathcal { U } _ { \\ge } \\displaystyle \\sum _ { u = 1 } ^ { N } \\mathbb { I } _ { \\ge u \\ne 1 } ^ \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 187, + 167, + 805, + 539 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "where the first inequality is by Chernoff bound and the last one is by Lemma 3.4. of Yin and Wang (2020) (Law of total variances). The rest of them are from Cauchy’s inequality. Similarly, for Theorem 3.3, we also have ", + "bbox": [ + 174, + 536, + 825, + 577 + ], + "page_idx": 34 + }, + { + "type": "equation", + "img_path": "images/40dc63a7f94f8b8104516a8524ef82f34bdccad0775dcb2696560fcab1e19f09.jpg", + "text": "$$\n\\begin{array} { r l } & { \\forall \\widehat { u } _ { \\xi } ^ { \\mu } = \\sqrt { \\frac { \\mu } { \\mu } \\sqrt { \\mu } \\frac { \\sqrt { \\mu } } { \\mu } \\left\\{ \\Gamma ^ { 2 } + \\widehat { u } _ { \\xi } ^ { \\mu } \\left[ \\Gamma ^ { 2 } + \\widehat { u } _ { \\xi } ^ { \\mu } - \\widehat { u } _ { \\xi } ^ { \\mu } \\right] \\right\\} } } \\\\ & = \\sqrt { \\widehat { u } _ { \\xi } ^ { \\mu } - \\widehat { u } _ { \\xi } ^ { \\mu } } \\sqrt { \\Gamma ^ { 2 } + \\widehat { u } _ { \\xi } ^ { \\mu } \\left[ \\Gamma ^ { 2 } + \\widehat { u } _ { \\xi } ^ { \\mu } - \\widehat { u } _ { \\xi } ^ { \\mu } \\right] \\right\\} \\sqrt { \\textnormal { w a r } \\Gamma ^ { 2 } + \\widehat { u } _ { \\xi } ^ { \\mu } \\left[ \\Gamma ^ { 2 } + \\widehat { u } _ { \\xi } ^ { \\mu } - \\widehat { u } _ { \\xi } ^ { \\mu } \\right] } } \\\\ & { \\quad - \\sqrt { \\widehat { u } _ { \\xi } ^ { \\mu } - \\widehat { u } _ { \\xi } ^ { \\mu } } \\sqrt { \\Gamma ^ { 2 } + \\widehat { u } _ { \\xi } ^ { \\mu } \\left[ \\Gamma ^ { 2 } + \\widehat { u } _ { \\xi } ^ { \\mu } - \\widehat { u } _ { \\xi } ^ { \\mu } \\right] } } \\\\ & = - \\sqrt { \\widehat { u } _ { \\xi } ^ { \\mu } - \\widehat { u } _ { \\xi } ^ { \\mu } } \\sqrt { \\frac { \\sqrt { \\mu } } { \\mu } \\sqrt { \\mu } \\frac { \\sqrt { \\mu } \\left\\{ \\Gamma ^ { 2 } + \\widehat { u } _ { \\xi } ^ { \\mu } \\left[ \\Gamma ^ { 2 } + \\widehat { u } _ { \\xi } ^ { \\mu } - \\widehat { u } _ { \\xi } ^ { \\mu } \\right] \\right\\} } } \\\\ & \\quad - \\sqrt { \\widehat { u } _ { \\xi } ^ { \\mu } - \\widehat { u } _ { \\xi } ^ { \\mu } } \\sqrt { \\frac { \\sqrt { \\mu } } { \\mu } \\sqrt { \\mu } \\frac { \\sqrt { \\mu } \\left\\{ \\Gamma ^ { 2 } + \\widehat { u } _ { \\xi } ^ { \\mu } \\left[ \\Gamma ^ { 2 } + \\widehat { u } _ { \\xi } ^ { \\mu } - \\widehat { u } _ { \\xi } ^ { \\mu } \\right] } { \\Delta \\xi \\sqrt { \\mu } } } \\\\ & \\quad \\leq \\sqrt { \\widehat { u } _ { \\xi } \\sqrt { \\mu } \\frac { \\mu } { \\mu } } \\sqrt \\ \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 272, + 582, + 727, + 890 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "H AUXILIARY LEMMAS ", + "text_level": 1, + "bbox": [ + 174, + 102, + 387, + 118 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "Lemma H.1 (Matrix McDiarmid inequality / Matrix Chernoff bound (Tropp, 2012)). Let $z _ { k }$ , $k =$ $1 , \\ldots , K$ be independent random vectors in $\\mathbb { R } ^ { d }$ , and let $H$ be a mapping that maps $K$ vectors to $a$ $d \\times d$ symmetric matrix. Assume there exists a sequence of fixed symmetric matrices $\\{ A _ { k } \\} _ { k \\in [ K ] }$ such that for $z _ { k } , z _ { k } ^ { \\prime }$ ranges over all possible values for each $k \\in [ K ]$ , it holds ", + "bbox": [ + 173, + 131, + 825, + 190 + ], + "page_idx": 35 + }, + { + "type": "equation", + "img_path": "images/949a1f83030c7027d3728a5f9ecccdce6b5b10af6a2f1c73f6d265939e0987cb.jpg", + "text": "$$\n( H ( z _ { 1 } , \\ldots , z _ { k } , \\ldots , z _ { K } ) - H ( z _ { 1 } , \\ldots , z _ { k } ^ { \\prime } , \\ldots , z _ { K } ) ) ^ { 2 } \\preceq A _ { k } ^ { 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 302, + 191, + 691, + 209 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "Define $\\textstyle \\sigma ^ { 2 } : = \\left\\| \\sum _ { k } A _ { k } ^ { 2 } \\right\\|$ . Then for any $t > 0$ , ", + "bbox": [ + 174, + 210, + 468, + 228 + ], + "page_idx": 35 + }, + { + "type": "equation", + "img_path": "images/6c35f93b60c307d1fffde5940ac40f54e84e605c61b81b2baa406e810d83b434.jpg", + "text": "$$\n\\begin{array} { r } { \\mathbb { P } \\left\\{ \\| H ( z _ { 1 } , \\ldots , z _ { K } ) - \\mathbb { E } H ( z _ { 1 } , \\ldots , z _ { K } ) \\| \\ge t \\right\\} \\le d \\cdot \\exp \\left( \\frac { - t ^ { 2 } } { 8 \\sigma ^ { 2 } } \\right) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 282, + 228, + 712, + 262 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "Lemma H.2 (Hoeffding inequality for self-normalized martingales (Abbasi-Yadkori et al., 2011)). Let $\\{ \\eta _ { t } \\} _ { t = 1 } ^ { \\infty }$ be a real-valued stochastic process. Let $\\{ \\mathcal { F } _ { t } \\} _ { t = 0 } ^ { \\infty }$ be a filtration, such that $\\eta _ { t }$ is $\\mathcal { F } _ { t }$ - measurable. Assume $\\eta _ { t }$ also satisfies $\\eta _ { t }$ given $\\mathcal { F } _ { t - 1 }$ is zero-mean and $R$ -subgaussian, i.e. ", + "bbox": [ + 173, + 263, + 825, + 305 + ], + "page_idx": 35 + }, + { + "type": "equation", + "img_path": "images/68d921c0d8d6baef147e6a1dfa7723a2ba24f54737488dc52fa5d76a4e30af23.jpg", + "text": "$$\n\\forall \\lambda \\in \\mathbb { R } , \\quad \\mathbb { E } \\left[ e ^ { \\lambda \\eta _ { t } } \\mid \\mathcal { F } _ { t - 1 } \\right] \\le e ^ { \\lambda ^ { 2 } R ^ { 2 } / 2 }\n$$", + "text_format": "latex", + "bbox": [ + 370, + 306, + 625, + 327 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "Let $\\begin{array} { r } { \\Lambda _ { t } = \\lambda I _ { d } + \\sum _ { s = 1 } ^ { t } x _ { s } x _ { s } ^ { \\top } } \\end{array}$ $\\{ x _ { t } \\} _ { t = 1 } ^ { \\infty }$ be an $\\mathbb { R } ^ { d }$ -valued stochastic process where . Then for any $\\delta > 0$ t , with probability is $\\mathcal { F } _ { t - 1 }$ $1 - \\delta _ { : }$ measurable and , for all $t > 0$ , $\\| x _ { t } \\| \\leq L$ . Let ", + "bbox": [ + 169, + 329, + 825, + 361 + ], + "page_idx": 35 + }, + { + "type": "equation", + "img_path": "images/bd4f9d67a95e63784fa2f0b8e402e452bbb2c83a6522a3507ae0cfc8784a5452.jpg", + "text": "$$\n\\left\\| \\sum _ { s = 1 } ^ { t } x _ { s } \\eta _ { s } \\right\\| _ { \\Lambda _ { t } ^ { - 1 } } ^ { 2 } \\leq 8 R ^ { 2 } \\cdot \\frac { d } { 2 } \\log \\left( \\frac { \\lambda + t L } { \\lambda \\delta } \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 354, + 362, + 642, + 410 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "Lemma H.3 (Bernstein inequality for self-normalized martingales (Zhou et al., 2021a)). Let $\\{ \\eta _ { t } \\} _ { t = 1 } ^ { \\infty }$ be a real-valued stochastic process. Let $\\{ \\mathcal { F } _ { t } \\} _ { t = 0 } ^ { \\infty }$ be a filtration, such that $\\eta _ { t }$ is $\\mathcal { F } _ { t }$ -measurable. Assume $\\eta _ { t }$ also satisfies ", + "bbox": [ + 173, + 410, + 826, + 452 + ], + "page_idx": 35 + }, + { + "type": "equation", + "img_path": "images/7a7bbe014b2ee9b6c8b953df095490e878534e3fac493e81f1e89ce7a64dbd66.jpg", + "text": "$$\n| \\eta _ { t } | \\leq R , \\mathbb { E } \\left[ \\eta _ { t } \\mid \\mathcal { F } _ { t - 1 } \\right] = 0 , \\mathbb { E } \\left[ \\eta _ { t } ^ { 2 } \\mid \\mathcal { F } _ { t - 1 } \\right] \\leq \\sigma ^ { 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 336, + 452, + 661, + 472 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "Let $\\{ x _ { t } \\} _ { t = 1 } ^ { \\infty }$ be an $\\mathbb { R } ^ { d }$ -valued stochastic process where $x _ { t }$ is $\\mathcal { F } _ { t - 1 }$ measurable and $\\| x _ { t } \\| \\leq L$ . Let $\\begin{array} { r } { \\Lambda _ { t } = \\lambda I _ { d } + \\sum _ { s = 1 } ^ { t } x _ { s } x _ { s } ^ { \\top } } \\end{array}$ . Then for any $\\delta > 0$ , with probability $1 - \\delta _ { : }$ , for all $t > 0$ , ", + "bbox": [ + 173, + 479, + 823, + 512 + ], + "page_idx": 35 + }, + { + "type": "equation", + "img_path": "images/da49e6c0d0b0c59d782ff1094cc677692fc85abe0900b483f4811e6b9db72872.jpg", + "text": "$$\n\\left\\| \\sum _ { s = 1 } ^ { t } \\mathbf { x } _ { s } \\eta _ { s } \\right\\| _ { \\Lambda _ { t } ^ { - 1 } } \\leq 8 \\sigma \\sqrt { d \\log \\left( 1 + \\frac { t L ^ { 2 } } { \\lambda d } \\right) \\cdot \\log \\left( \\frac { 4 t ^ { 2 } } { \\delta } \\right) } + 4 R \\log \\left( \\frac { 4 t ^ { 2 } } { \\delta } \\right)\n$$", + "text_format": "latex", + "bbox": [ + 256, + 513, + 738, + 560 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "Lemma H.4 (Converting the variance under the matrix norm). Let $\\Lambda _ { 1 }$ and $\\Lambda _ { 2 } \\in \\mathbb { R } ^ { d \\times d }$ are two positive semi-definite matrices. Then: ", + "bbox": [ + 173, + 561, + 825, + 589 + ], + "page_idx": 35 + }, + { + "type": "equation", + "img_path": "images/1f487bf7793819c8985870167abb327e7ea84fbbf1a0c59906a3cd47813a258e.jpg", + "text": "$$\n\\left\\| \\Lambda _ { 1 } ^ { - 1 } \\right\\| \\leq \\left\\| \\Lambda _ { 2 } ^ { - 1 } \\right\\| + \\left\\| \\Lambda _ { 1 } ^ { - 1 } \\right\\| \\cdot \\left\\| \\Lambda _ { 2 } ^ { - 1 } \\right\\| \\cdot \\left\\| \\Lambda _ { 1 } - \\Lambda _ { 2 } \\right\\|\n$$", + "text_format": "latex", + "bbox": [ + 331, + 589, + 665, + 609 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "and ", + "bbox": [ + 173, + 608, + 202, + 622 + ], + "page_idx": 35 + }, + { + "type": "equation", + "img_path": "images/94f1e095fee85caffb4e372ca330e4290ae863ac0f82a3e8a367aa09e4aaa5ed.jpg", + "text": "$$\n\\left\\| \\phi \\right\\| _ { \\Lambda _ { 1 } ^ { - 1 } } \\leq \\left[ 1 + \\sqrt { \\left\\| \\Lambda _ { 2 } ^ { - 1 } \\right\\| \\left\\| \\Lambda _ { 2 } \\right\\| \\cdot \\left\\| \\Lambda _ { 1 } ^ { - 1 } \\right\\| \\cdot \\left\\| \\Lambda _ { 1 } - \\Lambda _ { 2 } \\right\\| } \\right] \\cdot \\left\\| \\phi \\right\\| _ { \\Lambda _ { 2 } ^ { - 1 } } .\n$$", + "text_format": "latex", + "bbox": [ + 279, + 617, + 717, + 654 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "for all $\\phi \\in \\mathbb { R } ^ { d }$ ", + "bbox": [ + 171, + 652, + 272, + 667 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "Proof. For the first part, note ", + "bbox": [ + 174, + 683, + 367, + 696 + ], + "page_idx": 35 + }, + { + "type": "equation", + "img_path": "images/a08860c6f1ba7a119921ae9849fb7c91ac5b465f5c8c079f2704476b0a95601e.jpg", + "text": "$$\n\\left\\| \\Lambda _ { 1 } ^ { - 1 } \\right\\| \\leq \\left\\| \\Lambda _ { 2 } ^ { - 1 } \\right\\| + \\left\\| \\Lambda _ { 1 } ^ { - 1 } - \\Lambda _ { 2 } ^ { - 1 } \\right\\| \\leq \\left\\| \\Lambda _ { 2 } ^ { - 1 } \\right\\| + \\left\\| \\Lambda _ { 2 } ^ { - 1 } \\right\\| \\left\\| \\Lambda _ { 1 } - \\Lambda _ { 2 } \\right\\| \\left\\| \\Lambda _ { 1 } ^ { - 1 } \\right\\|\n$$", + "text_format": "latex", + "bbox": [ + 250, + 696, + 745, + 717 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "For the second one, ", + "bbox": [ + 173, + 717, + 303, + 729 + ], + "page_idx": 35 + }, + { + "type": "equation", + "img_path": "images/278f938886d95b8d6a970637e758173e4d6caebe9c15d31c9e1e7091e56a29a9.jpg", + "text": "$$\n\\begin{array} { r l } & { \\left\\| \\phi \\right\\| _ { \\Lambda _ { 1 } ^ { - 1 } } = \\sqrt { \\phi ^ { \\top } \\Lambda _ { 1 } ^ { - 1 } \\phi } = \\sqrt { \\phi ^ { \\top } \\left( \\Lambda _ { 1 } ^ { - 1 } - \\Lambda _ { 2 } ^ { - 1 } \\right) \\phi + \\phi ^ { \\top } \\Lambda _ { 2 } ^ { - 1 } \\phi } } \\\\ & { = \\sqrt { \\phi ^ { \\top } \\Lambda _ { 2 } ^ { - 1 / 2 } \\left( \\Lambda _ { 2 } ^ { 1 / 2 } \\Lambda _ { 1 } ^ { - 1 } \\Lambda _ { 2 } ^ { 1 / 2 } - I + I \\right) \\Lambda _ { 2 } ^ { - 1 / 2 } \\phi } \\leq \\sqrt { \\| \\phi \\| _ { \\Lambda _ { 2 } ^ { - 1 } } \\cdot \\left( 1 + \\left\\| \\Lambda _ { 2 } ^ { 1 / 2 } \\Lambda _ { 1 } ^ { - 1 } \\Lambda _ { 2 } ^ { 1 / 2 } - I \\right\\| \\right) \\| \\phi \\| _ { \\Lambda _ { 2 } ^ { - 1 } } } } \\\\ & { \\leq \\left( 1 + \\left\\| \\Lambda _ { 2 } ^ { 1 / 2 } \\Lambda _ { 1 } ^ { - 1 } \\Lambda _ { 2 } ^ { 1 / 2 } - I \\right\\| ^ { 1 / 2 } \\right) \\cdot \\left\\| \\phi \\right\\| _ { \\Lambda _ { 2 } ^ { - 1 } } = \\left( 1 + \\left\\| \\Lambda _ { 2 } ^ { 1 / 2 } \\Lambda _ { 1 } ^ { - 1 } \\left( \\Lambda _ { 2 } - \\Lambda _ { 1 } \\right) \\Lambda _ { 2 } ^ { - 1 } \\Lambda _ { 2 } ^ { 1 / 2 } \\right\\| ^ { 1 / 2 } \\right) \\cdot \\left\\| \\phi \\right\\| _ { \\Lambda _ { 2 } ^ { - 1 } } } \\\\ & { \\leq \\left( 1 + \\sqrt { \\| \\Lambda _ { 2 } \\| \\left\\| \\Lambda _ { 1 } ^ { - 1 } \\right\\| \\left\\| \\Lambda _ { 2 } ^ { - 1 } \\right\\| \\left\\| \\Lambda _ { 1 } ^ { - 1 } - \\Lambda _ { 2 } \\right\\| } \\right) \\cdot \\left\\| \\phi \\right\\| _ { \\Lambda _ { 2 } ^ { - 1 } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 748, + 836, + 886 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "Lemma H.5 (Lemma H.4 of Min et al. (2021)). let $\\phi : \\mathcal { S } \\times \\mathcal { A } \\mathbb { R } ^ { d }$ satisfies $\\| \\phi ( s , a ) \\| \\leq C$ for $s , a \\in \\mathcal S \\times \\mathcal A$ . For any $K > 0 , \\lambda > 0 ,$ , define $\\begin{array} { r } { \\bar { G } _ { K } = \\sum _ { k = 1 } ^ { K } \\phi ( s _ { k } , a _ { k } ) \\phi ( s _ { k } , \\underline { { a } } _ { k } ) ^ { \\top } + \\lambda I _ { d } } \\end{array}$ where $( s _ { k } , a _ { k } )$ $\\nu$ $1 - \\delta$ ", + "bbox": [ + 171, + 102, + 825, + 150 + ], + "page_idx": 36 + }, + { + "type": "equation", + "img_path": "images/82c9d4ef9825f2eae00ef9f22444d30d601b67065a1c9c0c21990661ed586735.jpg", + "text": "$$\n\\left\\| \\frac { { { { \\bar { G } } _ { K } } } } { K } - \\mathbb { E } _ { \\nu } \\left[ \\frac { { { { \\bar { G } } _ { K } } } } { K } \\right] \\right\\| \\le \\frac { 4 \\sqrt 2 C ^ { 2 } } { \\sqrt K } \\left( \\log \\frac { { 2 d } } { \\delta } \\right) ^ { 1 / 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 341, + 156, + 655, + 195 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "Proof of Lemma $H . 5$ . For completeness, we provide the proof of Lemma H.5. Let $x _ { k } = \\phi ( s _ { k } , a _ { k } )$ . Denote $\\widetilde { \\Sigma } _ { h }$ as the matrix obtained by replacing the $\\mathrm { k }$ -th vector $x _ { k }$ in $\\widehat { \\Sigma } _ { h }$ by $\\widetilde { x } _ { k }$ and leaving the rest $K - 1$ vectors unchanged. Then ", + "bbox": [ + 173, + 210, + 826, + 257 + ], + "page_idx": 36 + }, + { + "type": "equation", + "img_path": "images/f4fe0083ad0a6a5e2ab377728942168d2e1e466dfe1a2ef05637980295e2b9fe.jpg", + "text": "$$\n\\left( \\frac { \\widehat \\Sigma _ { h } } { K } - \\frac { \\widetilde \\Sigma _ { h } } { K } \\right) ^ { 2 } = \\left( \\frac { x _ { k } x _ { k } ^ { \\top } - \\widetilde x _ { k } \\widetilde x _ { k } ^ { \\top } } { K } \\right) \\preceq \\frac { 1 } { K ^ { 2 } } \\left( 2 x _ { k } x _ { k } ^ { \\top } x _ { k } x _ { k } ^ { \\top } + 2 \\widetilde x _ { k } \\widetilde x _ { k } ^ { \\top } \\widetilde x _ { k } \\widetilde x _ { k } ^ { \\top } \\right) \\preceq \\frac { 4 C ^ { 4 } } { K ^ { 2 } } I _ { d } : = A _ { k } ^ { 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 187, + 263, + 808, + 309 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "Notice that $\\begin{array} { r } { \\left\\| \\sum _ { k } ^ { K } A _ { k } ^ { 2 } \\right\\| = \\frac { 4 C ^ { 4 } } { K } } \\end{array}$ , by Lemma H.1 we have the result. ", + "bbox": [ + 173, + 316, + 604, + 342 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "Lemma H.6 (Lemma H.5. of Min et al. (2021)). Let $\\| \\phi \\| _ { 2 } \\leq C$ . Define $\\begin{array} { r } { \\bar { G } _ { K } = \\sum _ { k = 1 } ^ { K } \\phi ( s _ { k } , a _ { k } ) \\phi ( s _ { k } , a _ { k } ) ^ { \\top } + \\lambda I _ { d } } \\end{array}$ $\\phi : \\mathcal { S } \\times \\mathcal { A } \\mathbb { R } ^ { d }$ where $( s _ { k } , a _ { k } )$ be a bounded function s.t. ’s are i.i.d samples from some distribution $\\nu$ . Let $G = \\mathbb { E } _ { \\nu } [ \\phi ( s , a ) \\phi ( s , a ) ^ { \\top } ]$ . Then for any $\\delta \\in ( 0 , 1 )$ , if $K$ satisfies ", + "bbox": [ + 173, + 371, + 826, + 420 + ], + "page_idx": 36 + }, + { + "type": "equation", + "img_path": "images/57ce29ef2533968070df73fa05bf312de8ac10bc38f79d82783a479ff410cc79.jpg", + "text": "$$\nK \\geq \\operatorname* { m a x } \\left\\{ 5 1 2 C ^ { 4 } \\left\\| \\mathbf { G } ^ { - 1 } \\right\\| ^ { 2 } \\log \\left( \\frac { 2 d } { \\delta } \\right) , 4 \\lambda \\left\\| \\mathbf { G } ^ { - 1 } \\right\\| \\right\\} .\n$$", + "text_format": "latex", + "bbox": [ + 316, + 426, + 679, + 462 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "Then with probability at least $1 - \\delta$ , it holds simultaneously for all $u \\in \\mathbb { R } ^ { d }$ that ", + "bbox": [ + 171, + 469, + 692, + 486 + ], + "page_idx": 36 + }, + { + "type": "equation", + "img_path": "images/81d9ca9813a280ca04a49f7d1ffe4ec892f01ad64b50f7036474ac7715cfae63.jpg", + "text": "$$\n\\| u \\| _ { \\bar { G } _ { K } ^ { - 1 } } \\leq \\frac { 2 } { \\sqrt { K } } \\| u \\| _ { G ^ { - 1 } } .\n$$", + "text_format": "latex", + "bbox": [ + 413, + 492, + 584, + 525 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "Lemma H.7 (Extended Value Difference (Section B.1 in Cai et al. (2020))). Let $\\pi = \\{ \\pi _ { h } \\} _ { h = 1 } ^ { H }$ and $\\pi ^ { \\prime } = \\{ \\pi _ { h } ^ { \\prime } \\} _ { h = 1 } ^ { H }$ be two arbitrary policies and let $\\{ \\widehat { Q } _ { h } \\} _ { h = 1 } ^ { H }$ be any given $Q$ -functions. Then define $\\widehat { V } _ { h } ( s ) : = \\langle \\widehat { Q } _ { h } ( s , \\cdot ) , \\pi _ { h } ( \\cdot \\mid s ) \\rangle$ for all $s \\in S$ . Then for all $s \\in S$ , ", + "bbox": [ + 173, + 531, + 826, + 584 + ], + "page_idx": 36 + }, + { + "type": "equation", + "img_path": "images/0cfa6e7abc2146b988255e873e148cf9cdcb443f507cebbdece42ebd010c7e73.jpg", + "text": "$$\n\\begin{array} { r } { \\widehat { V } _ { 1 } ( s ) - V _ { 1 } ^ { \\pi ^ { \\prime } } ( s ) = \\displaystyle \\sum _ { h = 1 } ^ { H } \\mathbb { E } _ { \\pi ^ { \\prime } } \\left[ \\langle \\widehat { Q } _ { h } \\left( s _ { h } , \\cdot \\right) , \\pi _ { h } \\left( \\cdot \\mid s _ { h } \\right) - \\pi _ { h } ^ { \\prime } \\left( \\cdot \\mid s _ { h } \\right) \\rangle \\mid s _ { 1 } = s \\right] } \\\\ { + \\displaystyle \\sum _ { h = 1 } ^ { H } \\mathbb { E } _ { \\pi ^ { \\prime } } \\left[ \\widehat { Q } _ { h } \\left( s _ { h } , a _ { h } \\right) - \\left( \\mathcal { T } _ { h } \\widehat { V } _ { h + 1 } \\right) \\left( s _ { h } , a _ { h } \\right) \\mid s _ { 1 } = s \\right] } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 250, + 599, + 746, + 688 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "where $( T _ { h } V ) ( \\cdot , \\cdot ) : = r _ { h } ( \\cdot , \\cdot ) + ( P _ { h } V ) ( \\cdot , \\cdot ) .$ for any $V \\in \\mathbb { R } ^ { S }$ . ", + "bbox": [ + 173, + 691, + 568, + 710 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "Proof. Denote $\\xi _ { h } = \\widehat { Q } _ { h } - \\mathcal { T } _ { h } \\widehat { V } _ { h + 1 }$ . For any $h \\in [ H ]$ , we have ", + "bbox": [ + 171, + 726, + 586, + 746 + ], + "page_idx": 36 + }, + { + "type": "equation", + "img_path": "images/3df6de8b361023b7e22a36b8e374b087e87f77d1ad4fc806d7abeda99a14a231.jpg", + "text": "$$\n\\begin{array} { r l } & { \\widehat { V } _ { h } - V _ { h } ^ { \\pi ^ { \\prime } } = \\langle \\widehat { Q } _ { h } , \\pi _ { h } \\rangle - \\langle Q _ { h } ^ { \\pi ^ { \\prime } } , \\pi _ { h } ^ { \\prime } \\rangle } \\\\ & { \\qquad = \\langle \\widehat { Q } _ { h } , \\pi _ { h } - \\pi _ { h } ^ { \\prime } \\rangle + \\langle \\widehat { Q } _ { h } - Q _ { h } ^ { \\pi ^ { \\prime } } , \\pi _ { h } ^ { \\prime } \\rangle } \\\\ & { \\qquad = \\langle \\widehat { Q } _ { h } , \\pi _ { h } - \\pi _ { h } ^ { \\prime } \\rangle + \\langle P _ { h } ( \\widehat { V } _ { h + 1 } - V _ { h + 1 } ^ { \\pi ^ { \\prime } } ) + \\xi _ { h } , \\pi _ { h } ^ { \\prime } \\rangle } \\\\ & { \\qquad = \\langle \\widehat { Q } _ { h } , \\pi _ { h } - \\pi _ { h } ^ { \\prime } \\rangle + \\langle P _ { h } ( \\widehat { V } _ { h + 1 } - V _ { h + 1 } ^ { \\pi ^ { \\prime } } ) , \\pi _ { h } ^ { \\prime } \\rangle + \\langle \\xi _ { h } , \\pi _ { h } ^ { \\prime } \\rangle } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 277, + 750, + 718, + 839 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "reof y apply the above for we can finish the pro $\\widehat { V } _ { h + 1 } - V _ { h + 1 } ^ { \\pi ^ { \\prime } }$ and use the ma. $\\mathbb { E } _ { \\pi ^ { \\prime } }$ notation (instead of the inner product $P _ { h } , \\pi _ { h } ^ { \\prime } )$ ", + "bbox": [ + 171, + 845, + 825, + 878 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "Lemma H.8. Let $\\widehat { \\pi } = \\left\\{ \\widehat { \\pi } _ { h } \\right\\} _ { h = 1 } ^ { H }$ and $\\widehat { Q } _ { h } ( \\cdot , \\cdot )$ be the arbitrary policy and $Q$ -function and also $\\widehat { V } _ { h } ( s ) = \\langle \\widehat { Q } _ { h } ( s , \\cdot ) , \\widehat { \\pi } _ { h } ( \\cdot | s ) \\rangle \\vee _ { s } ^ { \\cdot } \\in \\mathcal { S }$ . and $\\zeta _ { h } ( s , a ) : = ( \\mathscr { T } _ { h } \\widehat { V } _ { h + 1 } ) ( s , a ) - \\widehat { Q } _ { h } ( s , a )$ (element-wisely) ", + "bbox": [ + 173, + 888, + 826, + 926 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "to be the Bellman update error. Then for any arbitrary $\\pi$ , we have ", + "bbox": [ + 173, + 103, + 607, + 119 + ], + "page_idx": 37 + }, + { + "type": "equation", + "img_path": "images/67417bc8f21fd1b43ef9a4a476ce6fa4232c33309bc8f685cf1e78494196e120.jpg", + "text": "$$\n\\begin{array} { c } { { { \\displaystyle V _ { 1 } ^ { \\pi } ( s ) - V _ { 1 } ^ { \\widehat { \\pi } } ( s ) = \\sum _ { h = 1 } ^ { H } \\mathbb { E } _ { \\pi } \\left[ \\zeta _ { h } ( s _ { h } , a _ { h } ) \\mid s _ { 1 } = s \\right] - \\sum _ { h = 1 } ^ { H } \\mathbb { E } _ { \\widehat { \\pi } } \\left[ \\zeta _ { h } ( s _ { h } , a _ { h } ) \\mid s _ { 1 } = s \\right] } } } \\\\ { { { + \\displaystyle \\sum _ { h = 1 } ^ { H } \\mathbb { E } _ { \\pi } \\left[ \\left. \\widehat { Q } _ { h } \\left( s _ { h } , \\cdot \\right) , \\pi _ { h } \\left( \\cdot | s _ { h } \\right) - \\widehat { \\pi } _ { h } \\left( \\cdot | s _ { h } \\right) \\right. \\mid s _ { 1 } = x \\right] } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 238, + 133, + 759, + 223 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "where the expectation are taken over $s _ { h } , a _ { h }$ . ", + "bbox": [ + 174, + 236, + 464, + 251 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "Proof. Note the gap can be rewritten as ", + "bbox": [ + 174, + 292, + 434, + 308 + ], + "page_idx": 37 + }, + { + "type": "equation", + "img_path": "images/d0b157725d446ee734981fa192e2d1aaf3a9bee3d716c71d27ee54b4a38f644b.jpg", + "text": "$$\n\\begin{array} { r } { V _ { 1 } ^ { \\pi } ( s ) - V _ { 1 } ^ { \\widehat { \\pi } } ( s ) = V _ { 1 } ^ { \\pi } ( s ) - \\widehat { V } _ { 1 } ( s ) + \\widehat { V } _ { 1 } ( s ) - V _ { 1 } ^ { \\widehat { \\pi } } ( s ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 321, + 323, + 674, + 343 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "By Lemma H.7 with $\\pi = { \\widehat { \\pi } }$ , $\\pi ^ { \\prime } = \\pi$ , we directly have ", + "bbox": [ + 173, + 357, + 527, + 373 + ], + "page_idx": 37 + }, + { + "type": "equation", + "img_path": "images/a3b17028ac9abc38cd618977334599ec09bdd3ee7aa6fb322892870650e381d9.jpg", + "text": "$$\n\\gamma _ { 1 } ^ { \\pi } ( s ) - \\widehat { V } _ { 1 } ( s ) = \\sum _ { h = 1 } ^ { H } \\mathbb { E } _ { \\pi } \\left[ \\zeta _ { h } ( s _ { h } , a _ { h } ) \\mid s _ { 1 } = s \\right] + \\sum _ { h = 1 } ^ { H } \\mathbb { E } _ { \\pi } \\left[ \\langle \\widehat { Q } _ { h } \\left( s _ { h } , \\cdot \\right) , \\pi _ { h } \\left( \\cdot | s _ { h } \\right) - \\widehat { \\pi } _ { h } \\left( \\cdot | s _ { h } \\right) \\rangle \\mid s _ { 1 } = s \\right]\n$$", + "text_format": "latex", + "bbox": [ + 181, + 387, + 834, + 431 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "Next apply Lemma H.7 again with $\\pi = \\pi ^ { \\prime } = \\widehat { \\pi }$ , we directly have ", + "bbox": [ + 173, + 441, + 601, + 458 + ], + "page_idx": 37 + }, + { + "type": "equation", + "img_path": "images/a1002b52c9057be9b8d21afce6c564fc9561e6a14660b37576f5e425fb964be7.jpg", + "text": "$$\n\\widehat { V } _ { 1 } ( s ) - V _ { 1 } ^ { \\widehat { \\pi } } ( s ) = - \\sum _ { h = 1 } ^ { H } \\mathbb { E } _ { \\widehat { \\pi } } \\left[ \\zeta _ { h } ( s _ { h } , a _ { h } ) \\mid s _ { 1 } = s \\right] .\n$$", + "text_format": "latex", + "bbox": [ + 331, + 472, + 665, + 516 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "Combine the above two results we prove the stated result. ", + "bbox": [ + 173, + 530, + 552, + 545 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "Lemma H.9. For a linear MDP, for any $0 \\le V ( \\cdot ) \\le H$ , then there exists a $w _ { h } \\in \\mathbb { R } ^ { d }$ s.t. $\\mathcal { T } _ { h } V =$ \n$\\langle \\phi , w _ { h } \\rangle$ and $\\| w _ { h } \\| _ { 2 } ~ \\le ~ 2 H \\sqrt { d }$ for all $h \\in [ H ]$ . Here ${ \\mathcal { T } } _ { h } ( V ) ( s , a ) = r _ { h } ( x , a ) + ( P _ { h } V ) ( s , a )$ . \nSimilarly, for any $\\pi$ , there exists $w _ { h } ^ { \\pi } \\in \\mathbb { R } ^ { d }$ , such that $Q _ { h } ^ { \\pi } = \\langle \\phi , w _ { h } ^ { \\pi } \\rangle$ with $\\| w _ { h } ^ { \\pi } \\| _ { 2 } \\leq 2 ( H - h + 1 ) \\sqrt { d }$ . ", + "bbox": [ + 173, + 578, + 826, + 630 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "Proof. By definition, ", + "bbox": [ + 173, + 670, + 313, + 685 + ], + "page_idx": 37 + }, + { + "type": "equation", + "img_path": "images/f4ae2ff946e4f8c01a455379722ad05b21aa49404e4833e76e3662916e99aa8a.jpg", + "text": "$$\n\\begin{array} { c } { { \\displaystyle { \\mathcal { T } } _ { h } V = r _ { h } + ( P _ { h } V ) = \\langle \\phi , \\theta _ { h } \\rangle + \\langle \\phi , \\displaystyle { \\int _ { S } V ( s ) d \\nu _ { h } ( s ) \\rangle } } } \\\\ { { \\displaystyle { \\Rightarrow } w _ { h } = \\theta _ { h } + \\int _ { S } V ( s ) d \\nu _ { h } ( s ) , } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 320, + 699, + 676, + 768 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "therefore $\\| w _ { h } \\| _ { 2 } \\leq \\| \\theta _ { h } \\| _ { 2 } + H \\cdot \\| \\nu _ { h } ( S ) \\| \\leq 1 + H { \\sqrt { d } } \\leq 2 H { \\sqrt { d } }$ . The proof of the second part is similar by backward induction and the fact $V _ { h } ^ { \\pi } \\leq H - h + 1$ for any $\\pi$ . □ ", + "bbox": [ + 173, + 781, + 825, + 814 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "Lemma H.10. For any pessimistic bonus design $\\Gamma _ { h }$ , suppose $K > \\operatorname* { m a x } \\{ \\mathcal { M } _ { 1 } , \\mathcal { M } _ { 2 } , \\mathcal { M } _ { 3 } , \\mathcal { M } _ { 4 } \\}$ , then with probability $1 - \\delta$ , Algorithm $^ { l }$ yields ", + "bbox": [ + 173, + 847, + 825, + 877 + ], + "page_idx": 37 + }, + { + "type": "equation", + "img_path": "images/aa01715208fe6f464e1410e6f782b6d2818cdf96751312360dc9ea78a29808f8.jpg", + "text": "$$\n\\left\\| \\mathcal { T } _ { h } \\widehat { V } _ { h + 1 } - \\widehat { \\mathcal { T } } _ { h } \\widehat { V } _ { h + 1 } \\right\\| _ { \\infty } \\leq \\widetilde { O } ( \\frac { H ^ { 2 } \\sqrt { d / \\kappa } } { \\sqrt { K } } )\n$$", + "text_format": "latex", + "bbox": [ + 361, + 892, + 635, + 929 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "Proof of Lemma H.10. Suppose $w _ { h }$ is the coefficient corresponding to the $\\mathcal { T } _ { h } \\widehat { V } _ { h + 1 }$ (such $w _ { h }$ exists by Lemma H.9), i.e. $\\mathcal { T } _ { h } \\widehat { V } _ { h + 1 } = \\phi ^ { \\top } w _ { h }$ , and recall $( \\widehat { T } _ { h } \\widehat { V } _ { h + 1 } ) ( s , a ) = \\phi ( s , a ) ^ { \\top } \\widehat { w } _ { h }$ , then: ", + "bbox": [ + 173, + 102, + 826, + 136 + ], + "page_idx": 38 + }, + { + "type": "equation", + "img_path": "images/1def6bbdbd348703f0ad5948b4059ab4ab3691a68049f97fcbb8dffe8f6060dc.jpg", + "text": "$$\n\\begin{array} { r l } & { \\quad \\left( \\mathcal { T } _ { h } \\widehat { V } _ { h + 1 } \\right) ( s , a ) - \\left( \\widehat { \\mathcal { T } } _ { h } \\widehat { V } _ { h + 1 } \\right) ( s , a ) = \\phi ( s , a ) ^ { \\top } ( w _ { h } - \\widehat { w } _ { h } ) } \\\\ & { = \\phi ( s , a ) ^ { \\top } w _ { h } - \\phi ( s , a ) ^ { \\top } \\widehat { \\Lambda } _ { h } ^ { - 1 } \\left( \\displaystyle \\sum _ { \\tau = 1 } ^ { K } \\phi \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\cdot \\left( r _ { h } ^ { \\tau } + \\widehat { V } _ { h + 1 } \\left( s _ { h + 1 } ^ { \\tau } \\right) \\right) / \\widehat { \\sigma } _ { h } ^ { 2 } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) \\right) } \\\\ & { = \\phi ( s , a ) ^ { \\top } w _ { h } - \\phi ( s , a ) ^ { \\top } \\widehat { \\Lambda } _ { h } ^ { - 1 } \\left( \\displaystyle \\sum _ { \\tau = 1 } ^ { K } \\phi \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\cdot \\left( \\mathcal { T } _ { h } \\widehat { V } _ { h + 1 } \\right) \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) / \\widehat { \\sigma } _ { h } ^ { 2 } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) \\right) } \\\\ & { \\enspace \\enspace \\overset { \\mathrm { ( ) } } { \\underset { \\mathrm { ( ) } } { = } } \\phi ( s , a ) ^ { \\top } w _ { h } - \\phi ( s , a ) ^ { \\top } \\widehat { \\Lambda } _ { h } ^ { - 1 } \\left( \\displaystyle \\sum _ { \\tau = 1 } ^ { K } \\phi \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\cdot \\left( \\mathcal { T } _ { h } \\widehat { V } _ { h + 1 } \\right) \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) / \\widehat { \\sigma } _ { h } ^ { 2 } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) \\right) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 171, + 141, + 725, + 258 + ], + "page_idx": 38 + }, + { + "type": "equation", + "img_path": "images/7806236ebe8f66ab0f58618dab2a7a0c83880ac3a72cf0e377023db89585dfd7.jpg", + "text": "$$\n+ \\phi ( s , a ) ^ { \\top } \\widehat { \\Lambda } _ { h } ^ { - 1 } \\left( \\sum _ { \\tau = 1 } ^ { K } \\phi \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\cdot \\left( r _ { h } ^ { \\tau } + \\widehat { V } _ { h + 1 } \\left( s _ { h + 1 } ^ { \\tau } \\right) - \\left( \\mathcal { T } _ { h } \\widehat { V } _ { h + 1 } \\right) \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\right) / \\widehat { \\sigma } _ { h } ^ { 2 } \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 215, + 277, + 833, + 321 + ], + "page_idx": 38 + }, + { + "type": "text", + "text": "For term (i), it is bounded by 2λH3 d/κ with probability 1 − δ by Lemma C.2. ", + "bbox": [ + 173, + 358, + 689, + 381 + ], + "page_idx": 38 + }, + { + "type": "text", + "text": "For term (ii), by Cauchy inequality it is bounded by ", + "bbox": [ + 173, + 385, + 514, + 400 + ], + "page_idx": 38 + }, + { + "type": "equation", + "img_path": "images/ce87fab67963f6b102fe3d6175e3c04fd8c9ae542f9f8f7c233e0cd08e5d0f37.jpg", + "text": "$$\n\\begin{array} { r l } & { \\quad \\left. \\phi ( s , a ) \\right. _ { \\widehat { \\Lambda } _ { h } ^ { - 1 } } \\cdot \\left. \\displaystyle \\sum _ { \\tau = 1 } ^ { K } \\phi \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\cdot \\left( r _ { h } ^ { \\tau } + \\widehat { V } _ { h + 1 } \\left( s _ { h + 1 } ^ { \\tau } \\right) - \\left( \\mathcal { T } _ { h } \\widehat { V } _ { h + 1 } \\right) \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\right) / \\widehat { \\sigma } _ { h } ^ { 2 } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) \\right. _ { \\widehat { \\Lambda } _ { h } ^ { - 1 } } } \\\\ & { \\leq \\frac { 2 H } { \\sqrt { K K } } \\left. \\displaystyle \\sum _ { \\tau = 1 } ^ { K } \\phi \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\cdot \\left( r _ { h } ^ { \\tau } + \\widehat { V } _ { h + 1 } \\left( s _ { h + 1 } ^ { \\tau } \\right) - \\left( \\mathcal { T } _ { h } \\widehat { V } _ { h + 1 } \\right) \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\right) / \\widehat { \\sigma } _ { h } ^ { 2 } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) \\right. _ { \\widehat { \\Lambda } _ { h } ^ { - 1 } } } \\\\ & { \\leq \\frac { 2 H } { \\sqrt { K K } } \\cdot \\tilde { O } ( H \\sqrt { d } ) = \\widetilde { O } ( \\frac { H ^ { 2 } \\sqrt { d / \\kappa } } { \\sqrt { K } } ) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 178, + 406, + 815, + 540 + ], + "page_idx": 38 + }, + { + "type": "text", + "text": "where the first inequality is by Lemma H.6 (with $\\phi ^ { \\prime } = \\phi / \\widehat { \\sigma } _ { h }$ and $\\| \\phi / { \\widehat { \\sigma } } _ { h } \\| \\leq \\| \\phi \\| \\leq 1 : = C )$ and the third inequality uses ${ \\sqrt { a ^ { \\top } \\cdot A \\cdot a } } \\leq { \\sqrt { \\| a \\| _ { 2 } \\| A \\| _ { 2 } \\| a \\| _ { 2 } } } = \\| a \\| _ { 2 } { \\sqrt { \\| A \\| _ { 2 } } }$ with $a$ to be either $\\phi$ or $w _ { h }$ . Moreover, $\\begin{array} { r } { \\lambda _ { \\operatorname* { m i n } } ( \\widetilde { \\Lambda } _ { h } ^ { p } ) \\ge \\kappa / \\operatorname* { m a x } _ { h , s , a } \\widehat { \\sigma } _ { h } ( s , a ) ^ { 2 } \\ge \\kappa / H ^ { 2 } } \\end{array}$ implies $\\left\\| ( \\tilde { \\Lambda } _ { h } ^ { p } ) ^ { - 1 } \\right\\| \\leq H ^ { 2 } / \\kappa$ . The second inequality comes from Lemma H.2 with $R = H$ since $| \\eta _ { \\tau } | = | ( r _ { h } ^ { \\tau } + \\widehat { V } _ { h + 1 } \\left( s _ { h + 1 } ^ { \\tau } \\right) -$ $( \\mathcal T _ { h } \\widehat V _ { h + 1 } ) ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) ) / \\widehat \\sigma _ { h } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) | \\leq H$ and $| x _ { \\tau } | = | \\phi ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) / \\widehat { \\sigma } _ { h } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) | \\leq 1$ . ", + "bbox": [ + 171, + 544, + 826, + 638 + ], + "page_idx": 38 + }, + { + "type": "text", + "text": "The final result is obtained by absorbing the term (i) via the condition $K \\_ { > }$ $\\operatorname* { m a x } \\{ \\mathcal { M } _ { 1 } , \\mathcal { M } _ { 2 } , \\mathcal { M } _ { 3 } , \\mathcal { M } _ { 4 } \\}$ . ", + "bbox": [ + 169, + 642, + 807, + 672 + ], + "page_idx": 38 + }, + { + "type": "text", + "text": "Lemma H.11. Suppose random variables $\\| X \\| _ { \\infty } \\leq 2 H$ , $\\| Y \\| _ { \\infty } \\leq 2 H$ , then ", + "bbox": [ + 173, + 678, + 681, + 695 + ], + "page_idx": 38 + }, + { + "type": "equation", + "img_path": "images/a70be81e3043f9bd2d753874d2e5c086e8a96e3377ad884fc989daa62f577bb4.jpg", + "text": "$$\n\\left| \\operatorname { V a r } ( X ) - \\operatorname { V a r } ( Y ) \\right| \\leq 8 H \\cdot \\left\\| X - Y \\right\\| _ { \\infty } .\n$$", + "text_format": "latex", + "bbox": [ + 359, + 700, + 635, + 718 + ], + "page_idx": 38 + }, + { + "type": "text", + "text": "Proof of Lemma H.11. ", + "bbox": [ + 174, + 731, + 320, + 746 + ], + "page_idx": 38 + }, + { + "type": "equation", + "img_path": "images/93afdd061bb0f4286f320a147a6596d3e2bfe07bc9bb1efc675107da2ce5c070.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathrm { ) - V a r } ( Y ) \\vert = \\vert \\mathbb { E } [ X ^ { 2 } ] - \\mathbb { E } [ Y ^ { 2 } ] - ( \\mathbb { E } [ X ] ^ { 2 } - \\mathbb { E } [ Y ] ^ { 2 } ) \\vert = \\vert \\mathbb { E } [ ( X + Y ) ( X - Y ) ] - ( \\mathbb { E } [ X + Y ] ) ( \\mathbb { E } [ X - Y ] - \\mathbb { E } [ X - Y ] ) \\vert } \\\\ & { \\qquad \\le \\mathbb { E } [ \\vert X + Y \\vert \\cdot \\vert X - Y \\vert ] + 4 H \\cdot \\Vert X - Y \\Vert _ { \\infty } } \\\\ & { \\qquad \\le 4 H \\mathbb { E } [ \\vert X - Y \\vert ] + 4 H \\cdot \\Vert X - Y \\Vert _ { \\infty } = 8 H \\cdot \\Vert X - Y \\Vert _ { \\infty } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 225, + 750, + 882, + 809 + ], + "page_idx": 38 + } +] \ No newline at end of file diff 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On the empirical side, there are", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 637, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 505, + 650 + ], + "score": 1.0, + "content": "a huge body of deep RL-based algorithms (Mnih et al., 2015; Silver et al., 2017; Fujimoto et al.,", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 649, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 505, + 660 + ], + "score": 1.0, + "content": "2019; Kumar et al., 2019; Wu et al., 2019; Kidambi et al., 2020; Yu et al., 2020; Kumar et al., 2020;", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 659, + 506, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 506, + 672 + ], + "score": 1.0, + "content": "Janner et al., 2021; Chen et al., 2021a; Kostrikov et al., 2022) that utilize function approximations to", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 670, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 106, + 670, + 505, + 683 + ], + "score": 1.0, + "content": "achieve respective successes in the offline regime. However, it is also realized that practical function", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 681, + 505, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 505, + 695 + ], + "score": 1.0, + "content": "approximation schemes can be quite sample inefficient (e.g. millions of samples are needed for deep", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 692, + 346, + 705 + ], + "spans": [ + { + "bbox": [ + 106, + 693, + 115, + 704 + ], + "score": 0.72, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 692, + 346, + 705 + ], + "score": 1.0, + "content": "-network to solve certain Atari games Mnih et al. (2015)).", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 41, + "bbox_fs": [ + 105, + 516, + 506, + 705 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 160 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 505, + 95 + ], + "score": 1.0, + "content": "To understand this phenomenon, there are numerous studies consider how to achieve sample efficiency", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 105 + ], + "score": 1.0, + "content": "with function approximation from the theoretical side, as researchers find sample efficient algorithms", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 506, + 117 + ], + "score": 1.0, + "content": "are possible with particular model representations, in either online RL (e.g. Yang and Wang (2019;", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 506, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 506, + 127 + ], + "score": 1.0, + "content": "2020); Modi et al. (2020); Jin et al. (2020); Ayoub et al. (2020); Jiang et al. (2017); Du et al. (2019);", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 126, + 506, + 138 + ], + "spans": [ + { + "bbox": [ + 106, + 126, + 506, + 138 + ], + "score": 1.0, + "content": "Sun et al. (2019); Zanette et al. (2020); Zhou et al. (2021a); Jin et al. (2021a); Du et al. (2021)) or", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 137, + 506, + 149 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 506, + 149 + ], + "score": 1.0, + "content": "offline RL (e.g. Munos (2003); Chen and Jiang (2019); Xie and Jiang (2020); Jin et al. (2021b); Xie", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 148, + 506, + 160 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 506, + 160 + ], + "score": 1.0, + "content": "et al. (2021a); Min et al. (2021); Duan et al. (2021); Nguyen-Tang et al. (2021); Zanette et al. (2021)).", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 107, + 165, + 505, + 231 + ], + "lines": [ + { + "bbox": [ + 106, + 165, + 505, + 176 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 505, + 176 + ], + "score": 1.0, + "content": "Among them, the linear MDP model (Yang and Wang, 2020; Jin et al., 2020), where the transition is", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 505, + 188 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 298, + 188 + ], + "score": 1.0, + "content": "represented as a linear combinations of the given", + "type": "text" + }, + { + "bbox": [ + 298, + 177, + 304, + 186 + ], + "score": 0.76, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 176, + 505, + 188 + ], + "score": 1.0, + "content": "-dimensional feature, is (arguably) the most studied", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "score": 1.0, + "content": "setting in function approximation and there are plenty of extensions based upon it (e.g. generalized", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 197, + 505, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 505, + 211 + ], + "score": 1.0, + "content": "linear model (Wang et al., 2021b), reward-free RL (Wang et al., 2020), gap-dependent analysis (He", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 209, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 106, + 209, + 505, + 221 + ], + "score": 1.0, + "content": "et al., 2021) or generative adversarial learning (Liu et al., 2021)). Given its prosperity, however, there", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 220, + 500, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 500, + 232 + ], + "score": 1.0, + "content": "are still unknowns for understanding function representations in RL, especially in the offline case.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 133, + 241, + 505, + 361 + ], + "lines": [ + { + "bbox": [ + 133, + 240, + 506, + 254 + ], + "spans": [ + { + "bbox": [ + 133, + 240, + 506, + 254 + ], + "score": 1.0, + "content": "• While there are surging researches in showing provable sample efficiency (polynomial", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 252, + 505, + 264 + ], + "spans": [ + { + "bbox": [ + 141, + 252, + 505, + 264 + ], + "score": 1.0, + "content": "sample complexity is possible) under a variety of function approximation schemes, how", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 263, + 506, + 276 + ], + "spans": [ + { + "bbox": [ + 141, + 263, + 506, + 276 + ], + "score": 1.0, + "content": "to improve the sample efficiency for a given class of function representations remains", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 142, + 274, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 142, + 274, + 505, + 286 + ], + "score": 1.0, + "content": "understudied. For instance, given a neural network approximation class, an algorithm that", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 284, + 506, + 297 + ], + "spans": [ + { + "bbox": [ + 141, + 284, + 312, + 297 + ], + "score": 1.0, + "content": "learns the optimal policy with complexity", + "type": "text" + }, + { + "bbox": [ + 312, + 284, + 346, + 296 + ], + "score": 0.93, + "content": "{ \\cal O } ( H ^ { 1 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 284, + 506, + 297 + ], + "score": 1.0, + "content": "is far worse than the one that can learn", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 295, + 506, + 308 + ], + "spans": [ + { + "bbox": [ + 141, + 295, + 153, + 308 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 153, + 295, + 183, + 308 + ], + "score": 0.92, + "content": "O ( H ^ { 3 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 295, + 506, + 308 + ], + "score": 1.0, + "content": "sample complexity, despite that both algorithms are considered sample efficient.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 306, + 506, + 320 + ], + "spans": [ + { + "bbox": [ + 141, + 306, + 506, + 320 + ], + "score": 1.0, + "content": "Therefore, how to achieve the optimal/tight sample complexity when function approximation", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 318, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 141, + 318, + 505, + 330 + ], + "score": 1.0, + "content": "is provided is a valuable question to consider. On the other hand, it is known that tight", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 328, + 506, + 342 + ], + "spans": [ + { + "bbox": [ + 141, + 328, + 506, + 342 + ], + "score": 1.0, + "content": "sample complexity, due to the limit of the existing statistical analysis tools, can be very", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 142, + 340, + 505, + 352 + ], + "spans": [ + { + "bbox": [ + 142, + 340, + 505, + 352 + ], + "score": 1.0, + "content": "tough to establish when function representation has a very complicated form. However, does", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 141, + 351, + 454, + 363 + ], + "spans": [ + { + "bbox": [ + 141, + 351, + 454, + 363 + ], + "score": 1.0, + "content": "this mean tight analysis is not hopeful even when the representation is linear?", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 134, + 365, + 505, + 474 + ], + "lines": [ + { + "bbox": [ + 133, + 363, + 506, + 377 + ], + "spans": [ + { + "bbox": [ + 133, + 363, + 506, + 377 + ], + "score": 1.0, + "content": "• Second, in the existing analysis of offline RL (with function approximation or simply the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 141, + 375, + 506, + 388 + ], + "spans": [ + { + "bbox": [ + 141, + 375, + 506, + 388 + ], + "score": 1.0, + "content": "tabular MDPs), the learning bounds depend either explicitly on the data-coverage quantities", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 142, + 387, + 506, + 399 + ], + "spans": [ + { + "bbox": [ + 142, + 387, + 506, + 399 + ], + "score": 1.0, + "content": "(e.g. uniform concentrability coefficients Chen and Jiang (2019); Xie and Jiang (2020),", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 142, + 397, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 142, + 397, + 506, + 410 + ], + "score": 1.0, + "content": "uniform visitation measure Yin et al. (2021); Yin and Wang (2021a) and single concen-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 142, + 408, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 142, + 408, + 452, + 420 + ], + "score": 1.0, + "content": "trability Rashidinejad et al. (2021); Xie et al. (2021b)) or the horizon length", + "type": "text" + }, + { + "bbox": [ + 452, + 409, + 463, + 419 + ], + "score": 0.7, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 408, + 505, + 420 + ], + "score": 1.0, + "content": "(Jin et al.,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 142, + 419, + 505, + 431 + ], + "spans": [ + { + "bbox": [ + 142, + 419, + 505, + 431 + ], + "score": 1.0, + "content": "2021b; Uehara and Sun, 2021). While those results are valuable as they do not depend on", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 142, + 431, + 506, + 442 + ], + "spans": [ + { + "bbox": [ + 142, + 431, + 506, + 442 + ], + "score": 1.0, + "content": "the structure of the particular problem (therefore, remain valid even for pathological MDPs),", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 142, + 442, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 142, + 442, + 506, + 453 + ], + "score": 1.0, + "content": "in practice, the empirical performances of offline reinforcement learning are often far better", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 141, + 452, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 141, + 452, + 506, + 464 + ], + "score": 1.0, + "content": "than those non-adaptive bounds would indicate. Can the learning bounds reflect the nature", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 142, + 464, + 501, + 475 + ], + "spans": [ + { + "bbox": [ + 142, + 464, + 501, + 475 + ], + "score": 1.0, + "content": "of individual MDP instances when the MDP model has a certain function representation?", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 28.5 + }, + { + "type": "text", + "bbox": [ + 108, + 484, + 502, + 506 + ], + "lines": [ + { + "bbox": [ + 105, + 483, + 504, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 504, + 497 + ], + "score": 1.0, + "content": "In this work, we think about offline RL from the above two aspects. In particular, we consider the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 495, + 444, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 444, + 507 + ], + "score": 1.0, + "content": "fundamental linear model representations and ask the following question of interest:", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34.5 + }, + { + "type": "text", + "bbox": [ + 114, + 515, + 495, + 527 + ], + "lines": [ + { + "bbox": [ + 115, + 515, + 496, + 529 + ], + "spans": [ + { + "bbox": [ + 115, + 515, + 496, + 529 + ], + "score": 1.0, + "content": "Can we achieve the statistical limits for offline RL when models have linear representations?", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "title", + "bbox": [ + 108, + 541, + 205, + 551 + ], + "lines": [ + { + "bbox": [ + 106, + 540, + 207, + 553 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 207, + 553 + ], + "score": 1.0, + "content": "1.1 RELATED WORKS", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 561, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 106, + 561, + 504, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 504, + 573 + ], + "score": 1.0, + "content": "Offline RL with general function representations. The finite sample analysis of offline RL with", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 572, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 331, + 584 + ], + "score": 1.0, + "content": "function approximation is initially conducted by Fitted", + "type": "text" + }, + { + "bbox": [ + 332, + 573, + 340, + 583 + ], + "score": 0.77, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 572, + 505, + 584 + ], + "score": 1.0, + "content": "-Iteration (FQI) type algorithms and can", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 582, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 506, + 595 + ], + "score": 1.0, + "content": "be dated back to (Munos, 2003; Szepesvari and Munos ´ , 2005; Antos et al., 2008a;b). Later, Chen", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 594, + 504, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 504, + 606 + ], + "score": 1.0, + "content": "and Jiang (2019); Le et al. (2019); Xie and Jiang (2020) follow this line of research and derive the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 605, + 506, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 506, + 617 + ], + "score": 1.0, + "content": "improved learning results. However, owing to the aim for tackling general function approximation,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 615, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 506, + 628 + ], + "score": 1.0, + "content": "those learning bounds are expressed in terms of the stringent concentrability coefficients (therefore,", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "are less adaptive to individual instances) and are usually only information-theoretical, due to the", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 638, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 506, + 650 + ], + "score": 1.0, + "content": "computational intractability of the optimization procedure over the general function classes. Other", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 648, + 506, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 506, + 661 + ], + "score": 1.0, + "content": "works impose weaker assumptions (e.g. partial coverage (Liu et al., 2020; Kidambi et al., 2020;", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 659, + 506, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 483, + 672 + ], + "score": 1.0, + "content": "Uehara and Sun, 2021)), and their finite sample analysis are generally suboptimal in terms of", + "type": "text" + }, + { + "bbox": [ + 483, + 660, + 493, + 670 + ], + "score": 0.8, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 659, + 506, + 672 + ], + "score": 1.0, + "content": "or", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 670, + 236, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 191, + 684 + ], + "score": 1.0, + "content": "the effective horizon", + "type": "text" + }, + { + "bbox": [ + 191, + 671, + 233, + 683 + ], + "score": 0.94, + "content": "( 1 - \\gamma ) ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 670, + 236, + 684 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 43 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 687, + 506, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 506, + 699 + ], + "score": 1.0, + "content": "Offline RL with tabular models. For tabular MDPs, tight learning bounds can be achieved under", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "several data-coverage assumptions. 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(2020); Jin et al. (2020); Ayoub et al. (2020); Jiang et al. (2017); Du et al. (2019);", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 126, + 506, + 138 + ], + "spans": [ + { + "bbox": [ + 106, + 126, + 506, + 138 + ], + "score": 1.0, + "content": "Sun et al. (2019); Zanette et al. (2020); Zhou et al. (2021a); Jin et al. (2021a); Du et al. (2021)) or", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 137, + 506, + 149 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 506, + 149 + ], + "score": 1.0, + "content": "offline RL (e.g. Munos (2003); Chen and Jiang (2019); Xie and Jiang (2020); Jin et al. (2021b); Xie", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 148, + 506, + 160 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 506, + 160 + ], + "score": 1.0, + "content": "et al. (2021a); Min et al. (2021); Duan et al. (2021); Nguyen-Tang et al. (2021); Zanette et al. (2021)).", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3, + "bbox_fs": [ + 105, + 81, + 506, + 160 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 165, + 505, + 231 + ], + "lines": [ + { + "bbox": [ + 106, + 165, + 505, + 176 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 505, + 176 + ], + "score": 1.0, + "content": "Among them, the linear MDP model (Yang and Wang, 2020; Jin et al., 2020), where the transition is", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 505, + 188 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 298, + 188 + ], + "score": 1.0, + "content": "represented as a linear combinations of the given", + "type": "text" + }, + { + "bbox": [ + 298, + 177, + 304, + 186 + ], + "score": 0.76, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 176, + 505, + 188 + ], + "score": 1.0, + "content": "-dimensional feature, is (arguably) the most studied", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "score": 1.0, + "content": "setting in function approximation and there are plenty of extensions based upon it (e.g. generalized", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 197, + 505, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 505, + 211 + ], + "score": 1.0, + "content": "linear model (Wang et al., 2021b), reward-free RL (Wang et al., 2020), gap-dependent analysis (He", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 209, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 106, + 209, + 505, + 221 + ], + "score": 1.0, + "content": "et al., 2021) or generative adversarial learning (Liu et al., 2021)). Given its prosperity, however, there", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 220, + 500, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 500, + 232 + ], + "score": 1.0, + "content": "are still unknowns for understanding function representations in RL, especially in the offline case.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9.5, + "bbox_fs": [ + 105, + 165, + 505, + 232 + ] + }, + { + "type": "text", + "bbox": [ + 133, + 241, + 505, + 361 + ], + "lines": [ + { + "bbox": [ + 133, + 240, + 506, + 254 + ], + "spans": [ + { + "bbox": [ + 133, + 240, + 506, + 254 + ], + "score": 1.0, + "content": "• While there are surging researches in showing provable sample efficiency (polynomial", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 252, + 505, + 264 + ], + "spans": [ + { + "bbox": [ + 141, + 252, + 505, + 264 + ], + "score": 1.0, + "content": "sample complexity is possible) under a variety of function approximation schemes, how", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 263, + 506, + 276 + ], + "spans": [ + { + "bbox": [ + 141, + 263, + 506, + 276 + ], + "score": 1.0, + "content": "to improve the sample efficiency for a given class of function representations remains", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 142, + 274, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 142, + 274, + 505, + 286 + ], + "score": 1.0, + "content": "understudied. 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On the other hand, it is known that tight", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 328, + 506, + 342 + ], + "spans": [ + { + "bbox": [ + 141, + 328, + 506, + 342 + ], + "score": 1.0, + "content": "sample complexity, due to the limit of the existing statistical analysis tools, can be very", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 142, + 340, + 505, + 352 + ], + "spans": [ + { + "bbox": [ + 142, + 340, + 505, + 352 + ], + "score": 1.0, + "content": "tough to establish when function representation has a very complicated form. However, does", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 141, + 351, + 454, + 363 + ], + "spans": [ + { + "bbox": [ + 141, + 351, + 454, + 363 + ], + "score": 1.0, + "content": "this mean tight analysis is not hopeful even when the representation is linear?", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 18, + "bbox_fs": [ + 133, + 240, + 506, + 363 + ] + }, + { + "type": "text", + "bbox": [ + 134, + 365, + 505, + 474 + ], + "lines": [ + { + "bbox": [ + 133, + 363, + 506, + 377 + ], + "spans": [ + { + "bbox": [ + 133, + 363, + 506, + 377 + ], + "score": 1.0, + "content": "• Second, in the existing analysis of offline RL (with function approximation or simply the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 141, + 375, + 506, + 388 + ], + "spans": [ + { + "bbox": [ + 141, + 375, + 506, + 388 + ], + "score": 1.0, + "content": "tabular MDPs), the learning bounds depend either explicitly on the data-coverage quantities", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 142, + 387, + 506, + 399 + ], + "spans": [ + { + "bbox": [ + 142, + 387, + 506, + 399 + ], + "score": 1.0, + "content": "(e.g. uniform concentrability coefficients Chen and Jiang (2019); Xie and Jiang (2020),", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 142, + 397, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 142, + 397, + 506, + 410 + ], + "score": 1.0, + "content": "uniform visitation measure Yin et al. 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While those results are valuable as they do not depend on", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 142, + 431, + 506, + 442 + ], + "spans": [ + { + "bbox": [ + 142, + 431, + 506, + 442 + ], + "score": 1.0, + "content": "the structure of the particular problem (therefore, remain valid even for pathological MDPs),", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 142, + 442, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 142, + 442, + 506, + 453 + ], + "score": 1.0, + "content": "in practice, the empirical performances of offline reinforcement learning are often far better", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 141, + 452, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 141, + 452, + 506, + 464 + ], + "score": 1.0, + "content": "than those non-adaptive bounds would indicate. Can the learning bounds reflect the nature", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 142, + 464, + 501, + 475 + ], + "spans": [ + { + "bbox": [ + 142, + 464, + 501, + 475 + ], + "score": 1.0, + "content": "of individual MDP instances when the MDP model has a certain function representation?", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 28.5, + "bbox_fs": [ + 133, + 363, + 506, + 475 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 484, + 502, + 506 + ], + "lines": [ + { + "bbox": [ + 105, + 483, + 504, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 504, + 497 + ], + "score": 1.0, + "content": "In this work, we think about offline RL from the above two aspects. In particular, we consider the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 495, + 444, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 444, + 507 + ], + "score": 1.0, + "content": "fundamental linear model representations and ask the following question of interest:", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34.5, + "bbox_fs": [ + 105, + 483, + 504, + 507 + ] + }, + { + "type": "text", + "bbox": [ + 114, + 515, + 495, + 527 + ], + "lines": [ + { + "bbox": [ + 115, + 515, + 496, + 529 + ], + "spans": [ + { + "bbox": [ + 115, + 515, + 496, + 529 + ], + "score": 1.0, + "content": "Can we achieve the statistical limits for offline RL when models have linear representations?", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36, + "bbox_fs": [ + 115, + 515, + 496, + 529 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 541, + 205, + 551 + ], + "lines": [ + { + "bbox": [ + 106, + 540, + 207, + 553 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 207, + 553 + ], + "score": 1.0, + "content": "1.1 RELATED WORKS", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 561, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 106, + 561, + 504, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 504, + 573 + ], + "score": 1.0, + "content": "Offline RL with general function representations. The finite sample analysis of offline RL with", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 572, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 331, + 584 + ], + "score": 1.0, + "content": "function approximation is initially conducted by Fitted", + "type": "text" + }, + { + "bbox": [ + 332, + 573, + 340, + 583 + ], + "score": 0.77, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 572, + 505, + 584 + ], + "score": 1.0, + "content": "-Iteration (FQI) type algorithms and can", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 582, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 506, + 595 + ], + "score": 1.0, + "content": "be dated back to (Munos, 2003; Szepesvari and Munos ´ , 2005; Antos et al., 2008a;b). Later, Chen", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 594, + 504, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 504, + 606 + ], + "score": 1.0, + "content": "and Jiang (2019); Le et al. (2019); Xie and Jiang (2020) follow this line of research and derive the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 605, + 506, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 506, + 617 + ], + "score": 1.0, + "content": "improved learning results. However, owing to the aim for tackling general function approximation,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 615, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 506, + 628 + ], + "score": 1.0, + "content": "those learning bounds are expressed in terms of the stringent concentrability coefficients (therefore,", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "are less adaptive to individual instances) and are usually only information-theoretical, due to the", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 638, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 506, + 650 + ], + "score": 1.0, + "content": "computational intractability of the optimization procedure over the general function classes. Other", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 648, + 506, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 506, + 661 + ], + "score": 1.0, + "content": "works impose weaker assumptions (e.g. partial coverage (Liu et al., 2020; Kidambi et al., 2020;", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 659, + 506, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 483, + 672 + ], + "score": 1.0, + "content": "Uehara and Sun, 2021)), and their finite sample analysis are generally suboptimal in terms of", + "type": "text" + }, + { + "bbox": [ + 483, + 660, + 493, + 670 + ], + "score": 0.8, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 659, + 506, + 672 + ], + "score": 1.0, + "content": "or", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 670, + 236, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 191, + 684 + ], + "score": 1.0, + "content": "the effective horizon", + "type": "text" + }, + { + "bbox": [ + 191, + 671, + 233, + 683 + ], + "score": 0.94, + "content": "( 1 - \\gamma ) ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 670, + 236, + 684 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 43, + "bbox_fs": [ + 105, + 561, + 506, + 684 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 687, + 506, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 506, + 699 + ], + "score": 1.0, + "content": "Offline RL with tabular models. For tabular MDPs, tight learning bounds can be achieved under", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "several data-coverage assumptions. For the class of problems with uniform data-visitation measure", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 107, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 107, + 710, + 120, + 721 + ], + "score": 0.88, + "content": "d _ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 120, + 709, + 346, + 722 + ], + "score": 1.0, + "content": ", the near-optimal sample complexity bound has the rate", + "type": "text" + }, + { + "bbox": [ + 347, + 710, + 402, + 722 + ], + "score": 0.92, + "content": "O ( H ^ { 3 } / d _ { m } \\epsilon ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "for time-inhomogeneous", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 220, + 733 + ], + "score": 1.0, + "content": "MDPs (Yin et al., 2021) and", + "type": "text" + }, + { + "bbox": [ + 221, + 720, + 272, + 732 + ], + "score": 0.93, + "content": "O ( H ^ { 2 } / d _ { m } \\dot { \\epsilon } )", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 720, + 506, + 733 + ], + "score": 1.0, + "content": "for time-homogeneous MDPs (Yin and Wang, 2021a; Ren", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 394, + 94 + ], + "score": 1.0, + "content": "et al., 2021). Under the single concentrability assumption, the tight rate", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 395, + 82, + 457, + 95 + ], + "score": 0.93, + "content": "{ \\cal O } ( H ^ { 3 } S C ^ { \\star } / \\epsilon ^ { 2 } )", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 457, + 82, + 505, + 94 + ], + "score": 1.0, + "content": "is obtained", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "by Xie et al. (2021b). In particular, the recent study Yin and Wang (2021b) introduces the intrinsic", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "offline learning bound that is not only instance-dependent but also subsumes previous optimal results.", + "type": "text", + "cross_page": true + } + ], + "index": 2 + } + ], + "index": 50.5, + "bbox_fs": [ + 105, + 687, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 115 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 394, + 94 + ], + "score": 1.0, + "content": "et al., 2021). Under the single concentrability assumption, the tight rate", + "type": "text" + }, + { + "bbox": [ + 395, + 82, + 457, + 95 + ], + "score": 0.93, + "content": "{ \\cal O } ( H ^ { 3 } S C ^ { \\star } / \\epsilon ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 82, + 505, + 94 + ], + "score": 1.0, + "content": "is obtained", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "by Xie et al. (2021b). In particular, the recent study Yin and Wang (2021b) introduces the intrinsic", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "offline learning bound that is not only instance-dependent but also subsumes previous optimal results.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 106, + 121, + 505, + 352 + ], + "lines": [ + { + "bbox": [ + 106, + 120, + 505, + 134 + ], + "spans": [ + { + "bbox": [ + 106, + 120, + 505, + 134 + ], + "score": 1.0, + "content": "Offline RL with linear model representations. Recently, there is more focus on studying the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 132, + 506, + 144 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 506, + 144 + ], + "score": 1.0, + "content": "provable efficient offline RL under the linear model representations. Jin et al. (2021b) first shows", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 143, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 106, + 143, + 506, + 156 + ], + "score": 1.0, + "content": "offline RL with linear MDP is provably efficient by the pessimistic value iteration. Their analysis", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 154, + 505, + 165 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 294, + 165 + ], + "score": 1.0, + "content": "deviates from their lower bound by a factor of", + "type": "text" + }, + { + "bbox": [ + 295, + 154, + 318, + 164 + ], + "score": 0.89, + "content": "d \\cdot H", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 154, + 505, + 165 + ], + "score": 1.0, + "content": "(check their Theorem 4.4 and 4.6). Later, Xie", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 164, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 506, + 178 + ], + "score": 1.0, + "content": "et al. (2021a) considers function approximation under the Bellman-consistent assumptions, and,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "when realized to linear MDP setting, improves the sample complexity guarantee of Jin et al. (2021b)", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 186, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 156, + 200 + ], + "score": 1.0, + "content": "by an order", + "type": "text" + }, + { + "bbox": [ + 156, + 187, + 178, + 199 + ], + "score": 0.9, + "content": "O ( d )", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 186, + 505, + 200 + ], + "score": 1.0, + "content": "(Theorem 3.2).2 However, their improvement only holds for finite action space", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 200, + 210 + ], + "score": 1.0, + "content": "(due to the dependence", + "type": "text" + }, + { + "bbox": [ + 200, + 198, + 232, + 210 + ], + "score": 0.89, + "content": "\\log \\left| { \\mathcal { A } } \\right| )", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 198, + 506, + 210 + ], + "score": 1.0, + "content": "and by the direct reduction (from Theorem 3.1) their result does not", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "score": 1.0, + "content": "imply a computationally tractable algorithm with the same guarantee. Concurrently, Zanette et al.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 219, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 505, + 232 + ], + "score": 1.0, + "content": "(2021) considers the Linear Bellman Complete model and designs the actor-critic style algorithm", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 230, + 505, + 242 + ], + "spans": [ + { + "bbox": [ + 106, + 230, + 505, + 242 + ], + "score": 1.0, + "content": "that achieves tight result under the assumption that the value function is bounded by 1. While their", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 241, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 241, + 505, + 254 + ], + "score": 1.0, + "content": "algorithm is efficient (which is based on solving a sequence of second-order cone programs), the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 253, + 504, + 264 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 504, + 264 + ], + "score": 1.0, + "content": "resulting learning bound requires the action space to be finite due to the mirror descent updates in", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 264, + 506, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 506, + 276 + ], + "score": 1.0, + "content": "the Actor procedure (Agarwal et al., 2021). Besides, assuming the value function to be less than 1", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 275, + 505, + 287 + ], + "spans": [ + { + "bbox": [ + 106, + 275, + 301, + 287 + ], + "score": 1.0, + "content": "simplifies the challenges in dealing with horizon", + "type": "text" + }, + { + "bbox": [ + 302, + 275, + 312, + 285 + ], + "score": 0.78, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 275, + 453, + 287 + ], + "score": 1.0, + "content": "since when rescaling their result to", + "type": "text" + }, + { + "bbox": [ + 454, + 275, + 479, + 286 + ], + "score": 0.91, + "content": "[ 0 , H ]", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 275, + 505, + 287 + ], + "score": 1.0, + "content": ", there", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 285, + 506, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 122, + 298 + ], + "score": 1.0, + "content": "is a", + "type": "text" + }, + { + "bbox": [ + 123, + 286, + 133, + 296 + ], + "score": 0.78, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 285, + 506, + 298 + ], + "score": 1.0, + "content": "factor blow-up, which makes no horizon improvement comparing to Jin et al. (2021b). As a", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 297, + 506, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 297, + 506, + 308 + ], + "score": 1.0, + "content": "result, none of the existing algorithms can achieve the statistical limit for the well-structured linear", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 306, + 506, + 321 + ], + "spans": [ + { + "bbox": [ + 104, + 306, + 506, + 321 + ], + "score": 1.0, + "content": "MDP model with the general (infinite or continuous) state-action spaces. On the other hand, Wang", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 317, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 505, + 331 + ], + "score": 1.0, + "content": "et al. (2021a); Zanette (2021) study the statistical hardness of offline RL with linear representations", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 329, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 505, + 342 + ], + "score": 1.0, + "content": "by proving the exponential lower bounds. Recently, Foster et al. (2021) shows realizability and", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 339, + 455, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 455, + 354 + ], + "score": 1.0, + "content": "concentrability are not sufficient for offline learning when state space is arbitrary large.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 357, + 505, + 446 + ], + "lines": [ + { + "bbox": [ + 106, + 356, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 106, + 356, + 505, + 369 + ], + "score": 1.0, + "content": "Variance-aware studies. Talebi and Maillard (2018) first incorporates the variance structure in online", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 369, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 106, + 369, + 505, + 379 + ], + "score": 1.0, + "content": "tabular MDPs and Zanette and Brunskill (2019) tightens the result. For linear mixture MDPs, Zhou", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 379, + 505, + 391 + ], + "spans": [ + { + "bbox": [ + 106, + 379, + 505, + 391 + ], + "score": 1.0, + "content": "et al. (2021a) first uses variance structure to achieve near-optimal result and the Weighted OFUL", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 391, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 106, + 391, + 505, + 402 + ], + "score": 1.0, + "content": "incorporates the variance structure explicitly in the regret bound. Recently, Variance-awareness is", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 401, + 506, + 414 + ], + "spans": [ + { + "bbox": [ + 106, + 401, + 506, + 414 + ], + "score": 1.0, + "content": "also considered in Zhang et al. (2021) for horizon-free setting and for OPE problem (Min et al., 2021).", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 411, + 506, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 506, + 425 + ], + "score": 1.0, + "content": "In particular, We point out that Min et al. (2021) is the first work that uses variance reweighting", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 423, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 106, + 423, + 505, + 435 + ], + "score": 1.0, + "content": "for policy evaluation in offline RL, which inspires our study for policy optimization problem. The", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 434, + 467, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 467, + 447 + ], + "score": 1.0, + "content": "guarantee of Min et al. (2021) strictly improves over Duan et al. (2020) for OPE problem.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 27.5 + }, + { + "type": "title", + "bbox": [ + 108, + 464, + 219, + 475 + ], + "lines": [ + { + "bbox": [ + 106, + 464, + 221, + 476 + ], + "spans": [ + { + "bbox": [ + 106, + 464, + 221, + 476 + ], + "score": 1.0, + "content": "1.2 OUR CONTRIBUTION", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 487, + 506, + 662 + ], + "lines": [ + { + "bbox": [ + 106, + 487, + 507, + 499 + ], + "spans": [ + { + "bbox": [ + 106, + 487, + 507, + 499 + ], + "score": 1.0, + "content": "In this work, we study offline RL for time-inhomogeneous episodic linear Markov decision processes.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 498, + 507, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 498, + 507, + 510 + ], + "score": 1.0, + "content": "Linear MDPs serve as one critical step towards understanding function approximation in RL since: 1.", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 508, + 506, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 508, + 506, + 522 + ], + "score": 1.0, + "content": "unlike general function representation, linear MDP representation has the well-structured form by", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 520, + 506, + 532 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 506, + 532 + ], + "score": 1.0, + "content": "the given feature representors, which makes delicate statistical analysis hopeful; 2. unlike tabular", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 531, + 506, + 543 + ], + "spans": [ + { + "bbox": [ + 106, + 531, + 506, + 543 + ], + "score": 1.0, + "content": "representation, which only works for finite models, linear MDP provides generalization as it adapts to", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 541, + 506, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 506, + 554 + ], + "score": 1.0, + "content": "infinite or continuous state-action spaces. Especially, we design the variance-aware pessimistic value", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 553, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 106, + 553, + 505, + 564 + ], + "score": 1.0, + "content": "iteration (VAPVI, Algorithm 1) which incorporates the conditional variance information of the value", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 563, + 506, + 575 + ], + "spans": [ + { + "bbox": [ + 106, + 563, + 506, + 575 + ], + "score": 1.0, + "content": "function and, by the variance structure, Theorem 3.2 is able to improve over the aforementioned state-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 574, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 506, + 587 + ], + "score": 1.0, + "content": "of-the-art guarantees. In addition, we further improve the state-action guarantee by designing an even", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 586, + 506, + 598 + ], + "spans": [ + { + "bbox": [ + 106, + 586, + 506, + 598 + ], + "score": 1.0, + "content": "tighter bonus (4). VAPVI-Improved (Theorem 3.3) is near-minimax optimal as indicated by our lower", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 596, + 506, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 596, + 506, + 609 + ], + "score": 1.0, + "content": "bound (Theorem 3.5). Importantly, the resulting learning bounds from VAPVI/VAPVI-Improved", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 607, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 106, + 607, + 505, + 619 + ], + "score": 1.0, + "content": "are able to characterize the adaptive nature of individual instances and yield different convergence", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 618, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 505, + 631 + ], + "score": 1.0, + "content": "rates for different problems. Algorithmically, our algorithm builds upon the nice Min et al. (2021)", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 629, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 106, + 629, + 505, + 641 + ], + "score": 1.0, + "content": "with pessimism as we use the estimated variances to reweight the Bellman residual learning objective", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 641, + 504, + 652 + ], + "spans": [ + { + "bbox": [ + 106, + 641, + 504, + 652 + ], + "score": 1.0, + "content": "so that the (training) samples with high uncertainty get less attention (Section 3). This is the key to", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 651, + 266, + 664 + ], + "spans": [ + { + "bbox": [ + 105, + 651, + 266, + 664 + ], + "score": 1.0, + "content": "obtaining instance-adaptive guarantees.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 40.5 + }, + { + "type": "title", + "bbox": [ + 108, + 684, + 207, + 696 + ], + "lines": [ + { + "bbox": [ + 105, + 682, + 209, + 699 + ], + "spans": [ + { + "bbox": [ + 105, + 682, + 209, + 699 + ], + "score": 1.0, + "content": "2 PRELIMINARIES", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 49 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 115, + 721, + 502, + 732 + ], + "lines": [ + { + "bbox": [ + 118, + 718, + 504, + 735 + ], + "spans": [ + { + "bbox": [ + 118, + 718, + 504, + 735 + ], + "score": 1.0, + "content": "2This comparison is based on translating their infinite horizon discounted setting to the finite-horizon case.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "score": 1.0, + "content": "3", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 115 + ], + "lines": [], + "index": 1, + "bbox_fs": [ + 105, + 82, + 505, + 117 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 121, + 505, + 352 + ], + "lines": [ + { + "bbox": [ + 106, + 120, + 505, + 134 + ], + "spans": [ + { + "bbox": [ + 106, + 120, + 505, + 134 + ], + "score": 1.0, + "content": "Offline RL with linear model representations. Recently, there is more focus on studying the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 132, + 506, + 144 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 506, + 144 + ], + "score": 1.0, + "content": "provable efficient offline RL under the linear model representations. Jin et al. (2021b) first shows", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 143, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 106, + 143, + 506, + 156 + ], + "score": 1.0, + "content": "offline RL with linear MDP is provably efficient by the pessimistic value iteration. Their analysis", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 154, + 505, + 165 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 294, + 165 + ], + "score": 1.0, + "content": "deviates from their lower bound by a factor of", + "type": "text" + }, + { + "bbox": [ + 295, + 154, + 318, + 164 + ], + "score": 0.89, + "content": "d \\cdot H", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 154, + 505, + 165 + ], + "score": 1.0, + "content": "(check their Theorem 4.4 and 4.6). Later, Xie", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 164, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 506, + 178 + ], + "score": 1.0, + "content": "et al. (2021a) considers function approximation under the Bellman-consistent assumptions, and,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "when realized to linear MDP setting, improves the sample complexity guarantee of Jin et al. (2021b)", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 186, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 156, + 200 + ], + "score": 1.0, + "content": "by an order", + "type": "text" + }, + { + "bbox": [ + 156, + 187, + 178, + 199 + ], + "score": 0.9, + "content": "O ( d )", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 186, + 505, + 200 + ], + "score": 1.0, + "content": "(Theorem 3.2).2 However, their improvement only holds for finite action space", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 200, + 210 + ], + "score": 1.0, + "content": "(due to the dependence", + "type": "text" + }, + { + "bbox": [ + 200, + 198, + 232, + 210 + ], + "score": 0.89, + "content": "\\log \\left| { \\mathcal { A } } \\right| )", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 198, + 506, + 210 + ], + "score": 1.0, + "content": "and by the direct reduction (from Theorem 3.1) their result does not", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "score": 1.0, + "content": "imply a computationally tractable algorithm with the same guarantee. Concurrently, Zanette et al.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 219, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 505, + 232 + ], + "score": 1.0, + "content": "(2021) considers the Linear Bellman Complete model and designs the actor-critic style algorithm", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 230, + 505, + 242 + ], + "spans": [ + { + "bbox": [ + 106, + 230, + 505, + 242 + ], + "score": 1.0, + "content": "that achieves tight result under the assumption that the value function is bounded by 1. While their", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 241, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 241, + 505, + 254 + ], + "score": 1.0, + "content": "algorithm is efficient (which is based on solving a sequence of second-order cone programs), the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 253, + 504, + 264 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 504, + 264 + ], + "score": 1.0, + "content": "resulting learning bound requires the action space to be finite due to the mirror descent updates in", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 264, + 506, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 506, + 276 + ], + "score": 1.0, + "content": "the Actor procedure (Agarwal et al., 2021). Besides, assuming the value function to be less than 1", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 275, + 505, + 287 + ], + "spans": [ + { + "bbox": [ + 106, + 275, + 301, + 287 + ], + "score": 1.0, + "content": "simplifies the challenges in dealing with horizon", + "type": "text" + }, + { + "bbox": [ + 302, + 275, + 312, + 285 + ], + "score": 0.78, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 275, + 453, + 287 + ], + "score": 1.0, + "content": "since when rescaling their result to", + "type": "text" + }, + { + "bbox": [ + 454, + 275, + 479, + 286 + ], + "score": 0.91, + "content": "[ 0 , H ]", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 275, + 505, + 287 + ], + "score": 1.0, + "content": ", there", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 285, + 506, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 122, + 298 + ], + "score": 1.0, + "content": "is a", + "type": "text" + }, + { + "bbox": [ + 123, + 286, + 133, + 296 + ], + "score": 0.78, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 285, + 506, + 298 + ], + "score": 1.0, + "content": "factor blow-up, which makes no horizon improvement comparing to Jin et al. (2021b). As a", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 297, + 506, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 297, + 506, + 308 + ], + "score": 1.0, + "content": "result, none of the existing algorithms can achieve the statistical limit for the well-structured linear", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 306, + 506, + 321 + ], + "spans": [ + { + "bbox": [ + 104, + 306, + 506, + 321 + ], + "score": 1.0, + "content": "MDP model with the general (infinite or continuous) state-action spaces. On the other hand, Wang", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 317, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 505, + 331 + ], + "score": 1.0, + "content": "et al. (2021a); Zanette (2021) study the statistical hardness of offline RL with linear representations", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 329, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 505, + 342 + ], + "score": 1.0, + "content": "by proving the exponential lower bounds. Recently, Foster et al. (2021) shows realizability and", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 339, + 455, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 455, + 354 + ], + "score": 1.0, + "content": "concentrability are not sufficient for offline learning when state space is arbitrary large.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 13, + "bbox_fs": [ + 104, + 120, + 506, + 354 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 357, + 505, + 446 + ], + "lines": [ + { + "bbox": [ + 106, + 356, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 106, + 356, + 505, + 369 + ], + "score": 1.0, + "content": "Variance-aware studies. Talebi and Maillard (2018) first incorporates the variance structure in online", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 369, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 106, + 369, + 505, + 379 + ], + "score": 1.0, + "content": "tabular MDPs and Zanette and Brunskill (2019) tightens the result. For linear mixture MDPs, Zhou", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 379, + 505, + 391 + ], + "spans": [ + { + "bbox": [ + 106, + 379, + 505, + 391 + ], + "score": 1.0, + "content": "et al. (2021a) first uses variance structure to achieve near-optimal result and the Weighted OFUL", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 391, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 106, + 391, + 505, + 402 + ], + "score": 1.0, + "content": "incorporates the variance structure explicitly in the regret bound. Recently, Variance-awareness is", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 401, + 506, + 414 + ], + "spans": [ + { + "bbox": [ + 106, + 401, + 506, + 414 + ], + "score": 1.0, + "content": "also considered in Zhang et al. (2021) for horizon-free setting and for OPE problem (Min et al., 2021).", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 411, + 506, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 506, + 425 + ], + "score": 1.0, + "content": "In particular, We point out that Min et al. (2021) is the first work that uses variance reweighting", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 423, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 106, + 423, + 505, + 435 + ], + "score": 1.0, + "content": "for policy evaluation in offline RL, which inspires our study for policy optimization problem. The", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 434, + 467, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 467, + 447 + ], + "score": 1.0, + "content": "guarantee of Min et al. (2021) strictly improves over Duan et al. 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Especially, we design the variance-aware pessimistic value", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 553, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 106, + 553, + 505, + 564 + ], + "score": 1.0, + "content": "iteration (VAPVI, Algorithm 1) which incorporates the conditional variance information of the value", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 563, + 506, + 575 + ], + "spans": [ + { + "bbox": [ + 106, + 563, + 506, + 575 + ], + "score": 1.0, + "content": "function and, by the variance structure, Theorem 3.2 is able to improve over the aforementioned state-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 574, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 506, + 587 + ], + "score": 1.0, + "content": "of-the-art guarantees. In addition, we further improve the state-action guarantee by designing an even", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 586, + 506, + 598 + ], + "spans": [ + { + "bbox": [ + 106, + 586, + 506, + 598 + ], + "score": 1.0, + "content": "tighter bonus (4). VAPVI-Improved (Theorem 3.3) is near-minimax optimal as indicated by our lower", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 596, + 506, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 596, + 506, + 609 + ], + "score": 1.0, + "content": "bound (Theorem 3.5). Importantly, the resulting learning bounds from VAPVI/VAPVI-Improved", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 607, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 106, + 607, + 505, + 619 + ], + "score": 1.0, + "content": "are able to characterize the adaptive nature of individual instances and yield different convergence", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 618, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 505, + 631 + ], + "score": 1.0, + "content": "rates for different problems. Algorithmically, our algorithm builds upon the nice Min et al. 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Here for notation simplicity we abuse", + "type": "text" + }, + { + "bbox": [ + 483, + 429, + 505, + 441 + ], + "score": 0.9, + "content": "d _ { h } ^ { \\pi } ( \\cdot )", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 441, + 333, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 333, + 453 + ], + "score": 1.0, + "content": "to denote either probability measure or density function.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 106, + 457, + 506, + 516 + ], + "lines": [ + { + "bbox": [ + 102, + 456, + 506, + 490 + ], + "spans": [ + { + "bbox": [ + 102, + 456, + 233, + 490 + ], + "score": 1.0, + "content": "Offline learning setting. Offlprovided with the historical data", + "type": "text" + }, + { + "bbox": [ + 350, + 456, + 416, + 490 + ], + "score": 1.0, + "content": "learn the policy rolled out from s", + "type": "text" + }, + { + "bbox": [ + 416, + 460, + 424, + 468 + ], + "score": 0.74, + "content": "\\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 456, + 490, + 490 + ], + "score": 1.0, + "content": "that maximizes me behavior polic", + "type": "text" + }, + { + "bbox": [ + 491, + 459, + 502, + 468 + ], + "score": 0.87, + "content": "v ^ { \\pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 456, + 506, + 490 + ], + "score": 1.0, + "content": ",.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 234, + 468, + 503, + 484 + ], + "spans": [ + { + "bbox": [ + 234, + 468, + 349, + 484 + ], + "score": 0.92, + "content": "\\mathcal { D } = \\{ ( s _ { h } ^ { \\tau } , \\bar { a } _ { h } ^ { \\tau } , r _ { h } ^ { \\tau } , s _ { h + 1 } ^ { \\tau } ) \\} _ { \\tau \\in [ K ] } ^ { h \\in [ H ] }", + "type": "inline_equation" + }, + { + "bbox": [ + 496, + 472, + 503, + 482 + ], + "score": 0.78, + "content": "\\mu", + "type": "inline_equation" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 480, + 506, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 286, + 496 + ], + "score": 1.0, + "content": "The offline nature requires we cannot change", + "type": "text" + }, + { + "bbox": [ + 286, + 485, + 294, + 494 + ], + "score": 0.81, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 480, + 506, + 496 + ], + "score": 1.0, + "content": "and in particular we do not know the data generating", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 491, + 506, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 167, + 507 + ], + "score": 1.0, + "content": "distribution of", + "type": "text" + }, + { + "bbox": [ + 168, + 495, + 174, + 504 + ], + "score": 0.8, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 491, + 358, + 507 + ], + "score": 1.0, + "content": ". 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To begin", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 570, + 480, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 407, + 584 + ], + "score": 1.0, + "content": "with, we define the population covariance matrix under the behavior policy", + "type": "text" + }, + { + "bbox": [ + 407, + 573, + 415, + 583 + ], + "score": 0.81, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 570, + 443, + 584 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 443, + 571, + 476, + 583 + ], + "score": 0.91, + "content": "h \\in [ H ]", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 570, + 480, + 584 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35 + }, + { + "type": "interline_equation", + "bbox": [ + 241, + 587, + 368, + 603 + ], + "lines": [ + { + "bbox": [ + 241, + 587, + 368, + 603 + ], + "spans": [ + { + "bbox": [ + 241, + 587, + 368, + 603 + ], + "score": 0.93, + "content": "\\Sigma _ { h } ^ { p } : = \\mathbb { E } _ { \\mu , h } \\left[ \\phi ( s , a ) \\phi ( s , a ) ^ { \\top } \\right] ,", + "type": "interline_equation", + "image_path": "a392cf2fac77b1af545c26a2481b647c07aaca80dc834e4153aada685a1f50ae.jpg" + } + ] + } + ], + "index": 37, + "virtual_lines": [ + { + "bbox": [ + 241, + 587, + 368, + 603 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 111, + 606, + 504, + 619 + ], + "lines": [ + { + "bbox": [ + 109, + 605, + 506, + 621 + ], + "spans": [ + { + "bbox": [ + 109, + 605, + 129, + 621 + ], + "score": 1.0, + "content": "since", + "type": "text" + }, + { + "bbox": [ + 129, + 607, + 143, + 619 + ], + "score": 0.9, + "content": "\\Sigma _ { h } ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 605, + 349, + 621 + ], + "score": 1.0, + "content": "measure the coverage of state-action space for data", + "type": "text" + }, + { + "bbox": [ + 350, + 608, + 359, + 617 + ], + "score": 0.82, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 605, + 506, + 621 + ], + "score": 1.0, + "content": ", we make the following assumption.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 107, + 621, + 505, + 655 + ], + "lines": [ + { + "bbox": [ + 105, + 621, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 362, + 634 + ], + "score": 1.0, + "content": "Assumption 2.2 (Feature Coverage). The data distributions", + "type": "text" + }, + { + "bbox": [ + 362, + 623, + 370, + 633 + ], + "score": 0.79, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 621, + 505, + 634 + ], + "score": 1.0, + "content": "satisfy the minimum eigenvalue", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 150, + 645 + ], + "score": 1.0, + "content": "condition:", + "type": "text" + }, + { + "bbox": [ + 150, + 632, + 190, + 644 + ], + "score": 0.86, + "content": "\\forall h \\in [ H ]", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 632, + 194, + 645 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 194, + 632, + 279, + 645 + ], + "score": 0.87, + "content": "\\kappa _ { h } : = \\lambda _ { \\operatorname* { m i n } } ( \\bar { \\Sigma } _ { h } ^ { p } ) > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 632, + 327, + 645 + ], + "score": 1.0, + "content": "and denote", + "type": "text" + }, + { + "bbox": [ + 327, + 633, + 381, + 644 + ], + "score": 0.89, + "content": "\\kappa = \\operatorname* { m i n } _ { h } \\kappa _ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 632, + 407, + 645 + ], + "score": 1.0, + "content": ". Note", + "type": "text" + }, + { + "bbox": [ + 407, + 635, + 414, + 642 + ], + "score": 0.7, + "content": "\\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 632, + 505, + 645 + ], + "score": 1.0, + "content": "is a system-dependent", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 643, + 482, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 308, + 657 + ], + "score": 1.0, + "content": "(non-universal) quantity as it is upper bounded by", + "type": "text" + }, + { + "bbox": [ + 308, + 644, + 325, + 655 + ], + "score": 0.88, + "content": "1 / d", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 643, + 482, + 657 + ], + "score": 1.0, + "content": "(Assumption 2 in Wang et al. 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By Definition 2.1, the", + "type": "text" + }, + { + "bbox": [ + 466, + 384, + 474, + 395 + ], + "score": 0.86, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 383, + 505, + 396 + ], + "score": 1.0, + "content": "-values", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 394, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 233, + 408 + ], + "score": 1.0, + "content": "also admit linear structures, i.e.", + "type": "text" + }, + { + "bbox": [ + 233, + 396, + 291, + 408 + ], + "score": 0.93, + "content": "Q _ { h } ^ { \\pi } = \\langle \\phi , w _ { h } ^ { \\pi } \\rangle", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 394, + 330, + 408 + ], + "score": 1.0, + "content": "for some", + "type": "text" + }, + { + "bbox": [ + 330, + 395, + 368, + 408 + ], + "score": 0.92, + "content": "w _ { h } ^ { \\pi } \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 394, + 505, + 408 + ], + "score": 1.0, + "content": "(Lemma H.9). Lastly, for a policy", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 107, + 406, + 506, + 419 + ], + "spans": [ + { + "bbox": [ + 107, + 409, + 114, + 416 + ], + "score": 0.74, + "content": "\\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 406, + 446, + 419 + ], + "score": 1.0, + "content": ", we denote the induced occupancy measure over the state-action space at any time", + "type": "text" + }, + { + "bbox": [ + 446, + 407, + 480, + 418 + ], + "score": 0.9, + "content": "h \\in [ H ]", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 406, + 506, + 419 + ], + "score": 1.0, + "content": "to be:", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 416, + 506, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 137, + 431 + ], + "score": 1.0, + "content": "for any", + "type": "text" + }, + { + "bbox": [ + 137, + 417, + 186, + 428 + ], + "score": 0.81, + "content": "E \\subseteq S \\times A", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 416, + 189, + 431 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 190, + 417, + 506, + 430 + ], + "score": 0.79, + "content": "d _ { h } ^ { \\pi } ( E ) : = \\mathbb { E } [ ( s _ { h } , a _ { h } ) \\in E | s _ { 1 } \\sim d _ { 1 } , a _ { i } \\sim \\pi ( \\cdot | s _ { i } ) , s _ { i } \\sim P _ { i - 1 } ( \\cdot | s _ { i - 1 } , \\bar { a _ { i - 1 } } ) , 1 \\le h | s _ { i - 1 } \\sim \\pi ( \\cdot | s _ { i - 1 } , \\bar { a _ { i - 1 } } ) ]", + "type": "inline_equation" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 426, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 106, + 428, + 133, + 441 + ], + "score": 0.91, + "content": "i \\leq h \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 426, + 151, + 444 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 151, + 429, + 326, + 442 + ], + "score": 0.89, + "content": "\\begin{array} { r } { \\mathbb { E } _ { \\pi , h } [ f ( s , a ) ] : = \\int _ { \\mathcal { S } \\times \\mathcal { A } } f ( s , a ) d _ { h } ^ { \\pi } ( s , a ) d s d a } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 426, + 482, + 444 + ], + "score": 1.0, + "content": ". Here for notation simplicity we abuse", + "type": "text" + }, + { + "bbox": [ + 483, + 429, + 505, + 441 + ], + "score": 0.9, + "content": "d _ { h } ^ { \\pi } ( \\cdot )", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 441, + 333, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 333, + 453 + ], + "score": 1.0, + "content": "to denote either probability measure or density function.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 331, + 507, + 453 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 457, + 506, + 516 + ], + "lines": [ + { + "bbox": [ + 102, + 456, + 506, + 490 + ], + "spans": [ + { + "bbox": [ + 102, + 456, + 233, + 490 + ], + "score": 1.0, + "content": "Offline learning setting. Offlprovided with the historical data", + "type": "text" + }, + { + "bbox": [ + 350, + 456, + 416, + 490 + ], + "score": 1.0, + "content": "learn the policy rolled out from s", + "type": "text" + }, + { + "bbox": [ + 416, + 460, + 424, + 468 + ], + "score": 0.74, + "content": "\\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 456, + 490, + 490 + ], + "score": 1.0, + "content": "that maximizes me behavior polic", + "type": "text" + }, + { + "bbox": [ + 491, + 459, + 502, + 468 + ], + "score": 0.87, + "content": "v ^ { \\pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 456, + 506, + 490 + ], + "score": 1.0, + "content": ",.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 234, + 468, + 503, + 484 + ], + "spans": [ + { + "bbox": [ + 234, + 468, + 349, + 484 + ], + "score": 0.92, + "content": "\\mathcal { D } = \\{ ( s _ { h } ^ { \\tau } , \\bar { a } _ { h } ^ { \\tau } , r _ { h } ^ { \\tau } , s _ { h + 1 } ^ { \\tau } ) \\} _ { \\tau \\in [ K ] } ^ { h \\in [ H ] }", + "type": "inline_equation" + }, + { + "bbox": [ + 496, + 472, + 503, + 482 + ], + "score": 0.78, + "content": "\\mu", + "type": "inline_equation" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 480, + 506, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 286, + 496 + ], + "score": 1.0, + "content": "The offline nature requires we cannot change", + "type": "text" + }, + { + "bbox": [ + 286, + 485, + 294, + 494 + ], + "score": 0.81, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 480, + 506, + 496 + ], + "score": 1.0, + "content": "and in particular we do not know the data generating", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 491, + 506, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 167, + 507 + ], + "score": 1.0, + "content": "distribution of", + "type": "text" + }, + { + "bbox": [ + 168, + 495, + 174, + 504 + ], + "score": 0.8, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 491, + 358, + 507 + ], + "score": 1.0, + "content": ". 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The data distributions", + "type": "text" + }, + { + "bbox": [ + 362, + 623, + 370, + 633 + ], + "score": 0.79, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 621, + 505, + 634 + ], + "score": 1.0, + "content": "satisfy the minimum eigenvalue", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 150, + 645 + ], + "score": 1.0, + "content": "condition:", + "type": "text" + }, + { + "bbox": [ + 150, + 632, + 190, + 644 + ], + "score": 0.86, + "content": "\\forall h \\in [ H ]", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 632, + 194, + 645 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 194, + 632, + 279, + 645 + ], + "score": 0.87, + "content": "\\kappa _ { h } : = \\lambda _ { \\operatorname* { m i n } } ( \\bar { \\Sigma } _ { h } ^ { p } ) > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 632, + 327, + 645 + ], + "score": 1.0, + "content": "and denote", + "type": "text" + }, + { + "bbox": [ + 327, + 633, + 381, + 644 + ], + "score": 0.89, + "content": "\\kappa = \\operatorname* { m i n } _ { h } \\kappa _ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 632, + 407, + 645 + ], + "score": 1.0, + "content": ". Note", + "type": "text" + }, + { + "bbox": [ + 407, + 635, + 414, + 642 + ], + "score": 0.7, + "content": "\\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 632, + 505, + 645 + ], + "score": 1.0, + "content": "is a system-dependent", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 643, + 482, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 308, + 657 + ], + "score": 1.0, + "content": "(non-universal) quantity as it is upper bounded by", + "type": "text" + }, + { + "bbox": [ + 308, + 644, + 325, + 655 + ], + "score": 0.88, + "content": "1 / d", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 643, + 482, + 657 + ], + "score": 1.0, + "content": "(Assumption 2 in Wang et al. (2021a)).", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 40, + "bbox_fs": [ + 105, + 621, + 505, + 657 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 663, + 505, + 703 + ], + "lines": [ + { + "bbox": [ + 105, + 663, + 506, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 663, + 506, + 677 + ], + "score": 1.0, + "content": "We make this assumption for the following reasons. First of all, our offline learning guarantee", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 674, + 505, + 688 + ], + "spans": [ + { + "bbox": [ + 105, + 674, + 505, + 688 + ], + "score": 1.0, + "content": "(Theorem 3.2) provides simultaneously comparison to all the policies, which is stronger than only", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 104, + 685, + 504, + 704 + ], + "spans": [ + { + "bbox": [ + 104, + 686, + 410, + 704 + ], + "score": 1.0, + "content": "competing with the optimal policy (whereas relaxed assumption suffices, e.g.", + "type": "text" + }, + { + "bbox": [ + 411, + 685, + 504, + 704 + ], + "score": 0.83, + "content": "\\begin{array} { r } { \\operatorname* { s u p } _ { x \\in \\mathbb { R } ^ { d } } \\frac { \\bar { x } \\Sigma _ { \\pi ^ { \\star } } x ^ { \\top } } { x \\Sigma _ { \\mu } x ^ { \\top } } < \\infty } \\end{array}", + "type": "inline_equation" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 82, + 505, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 382, + 94 + ], + "score": 1.0, + "content": "(Uehara and Sun, 2021)). As a consequence, the behavior distribution", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 382, + 84, + 390, + 94 + ], + "score": 0.81, + "content": "\\mu", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 390, + 82, + 505, + 94 + ], + "score": 1.0, + "content": "must be able to explore each", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 505, + 105 + ], + "score": 1.0, + "content": "feature dimension for the result to be valid. 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In this scenario, learning the optimal policy cannot be guaranteed as a", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 506, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 506, + 149 + ], + "score": 1.0, + "content": "constant suboptimality gap needs to be suffered due to the lack of coverage and this is formed as", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 148, + 506, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 174, + 161 + ], + "score": 1.0, + "content": "assumption-free", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 174, + 149, + 188, + 159 + ], + "score": 0.56, + "content": "R L", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 188, + 148, + 506, + 161 + ], + "score": 1.0, + "content": "in Yin and Wang (2021b). 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(2021a) Theorem 3.2 where", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 406, + 158, + 424, + 172 + ], + "score": 0.92, + "content": "\\dot { \\Sigma _ { \\mathcal { D } } } ^ { - 1 }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 425, + 156, + 506, + 174 + ], + "score": 1.0, + "content": "exists and Min et al.", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 170, + 223, + 182 + ], + "spans": [ + { + "bbox": [ + 106, + 170, + 223, + 182 + ], + "score": 1.0, + "content": "(2021) for the OPE problem.", + "type": "text", + "cross_page": true + } + ], + "index": 8 + } + ], + "index": 43, + "bbox_fs": [ + 104, + 663, + 506, + 704 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 182 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 382, + 94 + ], + "score": 1.0, + "content": "(Uehara and Sun, 2021)). 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In this scenario, learning the optimal policy cannot be guaranteed as a", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 506, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 506, + 149 + ], + "score": 1.0, + "content": "constant suboptimality gap needs to be suffered due to the lack of coverage and this is formed as", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 148, + 506, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 174, + 161 + ], + "score": 1.0, + "content": "assumption-free", + "type": "text" + }, + { + "bbox": [ + 174, + 149, + 188, + 159 + ], + "score": 0.56, + "content": "R L", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 148, + 506, + 161 + ], + "score": 1.0, + "content": "in Yin and Wang (2021b). Lastly, previous works analyzing the linear MDPs", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 156, + 506, + 174 + ], + "spans": [ + { + "bbox": [ + 104, + 156, + 405, + 174 + ], + "score": 1.0, + "content": "impose very similar assumptions, e.g. Xie et al. (2021a) Theorem 3.2 where", + "type": "text" + }, + { + "bbox": [ + 406, + 158, + 424, + 172 + ], + "score": 0.92, + "content": "\\dot { \\Sigma _ { \\mathcal { D } } } ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 156, + 506, + 174 + ], + "score": 1.0, + "content": "exists and Min et al.", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 170, + 223, + 182 + ], + "spans": [ + { + "bbox": [ + 106, + 170, + 223, + 182 + ], + "score": 1.0, + "content": "(2021) for the OPE problem.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 107, + 186, + 505, + 237 + ], + "lines": [ + { + "bbox": [ + 105, + 186, + 505, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 195, + 201 + ], + "score": 1.0, + "content": "Next, for any function", + "type": "text" + }, + { + "bbox": [ + 195, + 187, + 279, + 199 + ], + "score": 0.92, + "content": "V _ { h + 1 } ( \\cdot ) \\in [ 0 , H - h ]", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 186, + 417, + 201 + ], + "score": 1.0, + "content": ", we define the conditional variance", + "type": "text" + }, + { + "bbox": [ + 418, + 187, + 505, + 200 + ], + "score": 0.91, + "content": "\\sigma _ { V _ { h + 1 } } : S \\times \\mathcal { A } \\to \\mathbb { R } _ { + }", + "type": "inline_equation" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 199, + 506, + 212 + ], + "spans": [ + { + "bbox": [ + 104, + 199, + 118, + 212 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 118, + 199, + 307, + 212 + ], + "score": 0.9, + "content": "\\sigma _ { V _ { h + 1 } } ( s , a ) ^ { 2 } : = \\operatorname* { m a x } \\{ 1 , \\operatorname { V a r } _ { P _ { h } } ( V _ { h + 1 } ) ( s , a ) \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 199, + 506, + 212 + ], + "score": 1.0, + "content": ".4 Based on this definition, we can define the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 211, + 506, + 226 + ], + "spans": [ + { + "bbox": [ + 105, + 211, + 322, + 226 + ], + "score": 1.0, + "content": "variance-involved population covariance matrices as:", + "type": "text" + }, + { + "bbox": [ + 331, + 212, + 491, + 225 + ], + "score": 0.79, + "content": "\\mathbf { \\Sigma } _ { h } ^ { p } : = \\mathbb { E } _ { \\mu , h } \\left[ \\sigma _ { V _ { h + 1 } } ( s , a ) ^ { - 2 } \\phi ( s , a ) \\phi ( s , a ) ^ { \\top } \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 491, + 211, + 506, + 226 + ], + "score": 1.0, + "content": ". In", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 223, + 347, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 173, + 237 + ], + "score": 1.0, + "content": "particular, when", + "type": "text" + }, + { + "bbox": [ + 173, + 224, + 211, + 237 + ], + "score": 0.93, + "content": "{ \\cal V } _ { h } = { \\cal V } _ { h } ^ { \\star }", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 223, + 295, + 237 + ], + "score": 1.0, + "content": ", we use the notation", + "type": "text" + }, + { + "bbox": [ + 295, + 224, + 312, + 237 + ], + "score": 0.91, + "content": "\\Lambda _ { h } ^ { \\star p }", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 223, + 347, + 237 + ], + "score": 1.0, + "content": "instead.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10.5 + }, + { + "type": "title", + "bbox": [ + 107, + 253, + 191, + 266 + ], + "lines": [ + { + "bbox": [ + 104, + 252, + 192, + 268 + ], + "spans": [ + { + "bbox": [ + 104, + 252, + 192, + 268 + ], + "score": 1.0, + "content": "3 ALGORITHM", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 106, + 278, + 505, + 330 + ], + "lines": [ + { + "bbox": [ + 105, + 277, + 505, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 505, + 290 + ], + "score": 1.0, + "content": "Least square regression is usually considered as one of the “default” tools for handling problems", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 289, + 505, + 300 + ], + "spans": [ + { + "bbox": [ + 106, + 289, + 505, + 300 + ], + "score": 1.0, + "content": "with linear structures (e.g. LinUCB algorithm for linear Bandits) and finds its popularity in RL as", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 298, + 506, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 506, + 311 + ], + "score": 1.0, + "content": "well since Least-Square Value Iteration (LSVI, Jin et al. (2020)) is shown to be provably efficient for", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 307, + 506, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 207, + 322 + ], + "score": 1.0, + "content": "linear MDPs, due to that", + "type": "text" + }, + { + "bbox": [ + 208, + 309, + 244, + 321 + ], + "score": 0.92, + "content": "V _ { h + 1 } ( s ^ { \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 307, + 356, + 322 + ], + "score": 1.0, + "content": "is an unbiased estimator of", + "type": "text" + }, + { + "bbox": [ + 356, + 309, + 417, + 321 + ], + "score": 0.92, + "content": "[ P _ { h } V _ { h + 1 } ] ( s , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 307, + 506, + 322 + ], + "score": 1.0, + "content": ". 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In offline RL, this has also been leveraged in pessimistic value", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 402, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 106, + 402, + 258, + 414 + ], + "score": 1.0, + "content": "iteration (Jin et al., 2021b) and fitted", + "type": "text" + }, + { + "bbox": [ + 259, + 403, + 267, + 414 + ], + "score": 0.78, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 402, + 505, + 414 + ], + "score": 1.0, + "content": "-evaluation (Duan et al., 2020). Nevertheless, LSVI could", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 414, + 402, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 414, + 402, + 425 + ], + "score": 1.0, + "content": "only yield suboptimal guarantees, as illustrated by the following example.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 106, + 428, + 506, + 494 + ], + "lines": [ + { + "bbox": [ + 106, + 427, + 506, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 427, + 395, + 441 + ], + "score": 1.0, + "content": "Example 3.1. Instantiate PEVI (Theorem 4.4 in Jin et al. (2021b)) with", + "type": "text" + }, + { + "bbox": [ + 395, + 429, + 453, + 441 + ], + "score": 0.91, + "content": "\\phi ( s , a ) = \\mathbf { 1 } _ { s , a }", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 427, + 506, + 441 + ], + "score": 1.0, + "content": "(i.e. tabular", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 439, + 509, + 460 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 138, + 453 + ], + "score": 0.8, + "content": "M D P s ) ^ { 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 439, + 349, + 458 + ], + "score": 1.0, + "content": ", by direct calculation the learning bound has the form", + "type": "text" + }, + { + "bbox": [ + 349, + 441, + 487, + 458 + ], + "score": 0.91, + "content": "\\begin{array} { r l } & { O ( d H \\cdot \\sum _ { \\boldsymbol { h } , \\boldsymbol { s } , \\boldsymbol { a } } { d _ { \\boldsymbol { h } } ^ { \\pi } } ^ { \\star } ( \\boldsymbol { s } , \\boldsymbol { a } ) \\sqrt { \\frac { 1 } { K \\cdot d _ { \\boldsymbol { h } } ^ { \\mu } ( \\boldsymbol { s } , \\boldsymbol { a } ) } } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 441, + 509, + 460 + ], + "score": 1.0, + "content": "q 1K·dµh(s,a) ) and", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 459, + 502, + 482 + ], + "spans": [ + { + "bbox": [ + 104, + 462, + 354, + 480 + ], + "score": 1.0, + "content": "the optimal result (Yin and Wang (2021b) Theorem 4.1) gives", + "type": "text" + }, + { + "bbox": [ + 355, + 459, + 502, + 482 + ], + "score": 0.92, + "content": "\\begin{array} { r l } & { O ( \\sum _ { h , s , a } { d _ { h } ^ { \\pi ^ { \\star } } ( s , a ) \\sqrt { \\frac { \\operatorname { V a r } _ { P _ { s , a } } ( r + V _ { h + 1 } ^ { \\star } ) } { K \\cdot d _ { h } ^ { \\mu } ( s , a ) } } } ) } \\end{array}", + "type": "inline_equation" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 478, + 473, + 496 + ], + "spans": [ + { + "bbox": [ + 104, + 478, + 266, + 496 + ], + "score": 1.0, + "content": "The former has the horizon dependence", + "type": "text" + }, + { + "bbox": [ + 267, + 481, + 281, + 492 + ], + "score": 0.88, + "content": "H ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 478, + 348, + 496 + ], + "score": 1.0, + "content": "and the latter is", + "type": "text" + }, + { + "bbox": [ + 348, + 480, + 371, + 492 + ], + "score": 0.86, + "content": "H ^ { 3 / 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 478, + 473, + 496 + ], + "score": 1.0, + "content": "by law of total variance.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27.5 + }, + { + "type": "text", + "bbox": [ + 106, + 503, + 506, + 614 + ], + "lines": [ + { + "bbox": [ + 106, + 503, + 506, + 516 + ], + "spans": [ + { + "bbox": [ + 106, + 503, + 506, + 516 + ], + "score": 1.0, + "content": "Motivation. By comparing the above two expressions, it can be seen that PEVI cannot get rid of", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 514, + 506, + 527 + ], + "spans": [ + { + "bbox": [ + 106, + 514, + 153, + 527 + ], + "score": 1.0, + "content": "the explicit", + "type": "text" + }, + { + "bbox": [ + 154, + 515, + 164, + 525 + ], + "score": 0.79, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 514, + 352, + 527 + ], + "score": 1.0, + "content": "factor due to missing the variance information", + "type": "text" + }, + { + "bbox": [ + 353, + 515, + 392, + 526 + ], + "score": 0.85, + "content": "( w . r . t ~ V ^ { \\star } )", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 514, + 506, + 527 + ], + "score": 1.0, + "content": ". If we go deeper, one could", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 525, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 506, + 538 + ], + "score": 1.0, + "content": "find that it might not be all that ideal to put equal weights on all the training samples in the least", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 537, + 506, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 506, + 549 + ], + "score": 1.0, + "content": "square objective (2), since, unlike linear regression where the randomness coming from one source", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 546, + 506, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 426, + 560 + ], + "score": 1.0, + "content": "distribution, we are regressing over a sequence of distributions in RL (i.e. each", + "type": "text" + }, + { + "bbox": [ + 426, + 549, + 452, + 559 + ], + "score": 0.87, + "content": "s _ { h } , a _ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 546, + 506, + 560 + ], + "score": 1.0, + "content": "corresponds", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 558, + 506, + 571 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 214, + 571 + ], + "score": 1.0, + "content": "to a different distribution", + "type": "text" + }, + { + "bbox": [ + 214, + 559, + 261, + 570 + ], + "score": 0.91, + "content": "P ( \\cdot | s _ { h } , a _ { h } )", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 558, + 506, + 571 + ], + "score": 1.0, + "content": "and there are possibly infinite many of them). Therefore,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 569, + 506, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 230, + 582 + ], + "score": 1.0, + "content": "conceptually, the sample piece", + "type": "text" + }, + { + "bbox": [ + 231, + 570, + 288, + 581 + ], + "score": 0.88, + "content": "( s _ { h } , a _ { h } , s _ { h + 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 569, + 433, + 582 + ], + "score": 1.0, + "content": "that has higher variance distribution", + "type": "text" + }, + { + "bbox": [ + 434, + 569, + 480, + 581 + ], + "score": 0.93, + "content": "P ( \\cdot | s _ { h } , a _ { h } )", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 569, + 506, + 582 + ], + "score": 1.0, + "content": "tends", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 578, + 507, + 596 + ], + "spans": [ + { + "bbox": [ + 104, + 578, + 242, + 596 + ], + "score": 1.0, + "content": "to be less “reliable” than the one", + "type": "text" + }, + { + "bbox": [ + 242, + 581, + 299, + 594 + ], + "score": 0.9, + "content": "( s _ { h } ^ { \\prime } , a _ { h } ^ { \\prime } , s _ { h + 1 } ^ { \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 578, + 507, + 596 + ], + "score": 1.0, + "content": "with lower variance (hence should not have equal", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 590, + 506, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 506, + 604 + ], + "score": 1.0, + "content": "weight in (2)). This suggests reweighting scheme might help improve the learning guarantee and", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 602, + 419, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 419, + 615 + ], + "score": 1.0, + "content": "reweighting over the variance of the value function stands as a natural choice.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 34.5 + }, + { + "type": "title", + "bbox": [ + 106, + 628, + 351, + 640 + ], + "lines": [ + { + "bbox": [ + 105, + 628, + 352, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 352, + 641 + ], + "score": 1.0, + "content": "3.1 VARIANCE-AWARE PESSIMISTIC VALUE ITERATION", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 107, + 649, + 505, + 698 + ], + "lines": [ + { + "bbox": [ + 106, + 649, + 506, + 661 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 506, + 661 + ], + "score": 1.0, + "content": "Now we explain our framework that incorporates the variance information. Our design is motivated", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 101, + 659, + 507, + 693 + ], + "spans": [ + { + "bbox": [ + 101, + 659, + 358, + 693 + ], + "score": 1.0, + "content": "by previous Zhou et al. 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In general, it suffices to think", + "type": "text" + }, + { + "bbox": [ + 426, + 708, + 501, + 722 + ], + "score": 0.91, + "content": "\\sigma _ { V _ { h + 1 } } ^ { 2 } \\approx \\mathrm { V a r } _ { h } V _ { h + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 705, + 506, + 725 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 20, + "width": 5 + } + ] + }, + { + "bbox": [ + 117, + 718, + 430, + 734 + ], + "spans": [ + { + "bbox": [ + 117, + 718, + 430, + 734 + ], + "score": 1.0, + "content": "5This provides a valid illustration since tabular MDP is a special case of linear MDPs.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "score": 1.0, + "content": "5", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 182 + ], + "lines": [], + "index": 4, + "bbox_fs": [ + 104, + 82, + 507, + 182 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 186, + 505, + 237 + ], + "lines": [ + { + "bbox": [ + 105, + 186, + 505, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 195, + 201 + ], + "score": 1.0, + "content": "Next, for any function", + "type": "text" + }, + { + "bbox": [ + 195, + 187, + 279, + 199 + ], + "score": 0.92, + "content": "V _ { h + 1 } ( \\cdot ) \\in [ 0 , H - h ]", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 186, + 417, + 201 + ], + "score": 1.0, + "content": ", we define the conditional variance", + "type": "text" + }, + { + "bbox": [ + 418, + 187, + 505, + 200 + ], + "score": 0.91, + "content": "\\sigma _ { V _ { h + 1 } } : S \\times \\mathcal { A } \\to \\mathbb { R } _ { + }", + "type": "inline_equation" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 199, + 506, + 212 + ], + "spans": [ + { + "bbox": [ + 104, + 199, + 118, + 212 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 118, + 199, + 307, + 212 + ], + "score": 0.9, + "content": "\\sigma _ { V _ { h + 1 } } ( s , a ) ^ { 2 } : = \\operatorname* { m a x } \\{ 1 , \\operatorname { V a r } _ { P _ { h } } ( V _ { h + 1 } ) ( s , a ) \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 199, + 506, + 212 + ], + "score": 1.0, + "content": ".4 Based on this definition, we can define the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 211, + 506, + 226 + ], + "spans": [ + { + "bbox": [ + 105, + 211, + 322, + 226 + ], + "score": 1.0, + "content": "variance-involved population covariance matrices as:", + "type": "text" + }, + { + "bbox": [ + 331, + 212, + 491, + 225 + ], + "score": 0.79, + "content": "\\mathbf { \\Sigma } _ { h } ^ { p } : = \\mathbb { E } _ { \\mu , h } \\left[ \\sigma _ { V _ { h + 1 } } ( s , a ) ^ { - 2 } \\phi ( s , a ) \\phi ( s , a ) ^ { \\top } \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 491, + 211, + 506, + 226 + ], + "score": 1.0, + "content": ". 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(2020)) is shown to be provably efficient for", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 307, + 506, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 207, + 322 + ], + "score": 1.0, + "content": "linear MDPs, due to that", + "type": "text" + }, + { + "bbox": [ + 208, + 309, + 244, + 321 + ], + "score": 0.92, + "content": "V _ { h + 1 } ( s ^ { \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 307, + 356, + 322 + ], + "score": 1.0, + "content": "is an unbiased estimator of", + "type": "text" + }, + { + "bbox": [ + 356, + 309, + 417, + 321 + ], + "score": 0.92, + "content": "[ P _ { h } V _ { h + 1 } ] ( s , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 307, + 506, + 322 + ], + "score": 1.0, + "content": ". 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(2021b)) with", + "type": "text" + }, + { + "bbox": [ + 395, + 429, + 453, + 441 + ], + "score": 0.91, + "content": "\\phi ( s , a ) = \\mathbf { 1 } _ { s , a }", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 427, + 506, + 441 + ], + "score": 1.0, + "content": "(i.e. tabular", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 439, + 509, + 460 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 138, + 453 + ], + "score": 0.8, + "content": "M D P s ) ^ { 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 439, + 349, + 458 + ], + "score": 1.0, + "content": ", by direct calculation the learning bound has the form", + "type": "text" + }, + { + "bbox": [ + 349, + 441, + 487, + 458 + ], + "score": 0.91, + "content": "\\begin{array} { r l } & { O ( d H \\cdot \\sum _ { \\boldsymbol { h } , \\boldsymbol { s } , \\boldsymbol { a } } { d _ { \\boldsymbol { h } } ^ { \\pi } } ^ { \\star } ( \\boldsymbol { s } , \\boldsymbol { a } ) \\sqrt { \\frac { 1 } { K \\cdot d _ { \\boldsymbol { h } } ^ { \\mu } ( \\boldsymbol { s } , \\boldsymbol { a } ) } } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 441, + 509, + 460 + ], + "score": 1.0, + "content": "q 1K·dµh(s,a) ) and", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 459, + 502, + 482 + ], + "spans": [ + { + "bbox": [ + 104, + 462, + 354, + 480 + ], + "score": 1.0, + "content": "the optimal result (Yin and Wang (2021b) Theorem 4.1) gives", + "type": "text" + }, + { + "bbox": [ + 355, + 459, + 502, + 482 + ], + "score": 0.92, + "content": "\\begin{array} { r l } & { O ( \\sum _ { h , s , a } { d _ { h } ^ { \\pi ^ { \\star } } ( s , a ) \\sqrt { \\frac { \\operatorname { V a r } _ { P _ { s , a } } ( r + V _ { h + 1 } ^ { \\star } ) } { K \\cdot d _ { h } ^ { \\mu } ( s , a ) } } } ) } \\end{array}", + "type": "inline_equation" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 478, + 473, + 496 + ], + "spans": [ + { + "bbox": [ + 104, + 478, + 266, + 496 + ], + "score": 1.0, + "content": "The former has the horizon dependence", + "type": "text" + }, + { + "bbox": [ + 267, + 481, + 281, + 492 + ], + "score": 0.88, + "content": "H ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 478, + 348, + 496 + ], + "score": 1.0, + "content": "and the latter is", + "type": "text" + }, + { + "bbox": [ + 348, + 480, + 371, + 492 + ], + "score": 0.86, + "content": "H ^ { 3 / 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 478, + 473, + 496 + ], + "score": 1.0, + "content": "by law of total variance.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27.5, + "bbox_fs": [ + 104, + 427, + 509, + 496 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 503, + 506, + 614 + ], + "lines": [ + { + "bbox": [ + 106, + 503, + 506, + 516 + ], + "spans": [ + { + "bbox": [ + 106, + 503, + 506, + 516 + ], + "score": 1.0, + "content": "Motivation. 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If we go deeper, one could", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 525, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 506, + 538 + ], + "score": 1.0, + "content": "find that it might not be all that ideal to put equal weights on all the training samples in the least", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 537, + 506, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 506, + 549 + ], + "score": 1.0, + "content": "square objective (2), since, unlike linear regression where the randomness coming from one source", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 546, + 506, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 426, + 560 + ], + "score": 1.0, + "content": "distribution, we are regressing over a sequence of distributions in RL (i.e. each", + "type": "text" + }, + { + "bbox": [ + 426, + 549, + 452, + 559 + ], + "score": 0.87, + "content": "s _ { h } , a _ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 546, + 506, + 560 + ], + "score": 1.0, + "content": "corresponds", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 558, + 506, + 571 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 214, + 571 + ], + "score": 1.0, + "content": "to a different distribution", + "type": "text" + }, + { + "bbox": [ + 214, + 559, + 261, + 570 + ], + "score": 0.91, + "content": "P ( \\cdot | s _ { h } , a _ { h } )", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 558, + 506, + 571 + ], + "score": 1.0, + "content": "and there are possibly infinite many of them). 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This suggests reweighting scheme might help improve the learning guarantee and", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 602, + 419, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 419, + 615 + ], + "score": 1.0, + "content": "reweighting over the variance of the value function stands as a natural choice.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 34.5, + "bbox_fs": [ + 104, + 503, + 507, + 615 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 628, + 351, + 640 + ], + "lines": [ + { + "bbox": [ + 105, + 628, + 352, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 352, + 641 + ], + "score": 1.0, + "content": "3.1 VARIANCE-AWARE PESSIMISTIC VALUE ITERATION", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 107, + 649, + 505, + 698 + ], + "lines": [ + { + "bbox": [ + 106, + 649, + 506, + 661 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 506, + 661 + ], + "score": 1.0, + "content": "Now we explain our framework that incorporates the variance information. Our design is motivated", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 101, + 659, + 507, + 693 + ], + "spans": [ + { + "bbox": [ + 101, + 659, + 358, + 693 + ], + "score": 1.0, + "content": "by previous Zhou et al. 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V _ { h + 1 } ^ { 2 } \\left( \\bar { s } _ { h + 1 } ^ { k } \\right) \\right] ^ { 2 } + \\lambda \\| \\beta \\| _ { 2 } ^ { 2 } = \\bar { \\Sigma } _ { h } ^ { - 1 } \\sum _ { k = 1 } ^ { K } \\phi ( \\bar { s } _ { h } ^ { k } , \\bar { a } _ { h } ^ { k } ) V _ { h + 1 } ^ { 2 } \\left( \\bar { s } _ { h + 1 } ^ { k } \\right)", + "type": "interline_equation", + "image_path": "ce73e95e1738e745e5014aff6a07fdb08e0fb63ba1af129801736b80d1b75d2e.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 121, + 137, + 489, + 147.66666666666666 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 121, + 147.66666666666666, + 489, + 158.33333333333331 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 121, + 158.33333333333331, + 489, + 168.99999999999997 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 170, + 424, + 183 + ], + "lines": [ + { + "bbox": [ + 106, + 170, + 423, + 184 + ], + "spans": [ + { + "bbox": [ + 106, + 170, + 257, + 184 + ], + "score": 1.0, + "content": "and, similarly, the first order moment", + "type": "text" + }, + { + "bbox": [ + 257, + 171, + 335, + 182 + ], + "score": 0.96, + "content": "P _ { h } V _ { h + 1 } : = \\left. \\phi , \\theta _ { h } \\right.", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 170, + 423, + 184 + ], + "score": 1.0, + "content": "can be estimated via:", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "interline_equation", + "bbox": [ + 123, + 183, + 488, + 215 + ], + "lines": [ + { + "bbox": [ + 123, + 183, + 488, + 215 + ], + "spans": [ + { + "bbox": [ + 123, + 183, + 488, + 215 + ], + "score": 0.9, + "content": "\\bar { \\theta } _ { h } = \\underset { \\theta \\in \\mathbb { R } ^ { d } } { \\operatorname { a r g m i n } } \\sum _ { k = 1 } ^ { K } \\left[ \\left. \\phi ( \\bar { s } _ { h } ^ { k } , \\bar { a } _ { h } ^ { k } ) , \\theta \\right. - 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Here", + "type": "text" + }, + { + "bbox": [ + 276, + 248, + 441, + 262 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\bar { \\Sigma } _ { h } = \\sum _ { \\tau = 1 } ^ { K } \\phi ( \\bar { s } _ { h } ^ { \\tau } , \\bar { a } _ { h } ^ { \\tau } ) \\phi ( \\bar { s } _ { h } ^ { \\tau } , \\bar { a } _ { h } ^ { \\tau } ) ^ { \\top } + \\lambda I _ { d } . } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 242, + 448, + 266 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 266, + 505, + 299 + ], + "lines": [ + { + "bbox": [ + 105, + 265, + 507, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 507, + 279 + ], + "score": 1.0, + "content": "Variance-weighted LSVI. 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With variance estimator", + "type": "text" + }, + { + "bbox": [ + 369, + 277, + 380, + 288 + ], + "score": 0.89, + "content": "\\bar { \\widehat { \\sigma } } _ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 276, + 505, + 289 + ], + "score": 1.0, + "content": "at hand, we can modify (2) to", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 288, + 379, + 299 + ], + "spans": [ + { + "bbox": [ + 106, + 288, + 379, + 299 + ], + "score": 1.0, + "content": "bsolve the variance-weighted LSVI instead (Line 10 of Algorithm 1)", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14 + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 300, + 498, + 330 + ], + "lines": [ + { + "bbox": [ + 111, + 300, + 498, + 330 + ], + "spans": [ + { + "bbox": [ + 111, + 300, + 498, + 330 + ], + "score": 0.92, + "content": "\\widehat { w } _ { h } : = \\operatorname { a r g m i n } _ { w \\in \\mathbb { R } ^ { d } } \\lambda \\| w \\| _ { 2 } ^ { 2 } + \\sum _ { k = 1 } ^ { K } \\frac { \\left[ \\langle \\phi ( s _ { h } ^ { k } , a _ { h } ^ { k } ) , w \\rangle - r _ { h } ^ { k } - \\widehat { V } _ { h + 1 } ( s _ { h + 1 } ^ { \\prime k } ) \\right] ^ { 2 } } { \\widehat { \\sigma } _ { h } ^ { 2 } ( s _ { h } ^ { k } , a _ { h } ^ { k } ) } = \\widehat { \\Lambda } _ { h } ^ { - 1 } \\sum _ { k = 1 } ^ { K } \\frac { \\phi \\left( s _ { h } ^ { k } , a _ { h } ^ { k } \\right) \\cdot \\left[ r _ { h } ^ { k } + \\widehat { V } _ { h + 1 } \\left( s _ { h + 1 } ^ { k } \\right) \\right] } { \\widehat { \\sigma } ^ { 2 } ( s _ { h } ^ { k } , a _ { h } ^ { k } ) }", + "type": "interline_equation", + "image_path": "49aa8bea12f8c76be121be4caf0974c4e6d9ea60f56e192cc41a1844db85c256.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 111, + 300, + 498, + 330 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 332, + 505, + 373 + ], + "lines": [ + { + "bbox": [ + 101, + 325, + 509, + 350 + ], + "spans": [ + { + "bbox": [ + 101, + 325, + 133, + 350 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 333, + 345, + 347 + ], + "score": 0.85, + "content": "\\begin{array} { r } { \\widehat { \\Lambda } _ { h } = \\sum _ { k = 1 } ^ { K } \\phi \\big ( s _ { h } ^ { k } , a _ { h } ^ { k } \\big ) \\phi \\big ( s _ { h } ^ { k } , a _ { h } ^ { k } \\big ) ^ { \\top } \\big / \\widehat \\sigma _ { h } ^ { 2 } \\big ( s _ { h } ^ { k } , a _ { h } ^ { k } \\big ) + \\lambda I _ { d } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 325, + 471, + 350 + ], + "score": 1.0, + "content": ". 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Then VAPVI provides the following", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 533, + 325, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 325, + 546 + ], + "score": 1.0, + "content": "result. The complete proof is provided in Appendix C.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5 + }, + { + "type": "text", + "bbox": [ + 106, + 548, + 506, + 596 + ], + "lines": [ + { + "bbox": [ + 106, + 547, + 506, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 184, + 561 + ], + "score": 1.0, + "content": "Theorem 3.2. Let", + "type": "text" + }, + { + "bbox": [ + 185, + 549, + 195, + 559 + ], + "score": 0.61, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 547, + 318, + 561 + ], + "score": 1.0, + "content": "be the number of episodes. 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V _ { h + 1 } ^ { 2 } \\left( \\bar { s } _ { h + 1 } ^ { k } \\right) \\right] ^ { 2 } + \\lambda \\| \\beta \\| _ { 2 } ^ { 2 } = \\bar { \\Sigma } _ { h } ^ { - 1 } \\sum _ { k = 1 } ^ { K } \\phi ( \\bar { s } _ { h } ^ { k } , \\bar { a } _ { h } ^ { k } ) V _ { h + 1 } ^ { 2 } \\left( \\bar { s } _ { h + 1 } ^ { k } \\right)", + "type": "interline_equation", + "image_path": "ce73e95e1738e745e5014aff6a07fdb08e0fb63ba1af129801736b80d1b75d2e.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 121, + 137, + 489, + 147.66666666666666 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 121, + 147.66666666666666, + 489, + 158.33333333333331 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 121, + 158.33333333333331, + 489, + 168.99999999999997 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 170, + 424, + 183 + ], + "lines": [ + { + "bbox": [ + 106, + 170, + 423, + 184 + ], + "spans": [ + { + "bbox": [ + 106, + 170, + 257, + 184 + ], + "score": 1.0, + "content": "and, similarly, the first order moment", + "type": "text" + }, + { + "bbox": [ + 257, + 171, + 335, + 182 + ], + "score": 0.96, + "content": "P _ { h } V _ { h + 1 } : = \\left. \\phi , \\theta _ { h } \\right.", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 170, + 423, + 184 + ], + "score": 1.0, + "content": "can be estimated via:", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6, + "bbox_fs": [ + 106, + 170, + 423, + 184 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 123, + 183, + 488, + 215 + ], + "lines": [ + { + "bbox": [ + 123, + 183, + 488, + 215 + ], + "spans": [ + { + "bbox": [ + 123, + 183, + 488, + 215 + ], + "score": 0.9, + "content": "\\bar { \\theta } _ { h } = \\underset { \\theta \\in \\mathbb { R } ^ { d } } { \\operatorname { a r g m i n } } \\sum _ { k = 1 } ^ { K } \\left[ \\left. \\phi ( \\bar { s } _ { h } ^ { k } , \\bar { a } _ { h } ^ { k } ) , \\theta \\right. - V _ { h + 1 } \\left( \\bar { s } _ { h + 1 } ^ { k } \\right) \\right] ^ { 2 } + \\lambda \\| \\theta \\| _ { 2 } ^ { 2 } = \\bar { \\Sigma } _ { h } ^ { - 1 } \\sum _ { k = 1 } ^ { K } \\phi ( \\bar { s } _ { h } ^ { k } , \\bar { a } _ { h } ^ { k } ) V _ { h + 1 } \\left( \\bar { s } _ { h + 1 } ^ { k } \\right)", + "type": "interline_equation", + "image_path": "5181d298ba4d99bca1c9f238c2a8de43778265b249b79d9b927947b6bd45cf1e.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 123, + 183, + 488, + 193.66666666666666 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 123, + 193.66666666666666, + 488, + 204.33333333333331 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 123, + 204.33333333333331, + 488, + 214.99999999999997 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 218, + 506, + 263 + ], + "lines": [ + { + "bbox": [ + 105, + 218, + 505, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 249, + 233 + ], + "score": 1.0, + "content": "The final estimator is defined as", + "type": "text" + }, + { + "bbox": [ + 249, + 218, + 404, + 233 + ], + "score": 0.91, + "content": "\\widehat { \\sigma } _ { V _ { h } } ^ { 2 } ( \\cdot , \\cdot ) \\ : = \\ \\operatorname* { m a x } \\{ 1 , \\widehat { \\mathrm { V a r } } _ { h } V _ { h + 1 } ( \\cdot , \\cdot ) \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 218, + 429, + 233 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 429, + 218, + 505, + 232 + ], + "score": 0.84, + "content": "\\widehat { \\mathrm { V a r } _ { h } } V _ { h + 1 } ( \\cdot , \\cdot ) \\ =", + "type": "inline_equation" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 227, + 508, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 233, + 315, + 248 + ], + "score": 0.89, + "content": "\\langle \\phi ( \\cdot , \\cdot ) , \\bar { \\beta } _ { h } \\rangle _ { [ 0 , ( H - h + 1 ) ^ { 2 } ] } - \\left[ \\langle \\phi ( \\cdot , \\cdot ) , \\bar { \\theta } _ { h } \\rangle _ { [ 0 , H - h + 1 ] } \\right] ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 227, + 441, + 252 + ], + "score": 1.0, + "content": ".6 In particular, when setting", + "type": "text" + }, + { + "bbox": [ + 442, + 233, + 502, + 248 + ], + "score": 0.92, + "content": "V _ { h + 1 } = \\widehat { V } _ { h + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 227, + 508, + 252 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 101, + 242, + 448, + 266 + ], + "spans": [ + { + "bbox": [ + 101, + 242, + 150, + 266 + ], + "score": 1.0, + "content": "it recovers", + "type": "text" + }, + { + "bbox": [ + 150, + 250, + 162, + 261 + ], + "score": 0.88, + "content": "\\widehat { \\sigma } _ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 242, + 275, + 266 + ], + "score": 1.0, + "content": "in Algorithm 1 line 8. Here", + "type": "text" + }, + { + "bbox": [ + 276, + 248, + 441, + 262 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\bar { \\Sigma } _ { h } = \\sum _ { \\tau = 1 } ^ { K } \\phi ( \\bar { s } _ { h } ^ { \\tau } , \\bar { a } _ { h } ^ { \\tau } ) \\phi ( \\bar { s } _ { h } ^ { \\tau } , \\bar { a } _ { h } ^ { \\tau } ) ^ { \\top } + \\lambda I _ { d } . } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 242, + 448, + 266 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11, + "bbox_fs": [ + 101, + 218, + 508, + 266 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 266, + 505, + 299 + ], + "lines": [ + { + "bbox": [ + 105, + 265, + 507, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 507, + 279 + ], + "score": 1.0, + "content": "Variance-weighted LSVI. The idea of LSVI (2) is based on approximate the Bellman updates:", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 276, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 106, + 277, + 264, + 289 + ], + "score": 0.92, + "content": "{ \\mathcal { T } } _ { h } ( V ) ( s , a ) { \\stackrel { \\smile } { = } } r _ { h } ( s , a ) + ( P _ { h } V ) ( s , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 276, + 368, + 289 + ], + "score": 1.0, + "content": ". 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Then VAPVI provides the following", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 533, + 325, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 325, + 546 + ], + "score": 1.0, + "content": "result. The complete proof is provided in Appendix C.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 520, + 506, + 546 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 548, + 506, + 596 + ], + "lines": [ + { + "bbox": [ + 106, + 547, + 506, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 184, + 561 + ], + "score": 1.0, + "content": "Theorem 3.2. Let", + "type": "text" + }, + { + "bbox": [ + 185, + 549, + 195, + 559 + ], + "score": 0.61, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 547, + 318, + 561 + ], + "score": 1.0, + "content": "be the number of episodes. If", + "type": "text" + }, + { + "bbox": [ + 318, + 548, + 450, + 561 + ], + "score": 0.89, + "content": "K > \\operatorname* { m a x } \\{ \\mathcal { M } _ { 1 } , \\mathcal { M } _ { 2 } , \\mathcal { M } _ { 3 } , \\mathcal { M } _ { 4 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 547, + 469, + 561 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 470, + 547, + 503, + 560 + ], + "score": 0.9, + "content": "\\sqrt { d } > \\xi", + "type": "inline_equation" + }, + { + "bbox": [ + 504, + 547, + 506, + 561 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 134, + 560, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 134, + 560, + 366, + 586 + ], + "score": 0.82, + "content": "\\begin{array} { r } { \\xi : = \\operatorname* { s u p } _ { V \\in [ 0 , H ] , \\ : s ^ { \\prime } \\sim P _ { h } ( s , a ) , \\ : h \\in [ H ] } \\left| \\frac { r _ { h } + V \\left( s ^ { \\prime } \\right) - ( \\mathcal T _ { h } V ) ( s , a ) } { \\sigma _ { V } ( s , a ) } \\right| . } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 564, + 429, + 580 + ], + "score": 1.0, + "content": "Then for any", + "type": "text" + }, + { + "bbox": [ + 429, + 567, + 480, + 577 + ], + "score": 0.88, + "content": "0 < \\lambda < \\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 564, + 505, + 580 + ], + "score": 1.0, + "content": ", with", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 583, + 452, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 153, + 597 + ], + "score": 1.0, + "content": "probability", + "type": "text" + }, + { + "bbox": [ + 153, + 585, + 177, + 595 + ], + "score": 0.6, + "content": "1 - \\delta ,", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 583, + 234, + 597 + ], + "score": 1.0, + "content": ", for all policy", + "type": "text" + }, + { + "bbox": [ + 234, + 587, + 241, + 594 + ], + "score": 0.67, + "content": "\\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 583, + 349, + 597 + ], + "score": 1.0, + "content": "simultaneously, the output", + "type": "text" + }, + { + "bbox": [ + 349, + 585, + 356, + 594 + ], + "score": 0.73, + "content": "\\widehat { \\pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 583, + 452, + 597 + ], + "score": 1.0, + "content": "of Algorithm 1 satisfies", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 547, + 506, + 597 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 187, + 598, + 424, + 628 + ], + "lines": [ + { + "bbox": [ + 187, + 598, + 424, + 628 + ], + "spans": [ + { + "bbox": [ + 187, + 598, + 424, + 628 + ], + "score": 0.92, + "content": "v ^ { \\pi } - v ^ { \\widehat { \\pi } } \\leq \\widetilde { O } \\big ( \\sqrt { d } \\cdot \\sum _ { h = 1 } ^ { H } \\mathbb { E } _ { \\pi } \\bigg [ \\sqrt { \\phi ( \\cdot , \\cdot ) ^ { \\top } \\Lambda _ { h } ^ { - 1 } \\phi ( \\cdot , \\cdot ) } \\bigg ] \\big ) + \\frac { 2 H ^ { 4 } \\sqrt { d } } { K }", + "type": "interline_equation", + "image_path": "c196ec7edcba08a2d1b311144741788cc3c74e1c099af5706c9436f5a54a5cd1.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 187, + 598, + 424, + 628 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 630, + 473, + 655 + ], + "lines": [ + { + "bbox": [ + 133, + 628, + 473, + 655 + ], + "spans": [ + { + "bbox": [ + 133, + 630, + 285, + 655 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\Lambda _ { h } = \\sum _ { k = 1 } ^ { K } \\frac { \\phi ( s _ { h } ^ { k } , a _ { h } ^ { k } ) \\cdot \\phi ( s _ { h } ^ { k } , a _ { h } ^ { k } ) ^ { \\top } } { \\sigma _ { \\hat { V } _ { h + 1 } ( s _ { h } ^ { k } , a _ { h } ^ { k } ) } ^ { 2 } } + \\lambda I _ { d } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 628, + 446, + 646 + ], + "score": 1.0, + "content": ". 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(2021a). In the online regime, Zhou et al. (2021a) is the first result", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 388, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 106, + 389, + 265, + 401 + ], + "score": 1.0, + "content": "that achieves optimal regret rate with", + "type": "text" + }, + { + "bbox": [ + 266, + 388, + 312, + 401 + ], + "score": 0.93, + "content": "O ( d H \\sqrt { T } )", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 389, + 505, + 401 + ], + "score": 1.0, + "content": "in the linear (mixture) MDPs. 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In offline RL, VAPVI only", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 412, + 506, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 222, + 426 + ], + "score": 1.0, + "content": "requires a milder condition", + "type": "text" + }, + { + "bbox": [ + 222, + 412, + 259, + 424 + ], + "score": 0.91, + "content": "\\sqrt { d } > \\xi", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 412, + 318, + 426 + ], + "score": 1.0, + "content": "comparing to", + "type": "text" + }, + { + "bbox": [ + 318, + 413, + 351, + 424 + ], + "score": 0.91, + "content": "d \\ge H", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 412, + 436, + 426 + ], + "score": 1.0, + "content": "(since for any fixed", + "type": "text" + }, + { + "bbox": [ + 437, + 412, + 485, + 425 + ], + "score": 0.93, + "content": "V \\in [ 0 , H ]", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 412, + 506, + 426 + ], + "score": 1.0, + "content": ", the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 102, + 420, + 508, + 443 + ], + "spans": [ + { + "bbox": [ + 102, + 420, + 199, + 443 + ], + "score": 1.0, + "content": "standardized quantity", + "type": "text" + }, + { + "bbox": [ + 199, + 425, + 276, + 442 + ], + "score": 0.93, + "content": "\\frac { r + V ( s ^ { \\prime } ) - ( \\mathcal { T } _ { h } V ) ( s , a ) } { \\sigma _ { V } ( s , a ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 420, + 508, + 443 + ], + "score": 1.0, + "content": "is bounded by constant with high probability, e.g. by", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 439, + 451, + 453 + ], + "spans": [ + { + "bbox": [ + 106, + 439, + 451, + 453 + ], + "score": 1.0, + "content": "chebyshev inequality), which makes our result apply to a wider range of linear MDPs.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 106, + 456, + 506, + 543 + ], + "lines": [ + { + "bbox": [ + 106, + 456, + 506, + 469 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 506, + 469 + ], + "score": 1.0, + "content": "Comparing to Jin et al. (2021b). Jin et al. (2021b) first shows pessimistic value iteration (PEVI)", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 468, + 505, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 459, + 482 + ], + "score": 1.0, + "content": "is provably efficient for Linear MDPs in offline RL. VAPVI improves PEVI over", + "type": "text" + }, + { + "bbox": [ + 459, + 468, + 489, + 481 + ], + "score": 0.92, + "content": "O ( { \\sqrt { d } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 469, + 505, + 482 + ], + "score": 1.0, + "content": "on", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 479, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 104, + 479, + 379, + 496 + ], + "score": 1.0, + "content": "the feature dimension, and improves the horizon dependence as", + "type": "text" + }, + { + "bbox": [ + 379, + 481, + 435, + 495 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\Lambda _ { h } \\ \\succcurlyeq \\ \\frac { 1 } { H ^ { 2 } } \\sum _ { h } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 479, + 471, + 496 + ], + "score": 1.0, + "content": "implies", + "type": "text" + }, + { + "bbox": [ + 472, + 481, + 505, + 494 + ], + "score": 0.91, + "content": "\\Lambda _ { h } ^ { - 1 } \\preccurlyeq", + "type": "inline_equation" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 492, + 507, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 493, + 139, + 507 + ], + "score": 0.92, + "content": "H ^ { 2 } \\Sigma _ { h } ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 492, + 377, + 510 + ], + "score": 1.0, + "content": ". In addition, when instantiate to the tabular case, i.e.", + "type": "text" + }, + { + "bbox": [ + 377, + 495, + 442, + 507 + ], + "score": 0.91, + "content": "\\phi ( s , a ) \\ : = \\ : \\mathbf { 1 } _ { s , a }", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 492, + 507, + 510 + ], + "score": 1.0, + "content": ", VAPVI gives", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 505, + 508, + 531 + ], + "spans": [ + { + "bbox": [ + 106, + 507, + 268, + 531 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { O ( \\sqrt { d } \\sum _ { h , s , a } { d _ { h } ^ { \\pi } } ^ { \\star } ( s , a ) \\sqrt { \\frac { \\operatorname { V a r } _ { P _ { s , a } } ( r + V _ { h + 1 } ^ { \\star } ) } { K \\cdot d _ { h } ^ { \\mu } ( s , a ) } } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 505, + 508, + 529 + ], + "score": 1.0, + "content": "r VarPs,a (r+V ?h+1)K·dµ(s,a) ), which enjoys O(√H) improvement over PEVI (recall", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 529, + 470, + 543 + ], + "spans": [ + { + "bbox": [ + 106, + 529, + 218, + 543 + ], + "score": 1.0, + "content": "Example 3.1) and the order", + "type": "text" + }, + { + "bbox": [ + 218, + 530, + 256, + 543 + ], + "score": 0.9, + "content": "{ \\cal O } ( H ^ { 3 / 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 529, + 470, + 543 + ], + "score": 1.0, + "content": "is tight (check Section G for the detailed derivation).", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28.5 + }, + { + "type": "text", + "bbox": [ + 106, + 547, + 506, + 637 + ], + "lines": [ + { + "bbox": [ + 106, + 547, + 505, + 560 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 505, + 560 + ], + "score": 1.0, + "content": "Comparing to Xie et al. (2021a). Their linear MDP guarantee in Theorem 3.2. enjoys the same", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 558, + 506, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 506, + 570 + ], + "score": 1.0, + "content": "rate as VAPVI in feature dimension but the horizon dependence is essentially the same as Jin et al.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 568, + 506, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 200, + 584 + ], + "score": 1.0, + "content": "(2021b) (by translating", + "type": "text" + }, + { + "bbox": [ + 200, + 569, + 257, + 583 + ], + "score": 0.93, + "content": "\\begin{array} { r } { H \\approx O \\bigl ( \\frac { 1 } { 1 - \\gamma } \\bigr ) , } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 568, + 506, + 584 + ], + "score": 1.0, + "content": ") therefore is not optimal. The general function approximation", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 582, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 506, + 595 + ], + "score": 1.0, + "content": "scheme in Xie et al. (2021a) provides elegant characterizations for on-support error and off-support", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 593, + 506, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 506, + 604 + ], + "score": 1.0, + "content": "error, but the algorithmic framework is information-theoretical only (and the practical version PSPI", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 603, + 506, + 617 + ], + "spans": [ + { + "bbox": [ + 104, + 603, + 506, + 617 + ], + "score": 1.0, + "content": "will not yield the same learning guarantee). Also, due to the use finite function class and policy class,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 615, + 505, + 627 + ], + "spans": [ + { + "bbox": [ + 106, + 615, + 505, + 627 + ], + "score": 1.0, + "content": "the reduction to linear MDP only works with finite action space. As a comparison, VAPVI has no", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 626, + 218, + 638 + ], + "spans": [ + { + "bbox": [ + 106, + 626, + 218, + 638 + ], + "score": 1.0, + "content": "constraints on any of these.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 35.5 + }, + { + "type": "text", + "bbox": [ + 106, + 642, + 505, + 709 + ], + "lines": [ + { + "bbox": [ + 106, + 643, + 505, + 654 + ], + "spans": [ + { + "bbox": [ + 106, + 643, + 505, + 654 + ], + "score": 1.0, + "content": "Comparing to Zanette et al. (2021). Concurrently, Zanette et al. (2021) considers offline RL with", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 653, + 505, + 666 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 505, + 666 + ], + "score": 1.0, + "content": "the linear Bellman complete model, which is more general than linear MDPs and, with the assumption", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 664, + 506, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 665, + 139, + 676 + ], + "score": 0.91, + "content": "Q ^ { \\pi } \\leq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 664, + 506, + 677 + ], + "score": 1.0, + "content": ", their PACLE algorithm provides near-minimax optimal guarantee in this setting. However,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 675, + 506, + 688 + ], + "spans": [ + { + "bbox": [ + 105, + 675, + 262, + 688 + ], + "score": 1.0, + "content": "when recovering to the standard setting", + "type": "text" + }, + { + "bbox": [ + 263, + 676, + 313, + 688 + ], + "score": 0.93, + "content": "Q ^ { \\pi } \\in [ 0 , H ]", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 675, + 435, + 688 + ], + "score": 1.0, + "content": ", their bound will rescale by an", + "type": "text" + }, + { + "bbox": [ + 436, + 676, + 446, + 686 + ], + "score": 0.76, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 675, + 506, + 688 + ], + "score": 1.0, + "content": "factor,7 which", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 686, + 506, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 686, + 468, + 699 + ], + "score": 1.0, + "content": "could be suboptimal due to the variance-unawareness. The reason behind this is: when", + "type": "text" + }, + { + "bbox": [ + 468, + 687, + 502, + 698 + ], + "score": 0.91, + "content": "Q ^ { \\pi } \\leq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 686, + 506, + 699 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 697, + 506, + 710 + ], + "spans": [ + { + "bbox": [ + 105, + 697, + 390, + 710 + ], + "score": 1.0, + "content": "lack of variance information encoding will not matter, since in this case", + "type": "text" + }, + { + "bbox": [ + 390, + 698, + 453, + 710 + ], + "score": 0.92, + "content": "\\mathrm { V a r } _ { P } ( V ^ { \\pi } ) \\leq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 697, + 506, + 710 + ], + "score": 1.0, + "content": "has constant", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 42.5 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 120, + 721, + 244, + 732 + ], + "lines": [ + { + "bbox": [ + 119, + 719, + 245, + 734 + ], + "spans": [ + { + "bbox": [ + 119, + 719, + 245, + 734 + ], + "score": 1.0, + "content": "7Check their Footnote 2 in Page 9.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 106, + 82, + 374, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 374, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 374, + 96 + ], + "score": 1.0, + "content": "Algorithm 1 Variance-Aware Pessimistic Value Iteration (VAPVI)", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "index", + "bbox": [ + 107, + 95, + 487, + 326 + ], + "lines": [ + { + "bbox": [ + 105, + 91, + 457, + 114 + ], + "spans": [ + { + "bbox": [ + 105, + 91, + 177, + 114 + ], + "score": 1.0, + "content": "1: Input: Dataset", + "type": "text" + }, + { + "bbox": [ + 178, + 96, + 369, + 110 + ], + "score": 0.87, + "content": "\\begin{array} { r } { \\mathcal { D } = \\{ \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } , r _ { h } ^ { \\tau } \\right) \\} _ { \\tau , h = 1 } ^ { K , H } \\mathcal { D } ^ { \\prime } = \\{ \\left( \\bar { s } _ { h } ^ { \\tau } , \\bar { a } _ { h } ^ { \\tau } , \\bar { r } _ { h } ^ { \\tau } \\right) \\} _ { \\tau , h = 1 } ^ { K , H } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 91, + 442, + 114 + ], + "score": 1.0, + "content": ". 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h + 1 ) ^ { 2 } \\right] } - \\left[ \\left. \\phi ( \\cdot , \\cdot ) , \\bar { \\theta } _ { h } \\right. _ { \\left[ 0 , H - h + 1 \\right] } \\right] ^ { 2 } } \\end{array}", + "type": "inline_equation" + } + ], + "index": 8, + "is_list_start_line": true + }, + { + "bbox": [ + 109, + 193, + 286, + 209 + ], + "spans": [ + { + "bbox": [ + 109, + 196, + 120, + 207 + ], + "score": 1.0, + "content": "9:", + "type": "text" + }, + { + "bbox": [ + 129, + 193, + 145, + 209 + ], + "score": 1.0, + "content": "Set", + "type": "text" + }, + { + "bbox": [ + 145, + 195, + 286, + 208 + ], + "score": 0.69, + "content": "\\widehat { \\sigma } _ { h } ( \\cdot , \\cdot ) ^ { 2 } \\gets \\operatorname* { m a x } \\{ 1 , \\widehat { \\mathrm { V a r } } _ { P _ { h } } \\widehat { V } _ { h + 1 } ( \\cdot , \\cdot ) \\}", + "type": "inline_equation" + } + ], + "index": 9, + "is_list_start_line": true + }, + { + "bbox": [ + 106, + 206, + 394, + 218 + ], + "spans": [ + { + "bbox": [ + 106, + 206, + 121, + 218 + ], + "score": 1.0, + "content": "10:", + "type": "text" + }, + { + "bbox": [ + 131, + 208, + 138, + 216 + ], + "score": 0.69, + "content": "\\diamond", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 206, + 394, + 218 + ], + "score": 1.0, + "content": "bPhase2: Weighted Least-square Value Iteration for pessimistic updates", + "type": "text" + } + ], + "index": 10, + "is_list_start_line": true + }, + { + "bbox": [ + 106, + 216, + 485, + 306 + ], + "spans": [ + { + "bbox": [ + 106, + 218, + 120, + 230 + ], + "score": 1.0, + "content": "11:", + "type": "text" + }, + { + "bbox": [ + 120, + 216, + 485, + 306 + ], + "score": 0.62, + "content": "\\begin{array} { r l } & { \\mathrm { S e t } \\widehat { \\Lambda } _ { h } \\gets \\sum _ { \\tau = 1 } ^ { \\overline { { K } } } \\phi ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) ^ { \\widehat { \\circ } } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) ^ { \\top } / \\widehat { \\sigma } ^ { 2 } \\big ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\big ) ^ { \\widehat { \\mathbf { \\phi } } } + \\lambda \\cdot I , } \\\\ & { \\mathrm { S e t } \\widehat { w } _ { h } \\gets \\widehat { \\Lambda } _ { h } ^ { - 1 } ( \\sum _ { \\tau = 1 } ^ { K } \\phi ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) \\cdot ( r _ { h } ^ { \\tau } + \\widehat { V } _ { h + 1 } ( s _ { h + 1 } ^ { \\tau } ) ) / \\widehat { \\sigma } ^ { 2 } \\big ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\big ) ) } \\\\ & { \\mathrm { S e t } \\Gamma _ { h } ( \\cdot , \\cdot ) \\gets C \\sqrt { d } \\cdot ( \\phi ( \\cdot , \\cdot ) ^ { \\top } \\widehat { \\Lambda } _ { h } ^ { - 1 } \\phi ( \\cdot , \\cdot ) ) ^ { 1 / 2 } + \\frac { 2 H ^ { 3 } \\sqrt { d } } { K } \\qquad \\mathrm { ( U s e ~ \\Gamma } _ { h } ^ { I } \\mathrm { ~ f o r ~ t h e ~ i m p r o v e d ~ v e r s i o n ) } } \\\\ & { \\mathrm { S e t } \\widehat { \\bar { Q } } _ { h } ( \\cdot , \\cdot ) \\gets \\phi \\big ( \\cdot , \\cdot \\big ) ^ { \\top } \\widehat { w } _ { h } - \\Gamma _ { h } ( \\cdot , \\cdot ) } \\\\ & { \\mathrm { S e t } \\widehat { \\widehat { \\hat { Q } } } _ { h } ( \\cdot , \\cdot ) \\gets \\operatorname* { m i n } \\{ \\widehat { \\bar { Q } } _ { h } ( \\cdot , \\cdot ) , H - h + 1 \\} ^ { + } } \\\\ & \\mathrm { S e t } \\widehat { \\hat { \\mathcal { R } } _ { h } } ( \\cdot , \\cdot ) \\gets \\operatorname* { a r g m a x } _ { \\pi _ { h } } \\widehat { Q } _ { h } ( \\cdot , \\cdot ) , \\pi _ { h } ( \\cdot \\cdot ) _ { A } , \\widehat { V } _ { h } ( \\cdot ) \\gets \\operatorname* { m a x } _ { \\pi _ { h } } \\widehat { Q } _ { h } ( \\cdot , \\cdot ) , \\pi _ { h } \\end{array}", + "type": "inline_equation", + "image_path": "9c20d8fe91130b2bf91275123554e2d89798c0fc179a56098ef9b796ba067d2a.jpg" + } + ], + "index": 15, + "is_list_start_line": true + }, + { + "bbox": [ + 106, + 231, + 120, + 245 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 120, + 245 + ], + "score": 1.0, + "content": "12:", + "type": "text" + } + ], + "index": 11, + "is_list_start_line": true + }, + { + "bbox": [ + 106, + 251, + 120, + 263 + ], + "spans": [ + { + "bbox": [ + 106, + 251, + 120, + 263 + ], + "score": 1.0, + "content": "13:", + "type": "text" + } + ], + "index": 12, + "is_list_start_line": true + }, + { + "bbox": [ + 106, + 264, + 120, + 278 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 120, + 278 + ], + "score": 1.0, + "content": "14:", + "type": "text" + } + ], + "index": 13, + "is_list_start_line": true + }, + { + "bbox": [ + 106, + 278, + 120, + 290 + ], + "spans": [ + { + "bbox": [ + 106, + 278, + 120, + 290 + ], + "score": 1.0, + "content": "15:", + "type": "text" + } + ], + "index": 14, + "is_list_start_line": true + }, + { + "bbox": [ + 106, + 291, + 120, + 302 + ], + "spans": [ + { + "bbox": [ + 106, + 291, + 120, + 302 + ], + "score": 1.0, + "content": "16:", + "type": "text" + } + ], + "index": 16, + "is_list_start_line": true + }, + { + "bbox": [ + 102, + 301, + 198, + 330 + ], + "spans": [ + { + "bbox": [ + 102, + 301, + 156, + 330 + ], + "score": 1.0, + "content": "17: end for18: Output:", + "type": "text" + }, + { + "bbox": [ + 156, + 310, + 190, + 324 + ], + "score": 0.88, + "content": "\\left\\{ \\widehat { \\pi } _ { h } \\right\\} _ { h = 1 } ^ { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 301, + 198, + 330 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 17, + "is_list_start_line": true + } + ], + "index": 9, + "bbox_fs": [ + 102, + 91, + 485, + 330 + ] + }, + { + "type": "list", + "bbox": [ + 106, + 349, + 504, + 372 + ], + "lines": [ + { + "bbox": [ + 106, + 349, + 506, + 362 + ], + "spans": [ + { + "bbox": [ + 106, + 349, + 506, + 362 + ], + "score": 1.0, + "content": "Theorem 3.2 provides improvements over the existing best-known results and we now explain it.", + "type": "text" + } + ], + "index": 18, + "is_list_end_line": true + }, + { + "bbox": [ + 106, + 360, + 376, + 372 + ], + "spans": [ + { + "bbox": [ + 106, + 360, + 376, + 372 + ], + "score": 1.0, + "content": "However, before that, we first discuss about our theorem condition.", + "type": "text" + } + ], + "index": 19, + "is_list_start_line": true, + "is_list_end_line": true + } + ], + "index": 18.5, + "bbox_fs": [ + 106, + 349, + 506, + 372 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 377, + 505, + 452 + ], + "lines": [ + { + "bbox": [ + 106, + 376, + 506, + 390 + ], + "spans": [ + { + "bbox": [ + 106, + 376, + 506, + 390 + ], + "score": 1.0, + "content": "Comparing to Zhou et al. (2021a). In the online regime, Zhou et al. (2021a) is the first result", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 388, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 106, + 389, + 265, + 401 + ], + "score": 1.0, + "content": "that achieves optimal regret rate with", + "type": "text" + }, + { + "bbox": [ + 266, + 388, + 312, + 401 + ], + "score": 0.93, + "content": "O ( d H \\sqrt { T } )", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 389, + 505, + 401 + ], + "score": 1.0, + "content": "in the linear (mixture) MDPs. However, this", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 401, + 505, + 413 + ], + "spans": [ + { + "bbox": [ + 106, + 401, + 223, + 413 + ], + "score": 1.0, + "content": "result requires the condition", + "type": "text" + }, + { + "bbox": [ + 224, + 401, + 253, + 411 + ], + "score": 0.9, + "content": "d \\geq H", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 401, + 505, + 413 + ], + "score": 1.0, + "content": "(their Theorem 6 and Remark 7). In offline RL, VAPVI only", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 412, + 506, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 222, + 426 + ], + "score": 1.0, + "content": "requires a milder condition", + "type": "text" + }, + { + "bbox": [ + 222, + 412, + 259, + 424 + ], + "score": 0.91, + "content": "\\sqrt { d } > \\xi", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 412, + 318, + 426 + ], + "score": 1.0, + "content": "comparing to", + "type": "text" + }, + { + "bbox": [ + 318, + 413, + 351, + 424 + ], + "score": 0.91, + "content": "d \\ge H", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 412, + 436, + 426 + ], + "score": 1.0, + "content": "(since for any fixed", + "type": "text" + }, + { + "bbox": [ + 437, + 412, + 485, + 425 + ], + "score": 0.93, + "content": "V \\in [ 0 , H ]", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 412, + 506, + 426 + ], + "score": 1.0, + "content": ", the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 102, + 420, + 508, + 443 + ], + "spans": [ + { + "bbox": [ + 102, + 420, + 199, + 443 + ], + "score": 1.0, + "content": "standardized quantity", + "type": "text" + }, + { + "bbox": [ + 199, + 425, + 276, + 442 + ], + "score": 0.93, + "content": "\\frac { r + V ( s ^ { \\prime } ) - ( \\mathcal { T } _ { h } V ) ( s , a ) } { \\sigma _ { V } ( s , a ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 420, + 508, + 443 + ], + "score": 1.0, + "content": "is bounded by constant with high probability, e.g. by", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 439, + 451, + 453 + ], + "spans": [ + { + "bbox": [ + 106, + 439, + 451, + 453 + ], + "score": 1.0, + "content": "chebyshev inequality), which makes our result apply to a wider range of linear MDPs.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 22.5, + "bbox_fs": [ + 102, + 376, + 508, + 453 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 456, + 506, + 543 + ], + "lines": [ + { + "bbox": [ + 106, + 456, + 506, + 469 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 506, + 469 + ], + "score": 1.0, + "content": "Comparing to Jin et al. (2021b). Jin et al. (2021b) first shows pessimistic value iteration (PEVI)", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 468, + 505, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 459, + 482 + ], + "score": 1.0, + "content": "is provably efficient for Linear MDPs in offline RL. VAPVI improves PEVI over", + "type": "text" + }, + { + "bbox": [ + 459, + 468, + 489, + 481 + ], + "score": 0.92, + "content": "O ( { \\sqrt { d } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 469, + 505, + 482 + ], + "score": 1.0, + "content": "on", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 479, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 104, + 479, + 379, + 496 + ], + "score": 1.0, + "content": "the feature dimension, and improves the horizon dependence as", + "type": "text" + }, + { + "bbox": [ + 379, + 481, + 435, + 495 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\Lambda _ { h } \\ \\succcurlyeq \\ \\frac { 1 } { H ^ { 2 } } \\sum _ { h } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 479, + 471, + 496 + ], + "score": 1.0, + "content": "implies", + "type": "text" + }, + { + "bbox": [ + 472, + 481, + 505, + 494 + ], + "score": 0.91, + "content": "\\Lambda _ { h } ^ { - 1 } \\preccurlyeq", + "type": "inline_equation" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 492, + 507, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 493, + 139, + 507 + ], + "score": 0.92, + "content": "H ^ { 2 } \\Sigma _ { h } ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 492, + 377, + 510 + ], + "score": 1.0, + "content": ". In addition, when instantiate to the tabular case, i.e.", + "type": "text" + }, + { + "bbox": [ + 377, + 495, + 442, + 507 + ], + "score": 0.91, + "content": "\\phi ( s , a ) \\ : = \\ : \\mathbf { 1 } _ { s , a }", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 492, + 507, + 510 + ], + "score": 1.0, + "content": ", VAPVI gives", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 505, + 508, + 531 + ], + "spans": [ + { + "bbox": [ + 106, + 507, + 268, + 531 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { O ( \\sqrt { d } \\sum _ { h , s , a } { d _ { h } ^ { \\pi } } ^ { \\star } ( s , a ) \\sqrt { \\frac { \\operatorname { V a r } _ { P _ { s , a } } ( r + V _ { h + 1 } ^ { \\star } ) } { K \\cdot d _ { h } ^ { \\mu } ( s , a ) } } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 505, + 508, + 529 + ], + "score": 1.0, + "content": "r VarPs,a (r+V ?h+1)K·dµ(s,a) ), which enjoys O(√H) improvement over PEVI (recall", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 529, + 470, + 543 + ], + "spans": [ + { + "bbox": [ + 106, + 529, + 218, + 543 + ], + "score": 1.0, + "content": "Example 3.1) and the order", + "type": "text" + }, + { + "bbox": [ + 218, + 530, + 256, + 543 + ], + "score": 0.9, + "content": "{ \\cal O } ( H ^ { 3 / 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 529, + 470, + 543 + ], + "score": 1.0, + "content": "is tight (check Section G for the detailed derivation).", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28.5, + "bbox_fs": [ + 104, + 456, + 508, + 543 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 547, + 506, + 637 + ], + "lines": [ + { + "bbox": [ + 106, + 547, + 505, + 560 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 505, + 560 + ], + "score": 1.0, + "content": "Comparing to Xie et al. (2021a). Their linear MDP guarantee in Theorem 3.2. enjoys the same", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 558, + 506, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 506, + 570 + ], + "score": 1.0, + "content": "rate as VAPVI in feature dimension but the horizon dependence is essentially the same as Jin et al.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 568, + 506, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 200, + 584 + ], + "score": 1.0, + "content": "(2021b) (by translating", + "type": "text" + }, + { + "bbox": [ + 200, + 569, + 257, + 583 + ], + "score": 0.93, + "content": "\\begin{array} { r } { H \\approx O \\bigl ( \\frac { 1 } { 1 - \\gamma } \\bigr ) , } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 568, + 506, + 584 + ], + "score": 1.0, + "content": ") therefore is not optimal. The general function approximation", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 582, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 506, + 595 + ], + "score": 1.0, + "content": "scheme in Xie et al. (2021a) provides elegant characterizations for on-support error and off-support", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 593, + 506, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 506, + 604 + ], + "score": 1.0, + "content": "error, but the algorithmic framework is information-theoretical only (and the practical version PSPI", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 603, + 506, + 617 + ], + "spans": [ + { + "bbox": [ + 104, + 603, + 506, + 617 + ], + "score": 1.0, + "content": "will not yield the same learning guarantee). Also, due to the use finite function class and policy class,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 615, + 505, + 627 + ], + "spans": [ + { + "bbox": [ + 106, + 615, + 505, + 627 + ], + "score": 1.0, + "content": "the reduction to linear MDP only works with finite action space. As a comparison, VAPVI has no", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 626, + 218, + 638 + ], + "spans": [ + { + "bbox": [ + 106, + 626, + 218, + 638 + ], + "score": 1.0, + "content": "constraints on any of these.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 35.5, + "bbox_fs": [ + 104, + 547, + 506, + 638 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 642, + 505, + 709 + ], + "lines": [ + { + "bbox": [ + 106, + 643, + 505, + 654 + ], + "spans": [ + { + "bbox": [ + 106, + 643, + 505, + 654 + ], + "score": 1.0, + "content": "Comparing to Zanette et al. (2021). Concurrently, Zanette et al. (2021) considers offline RL with", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 653, + 505, + 666 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 505, + 666 + ], + "score": 1.0, + "content": "the linear Bellman complete model, which is more general than linear MDPs and, with the assumption", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 664, + 506, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 665, + 139, + 676 + ], + "score": 0.91, + "content": "Q ^ { \\pi } \\leq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 664, + 506, + 677 + ], + "score": 1.0, + "content": ", their PACLE algorithm provides near-minimax optimal guarantee in this setting. However,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 675, + 506, + 688 + ], + "spans": [ + { + "bbox": [ + 105, + 675, + 262, + 688 + ], + "score": 1.0, + "content": "when recovering to the standard setting", + "type": "text" + }, + { + "bbox": [ + 263, + 676, + 313, + 688 + ], + "score": 0.93, + "content": "Q ^ { \\pi } \\in [ 0 , H ]", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 675, + 435, + 688 + ], + "score": 1.0, + "content": ", their bound will rescale by an", + "type": "text" + }, + { + "bbox": [ + 436, + 676, + 446, + 686 + ], + "score": 0.76, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 675, + 506, + 688 + ], + "score": 1.0, + "content": "factor,7 which", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 686, + 506, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 686, + 468, + 699 + ], + "score": 1.0, + "content": "could be suboptimal due to the variance-unawareness. The reason behind this is: when", + "type": "text" + }, + { + "bbox": [ + 468, + 687, + 502, + 698 + ], + "score": 0.91, + "content": "Q ^ { \\pi } \\leq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 686, + 506, + 699 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 697, + 506, + 710 + ], + "spans": [ + { + "bbox": [ + 105, + 697, + 390, + 710 + ], + "score": 1.0, + "content": "lack of variance information encoding will not matter, since in this case", + "type": "text" + }, + { + "bbox": [ + 390, + 698, + 453, + 710 + ], + "score": 0.92, + "content": "\\mathrm { V a r } _ { P } ( V ^ { \\pi } ) \\leq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 697, + 506, + 710 + ], + "score": 1.0, + "content": "has constant", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 81, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 330, + 95 + ], + "score": 1.0, + "content": "order (therefore will not affect the optimal rate); when", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 330, + 82, + 381, + 95 + ], + "score": 0.89, + "content": "Q ^ { \\pi } \\in [ 0 , H ]", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 381, + 81, + 385, + 95 + ], + "score": 1.0, + "content": ",", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 385, + 82, + 430, + 95 + ], + "score": 0.9, + "content": "\\operatorname { V a r } _ { P } ( V ^ { \\pi } )", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 430, + 81, + 505, + 95 + ], + "score": 1.0, + "content": "can be as large as", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 107, + 91, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 107, + 93, + 121, + 104 + ], + "score": 0.87, + "content": "H ^ { 2 }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 121, + 91, + 506, + 107 + ], + "score": 1.0, + "content": ", effectively leveraging the variance information can help improve the sample efficiency, e.g. via", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "law of total variances, just like VAPVI does. On the other hand, their guarantee also requires finite", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "action space, due to the mirror descent style analysis. Nevertheless, we do point out Zanette et al.", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 338, + 140 + ], + "score": 1.0, + "content": "(2021) has improved state-action measure than VAPVI, as", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 339, + 126, + 492, + 139 + ], + "score": 0.91, + "content": "\\| E _ { \\pi } [ \\phi ( \\cdot , \\cdot ) ] \\| _ { M ^ { - 1 } } \\leq \\mathbb { E } _ { \\pi } [ \\| \\phi ( \\cdot , \\cdot ) \\| _ { M ^ { - 1 } } ]", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 492, + 126, + 506, + 140 + ], + "score": 1.0, + "content": "by", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 136, + 465, + 151 + ], + "spans": [ + { + "bbox": [ + 104, + 136, + 242, + 151 + ], + "score": 1.0, + "content": "Jensen’s inequality and that norm", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 243, + 137, + 274, + 150 + ], + "score": 0.92, + "content": "\\lVert \\cdot \\rVert _ { M ^ { - 1 } }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 274, + 136, + 448, + 151 + ], + "score": 1.0, + "content": "is convex for some positive-definite matrix", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 448, + 138, + 460, + 147 + ], + "score": 0.84, + "content": "M", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 460, + 136, + 465, + 151 + ], + "score": 1.0, + "content": ".", + "type": "text", + "cross_page": true + } + ], + "index": 5 + } + ], + "index": 42.5, + "bbox_fs": [ + 105, + 643, + 506, + 710 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 505, + 150 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 330, + 95 + ], + "score": 1.0, + "content": "order (therefore will not affect the optimal rate); when", + "type": "text" + }, + { + "bbox": [ + 330, + 82, + 381, + 95 + ], + "score": 0.89, + "content": "Q ^ { \\pi } \\in [ 0 , H ]", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 81, + 385, + 95 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 385, + 82, + 430, + 95 + ], + "score": 0.9, + "content": "\\operatorname { V a r } _ { P } ( V ^ { \\pi } )", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 81, + 505, + 95 + ], + "score": 1.0, + "content": "can be as large as", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 107, + 91, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 107, + 93, + 121, + 104 + ], + "score": 0.87, + "content": "H ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 121, + 91, + 506, + 107 + ], + "score": 1.0, + "content": ", effectively leveraging the variance information can help improve the sample efficiency, e.g. via", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "law of total variances, just like VAPVI does. 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Comparing to existing works, one major", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 102, + 163, + 508, + 188 + ], + "spans": [ + { + "bbox": [ + 102, + 163, + 305, + 188 + ], + "score": 1.0, + "content": "improvement is that the main term for VAPVI", + "type": "text" + }, + { + "bbox": [ + 305, + 165, + 458, + 185 + ], + "score": 0.94, + "content": "\\begin{array} { r l } { { \\sqrt { d } \\sum _ { h = 1 } ^ { H } \\mathbb { E } _ { \\pi ^ { \\star } } \\bigl [ \\sqrt { \\phi ( \\cdot , \\cdot ) ^ { \\top } \\Lambda _ { h } ^ { \\star - 1 } \\phi ( \\cdot , \\cdot ) } \\bigr ] } \\quad } & { { } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 163, + 508, + 188 + ], + "score": 1.0, + "content": "admits no", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 183, + 506, + 196 + ], + "spans": [ + { + "bbox": [ + 106, + 183, + 200, + 196 + ], + "score": 1.0, + "content": "explicit dependence on", + "type": "text" + }, + { + "bbox": [ + 200, + 184, + 210, + 194 + ], + "score": 0.76, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 183, + 506, + 196 + ], + "score": 1.0, + "content": ", which provides a more adaptive/instance-dependent characterization. For", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 192, + 507, + 210 + ], + "spans": [ + { + "bbox": [ + 104, + 192, + 331, + 210 + ], + "score": 1.0, + "content": "instance, if we ignore the technical treatment by taking", + "type": "text" + }, + { + "bbox": [ + 332, + 195, + 357, + 205 + ], + "score": 0.9, + "content": "\\lambda = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 192, + 375, + 210 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 376, + 194, + 452, + 207 + ], + "score": 0.91, + "content": "\\sigma _ { h } ^ { \\star } \\widetilde { \\approx } \\mathrm { V a r } _ { P } ( V _ { h + 1 } ^ { \\star } )", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 192, + 507, + 210 + ], + "score": 1.0, + "content": ", then for the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 205, + 505, + 218 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 301, + 218 + ], + "score": 1.0, + "content": "partially deterministic systems (where there are", + "type": "text" + }, + { + "bbox": [ + 302, + 207, + 307, + 216 + ], + "score": 0.78, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 205, + 348, + 218 + ], + "score": 1.0, + "content": "stochastic", + "type": "text" + }, + { + "bbox": [ + 349, + 206, + 361, + 217 + ], + "score": 0.87, + "content": "P _ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 205, + 385, + 218 + ], + "score": 1.0, + "content": "’s and", + "type": "text" + }, + { + "bbox": [ + 385, + 207, + 410, + 216 + ], + "score": 0.88, + "content": "H - t", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 205, + 464, + 218 + ], + "score": 1.0, + "content": "deterministic", + "type": "text" + }, + { + "bbox": [ + 465, + 206, + 477, + 217 + ], + "score": 0.86, + "content": "P _ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 205, + 505, + 218 + ], + "score": 1.0, + "content": "’s), the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 217, + 504, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 203, + 236 + ], + "score": 1.0, + "content": "main term diminishes to", + "type": "text" + }, + { + "bbox": [ + 203, + 217, + 354, + 236 + ], + "score": 0.94, + "content": "\\begin{array} { r l } { \\sqrt { d } \\sum _ { i = 1 } ^ { t } \\mathbb { E } _ { \\pi ^ { \\star } } \\left[ \\sqrt { \\phi ( \\cdot , \\cdot ) ^ { \\top } \\Lambda _ { h _ { i } } ^ { \\star - 1 } \\phi ( \\cdot , \\cdot ) } \\right] } & { { } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 219, + 375, + 236 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 375, + 220, + 447, + 233 + ], + "score": 0.88, + "content": "h _ { i } \\in \\{ h : ~ s . t . ~ P _ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 219, + 498, + 236 + ], + "score": 1.0, + "content": "is stochastic", + "type": "text" + }, + { + "bbox": [ + 498, + 221, + 504, + 233 + ], + "score": 0.34, + "content": "\\}", + "type": "inline_equation" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 233, + 506, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 233, + 274, + 248 + ], + "score": 1.0, + "content": "and can be a much smaller quantity when", + "type": "text" + }, + { + "bbox": [ + 275, + 236, + 304, + 245 + ], + "score": 0.87, + "content": "t \\ll H", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 233, + 506, + 248 + ], + "score": 1.0, + "content": ". Furthermore, for the fully deterministic system,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 244, + 506, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 244, + 325, + 259 + ], + "score": 1.0, + "content": "VAPVI automatically provides faster convergence rate", + "type": "text" + }, + { + "bbox": [ + 325, + 245, + 351, + 258 + ], + "score": 0.93, + "content": "O ( \\textstyle { \\frac { 1 } { K } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 244, + 506, + 259 + ], + "score": 1.0, + "content": "from the higher order term, given that", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 255, + 506, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 255, + 506, + 270 + ], + "score": 1.0, + "content": "the main term degenerates to 0. Those adaptive/instance-dependent features are not enjoyed by (Xie", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 266, + 507, + 284 + ], + "spans": [ + { + "bbox": [ + 104, + 266, + 449, + 284 + ], + "score": 1.0, + "content": "et al., 2021a; Zanette et al., 2021), as they always provide the standard statistical rate", + "type": "text" + }, + { + "bbox": [ + 449, + 267, + 482, + 283 + ], + "score": 0.94, + "content": "\\scriptstyle { \\dot { O ( } } { \\frac { 1 } { \\sqrt { K } } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 266, + 507, + 284 + ], + "score": 1.0, + "content": "(also", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 280, + 282, + 293 + ], + "spans": [ + { + "bbox": [ + 106, + 280, + 282, + 293 + ], + "score": 1.0, + "content": "check Remark C.9 for a related discussion).", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 11 + }, + { + "type": "title", + "bbox": [ + 108, + 305, + 453, + 317 + ], + "lines": [ + { + "bbox": [ + 105, + 305, + 455, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 305, + 455, + 318 + ], + "score": 1.0, + "content": "3.3 VAPVI-IMPROVED: FURTHER IMPROVEMENT IN STATE-ACTION DIMENSION", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 325, + 505, + 369 + ], + "lines": [ + { + "bbox": [ + 105, + 325, + 506, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 506, + 338 + ], + "score": 1.0, + "content": "Can we further improve the VAPVI? Indeed, by deploying a carefully tuned tighter penalty, we are", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 334, + 507, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 425, + 348 + ], + "score": 1.0, + "content": "able to further improve the state-action dependence if the feature is non-negative", + "type": "text" + }, + { + "bbox": [ + 425, + 336, + 452, + 347 + ], + "score": 0.87, + "content": "( \\phi \\ge 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 334, + 507, + 348 + ], + "score": 1.0, + "content": "). Concretely,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 345, + 506, + 361 + ], + "spans": [ + { + "bbox": [ + 104, + 345, + 210, + 361 + ], + "score": 1.0, + "content": "we replace the following", + "type": "text" + }, + { + "bbox": [ + 210, + 346, + 223, + 359 + ], + "score": 0.89, + "content": "\\Gamma _ { h } ^ { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 345, + 506, + 361 + ], + "score": 1.0, + "content": "in Algorithm 1 instead, and call the algorithm VAPVI-Improved (or", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 357, + 347, + 369 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 347, + 369 + ], + "score": 1.0, + "content": "VAPVI-I for short). The proof can be found in Appendix D.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19.5 + }, + { + "type": "interline_equation", + "bbox": [ + 110, + 373, + 491, + 408 + ], + "lines": [ + { + "bbox": [ + 110, + 373, + 491, + 408 + ], + "spans": [ + { + "bbox": [ + 110, + 373, + 491, + 408 + ], + "score": 0.94, + "content": "\\Gamma _ { h } ^ { I } ( s , a ) \\gets \\phi ( s , a ) ^ { \\top } \\left| \\widehat { \\Lambda } _ { h } ^ { - 1 } \\sum _ { \\tau = 1 } ^ { K } \\frac { \\phi \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\cdot \\left( r _ { h } ^ { \\tau } + \\widehat { V } _ { h + 1 } \\left( s _ { h + 1 } ^ { \\tau } \\right) - \\left( \\widehat { T } _ { h } \\widehat { V } _ { h + 1 } \\right) \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\right) } { \\widehat { \\sigma } _ { h } ^ { 2 } \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) } \\right| + \\widetilde { O } ( \\frac { H ^ { 3 } d / \\kappa } { K } )", + "type": "interline_equation", + "image_path": "7631a8a82c712d14a00c564fc4509f76e658c972c927b874120be0ca5bf51fe5.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 110, + 373, + 491, + 384.6666666666667 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 110, + 384.6666666666667, + 491, + 396.33333333333337 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 110, + 396.33333333333337, + 491, + 408.00000000000006 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 419, + 506, + 465 + ], + "lines": [ + { + "bbox": [ + 105, + 419, + 507, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 319, + 433 + ], + "score": 1.0, + "content": "Theorem 3.3. Suppose the feature is non-negative √", + "type": "text" + }, + { + "bbox": [ + 319, + 420, + 347, + 432 + ], + "score": 0.87, + "content": "\\left( \\phi \\geq 0 \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 419, + 371, + 433 + ], + "score": 1.0, + "content": "). Let", + "type": "text" + }, + { + "bbox": [ + 371, + 421, + 381, + 430 + ], + "score": 0.72, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 419, + 507, + 433 + ], + "score": 1.0, + "content": "be the number of episodes. 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Deploying", + "type": "text" + }, + { + "bbox": [ + 350, + 432, + 363, + 445 + ], + "score": 0.9, + "content": "\\Gamma _ { h } ^ { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 431, + 435, + 446 + ], + "score": 1.0, + "content": "(4) in Algorithm", + "type": "text" + }, + { + "bbox": [ + 436, + 433, + 442, + 442 + ], + "score": 0.34, + "content": "^ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 431, + 505, + 446 + ], + "score": 1.0, + "content": ". 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v ^ { \\widehat { \\pi } } \\leq \\widetilde { O } \\big ( \\sqrt { d } \\cdot \\sum _ { h = 1 } ^ { H } \\sqrt { \\mathbb { E } _ { \\pi } [ \\phi ( \\cdot , \\cdot ) ] ^ { \\top } \\Lambda _ { h } ^ { - 1 } \\mathbb { E } _ { \\pi } [ \\phi ( \\cdot , \\cdot ) ] } \\big ) + \\widetilde { O } ( \\frac { H ^ { 4 } d / \\kappa } { K } )", + "type": "interline_equation", + "image_path": "657dc53c523f18ef592ea469bd94629f3f49c99e6d3f5fc110c83d5f0282e20f.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 176, + 470, + 435, + 480.3333333333333 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 176, + 480.3333333333333, + 435, + 490.66666666666663 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 176, + 490.66666666666663, + 435, + 500.99999999999994 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 507, + 504, + 533 + ], + "lines": [ + { + "bbox": [ + 104, + 504, + 502, + 524 + ], + "spans": [ + { + "bbox": [ + 104, + 504, + 225, + 524 + ], + "score": 1.0, + "content": "In particular, when choosing", + "type": "text" + }, + { + "bbox": [ + 226, + 509, + 257, + 519 + ], + "score": 0.89, + "content": "\\pi = \\pi ^ { \\star }", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 504, + 410, + 524 + ], + "score": 1.0, + "content": ", the above guarantee holds true with", + "type": "text" + }, + { + "bbox": [ + 410, + 507, + 429, + 521 + ], + "score": 0.92, + "content": "\\Lambda _ { h } ^ { - 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Clearly, the two bounds", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 565, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 565, + 505, + 578 + ], + "score": 1.0, + "content": "differ by the magnitude of Jensen’s inequality. To provide a concrete view of how much improvement", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 576, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 104, + 576, + 505, + 590 + ], + "score": 1.0, + "content": "is made, we check the parameter dependence in the context of tabular MDPs (where we ignore the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 587, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 505, + 601 + ], + "score": 1.0, + "content": "higher order term for conciseness). 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Furthermore, for the fully deterministic system,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 244, + 506, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 244, + 325, + 259 + ], + "score": 1.0, + "content": "VAPVI automatically provides faster convergence rate", + "type": "text" + }, + { + "bbox": [ + 325, + 245, + 351, + 258 + ], + "score": 0.93, + "content": "O ( \\textstyle { \\frac { 1 } { K } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 244, + 506, + 259 + ], + "score": 1.0, + "content": "from the higher order term, given that", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 255, + 506, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 255, + 506, + 270 + ], + "score": 1.0, + "content": "the main term degenerates to 0. Those adaptive/instance-dependent features are not enjoyed by (Xie", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 266, + 507, + 284 + ], + "spans": [ + { + "bbox": [ + 104, + 266, + 449, + 284 + ], + "score": 1.0, + "content": "et al., 2021a; Zanette et al., 2021), as they always provide the standard statistical rate", + "type": "text" + }, + { + "bbox": [ + 449, + 267, + 482, + 283 + ], + "score": 0.94, + "content": "\\scriptstyle { \\dot { O ( } } { \\frac { 1 } { \\sqrt { K } } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 266, + 507, + 284 + ], + "score": 1.0, + "content": "(also", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 280, + 282, + 293 + ], + "spans": [ + { + "bbox": [ + 106, + 280, + 282, + 293 + ], + "score": 1.0, + "content": "check Remark C.9 for a related discussion).", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 11, + "bbox_fs": [ + 102, + 152, + 508, + 293 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 305, + 453, + 317 + ], + "lines": [ + { + "bbox": [ + 105, + 305, + 455, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 305, + 455, + 318 + ], + "score": 1.0, + "content": "3.3 VAPVI-IMPROVED: FURTHER IMPROVEMENT IN STATE-ACTION DIMENSION", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 325, + 505, + 369 + ], + "lines": [ + { + "bbox": [ + 105, + 325, + 506, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 506, + 338 + ], + "score": 1.0, + "content": "Can we further improve the VAPVI? Indeed, by deploying a carefully tuned tighter penalty, we are", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 334, + 507, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 425, + 348 + ], + "score": 1.0, + "content": "able to further improve the state-action dependence if the feature is non-negative", + "type": "text" + }, + { + "bbox": [ + 425, + 336, + 452, + 347 + ], + "score": 0.87, + "content": "( \\phi \\ge 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 334, + 507, + 348 + ], + "score": 1.0, + "content": "). Concretely,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 345, + 506, + 361 + ], + "spans": [ + { + "bbox": [ + 104, + 345, + 210, + 361 + ], + "score": 1.0, + "content": "we replace the following", + "type": "text" + }, + { + "bbox": [ + 210, + 346, + 223, + 359 + ], + "score": 0.89, + "content": "\\Gamma _ { h } ^ { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 345, + 506, + 361 + ], + "score": 1.0, + "content": "in Algorithm 1 instead, and call the algorithm VAPVI-Improved (or", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 357, + 347, + 369 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 347, + 369 + ], + "score": 1.0, + "content": "VAPVI-I for short). The proof can be found in Appendix D.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19.5, + "bbox_fs": [ + 104, + 325, + 507, + 369 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 110, + 373, + 491, + 408 + ], + "lines": [ + { + "bbox": [ + 110, + 373, + 491, + 408 + ], + "spans": [ + { + "bbox": [ + 110, + 373, + 491, + 408 + ], + "score": 0.94, + "content": "\\Gamma _ { h } ^ { I } ( s , a ) \\gets \\phi ( s , a ) ^ { \\top } \\left| \\widehat { \\Lambda } _ { h } ^ { - 1 } \\sum _ { \\tau = 1 } ^ { K } \\frac { \\phi \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\cdot \\left( r _ { h } ^ { \\tau } + \\widehat { V } _ { h + 1 } \\left( s _ { h + 1 } ^ { \\tau } \\right) - \\left( \\widehat { T } _ { h } \\widehat { V } _ { h + 1 } \\right) \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\right) } { \\widehat { \\sigma } _ { h } ^ { 2 } \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) } \\right| + \\widetilde { O } ( \\frac { H ^ { 3 } d / \\kappa } { K } )", + "type": "interline_equation", + "image_path": "7631a8a82c712d14a00c564fc4509f76e658c972c927b874120be0ca5bf51fe5.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 110, + 373, + 491, + 384.6666666666667 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 110, + 384.6666666666667, + 491, + 396.33333333333337 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 110, + 396.33333333333337, + 491, + 408.00000000000006 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 419, + 506, + 465 + ], + "lines": [ + { + "bbox": [ + 105, + 419, + 507, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 319, + 433 + ], + "score": 1.0, + "content": "Theorem 3.3. Suppose the feature is non-negative √", + "type": "text" + }, + { + "bbox": [ + 319, + 420, + 347, + 432 + ], + "score": 0.87, + "content": "\\left( \\phi \\geq 0 \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 419, + 371, + 433 + ], + "score": 1.0, + "content": "). Let", + "type": "text" + }, + { + "bbox": [ + 371, + 421, + 381, + 430 + ], + "score": 0.72, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 419, + 507, + 433 + ], + "score": 1.0, + "content": "be the number of episodes. If", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 431, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 106, + 432, + 241, + 444 + ], + "score": 0.91, + "content": "K > \\operatorname* { m a x } \\{ \\mathcal { M } _ { 1 } , \\mathcal { M } _ { 2 } , \\mathcal { M } _ { 3 } , \\mathcal { M } _ { 4 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 431, + 261, + 446 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 262, + 431, + 297, + 444 + ], + "score": 0.92, + "content": "\\sqrt { d } > \\xi", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 431, + 349, + 446 + ], + "score": 1.0, + "content": ". Deploying", + "type": "text" + }, + { + "bbox": [ + 350, + 432, + 363, + 445 + ], + "score": 0.9, + "content": "\\Gamma _ { h } ^ { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 431, + 435, + 446 + ], + "score": 1.0, + "content": "(4) in Algorithm", + "type": "text" + }, + { + "bbox": [ + 436, + 433, + 442, + 442 + ], + "score": 0.34, + "content": "^ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 431, + 505, + 446 + ], + "score": 1.0, + "content": ". Then for any", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 107, + 443, + 504, + 456 + ], + "spans": [ + { + "bbox": [ + 107, + 444, + 155, + 454 + ], + "score": 0.89, + "content": "0 < \\lambda < \\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 443, + 228, + 456 + ], + "score": 1.0, + "content": ", with probability", + "type": "text" + }, + { + "bbox": [ + 228, + 444, + 252, + 455 + ], + "score": 0.84, + "content": "1 - \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 443, + 313, + 456 + ], + "score": 1.0, + "content": ", for all policy", + "type": "text" + }, + { + "bbox": [ + 313, + 446, + 321, + 453 + ], + "score": 0.73, + "content": "\\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 443, + 433, + 456 + ], + "score": 1.0, + "content": "simultaneously, the output", + "type": "text" + }, + { + "bbox": [ + 434, + 445, + 441, + 453 + ], + "score": 0.76, + "content": "\\widehat { \\pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 443, + 498, + 456 + ], + "score": 1.0, + "content": "of Algorithm", + "type": "text" + }, + { + "bbox": [ + 499, + 446, + 504, + 453 + ], + "score": 0.41, + "content": "\\boldsymbol { l }", + "type": "inline_equation" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 453, + 183, + 466 + ], + "spans": [ + { + "bbox": [ + 104, + 453, + 183, + 466 + ], + "score": 1.0, + "content": "(VAPVI-I) satisfies", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26.5, + "bbox_fs": [ + 104, + 419, + 507, + 466 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 176, + 470, + 435, + 501 + ], + "lines": [ + { + "bbox": [ + 176, + 470, + 435, + 501 + ], + "spans": [ + { + "bbox": [ + 176, + 470, + 435, + 501 + ], + "score": 0.93, + "content": "v ^ { \\pi } - v ^ { \\widehat { \\pi } } \\leq \\widetilde { O } \\big ( \\sqrt { d } \\cdot \\sum _ { h = 1 } ^ { H } \\sqrt { \\mathbb { E } _ { \\pi } [ \\phi ( \\cdot , \\cdot ) ] ^ { \\top } \\Lambda _ { h } ^ { - 1 } \\mathbb { E } _ { \\pi } [ \\phi ( \\cdot , \\cdot ) ] } \\big ) + \\widetilde { O } ( \\frac { H ^ { 4 } d / \\kappa } { K } )", + "type": "interline_equation", + "image_path": "657dc53c523f18ef592ea469bd94629f3f49c99e6d3f5fc110c83d5f0282e20f.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 176, + 470, + 435, + 480.3333333333333 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 176, + 480.3333333333333, + 435, + 490.66666666666663 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 176, + 490.66666666666663, + 435, + 500.99999999999994 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 507, + 504, + 533 + ], + "lines": [ + { + "bbox": [ + 104, + 504, + 502, + 524 + ], + "spans": [ + { + "bbox": [ + 104, + 504, + 225, + 524 + ], + "score": 1.0, + "content": "In particular, when choosing", + "type": "text" + }, + { + "bbox": [ + 226, + 509, + 257, + 519 + ], + "score": 0.89, + "content": "\\pi = \\pi ^ { \\star }", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 504, + 410, + 524 + ], + "score": 1.0, + "content": ", the above guarantee holds true with", + "type": "text" + }, + { + "bbox": [ + 410, + 507, + 429, + 521 + ], + "score": 0.92, + "content": "\\Lambda _ { h } ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 504, + 480, + 524 + ], + "score": 1.0, + "content": "replaced by", + "type": "text" + }, + { + "bbox": [ + 480, + 507, + 502, + 521 + ], + "score": 0.92, + "content": "\\Lambda _ { h } ^ { \\star - 1 }", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 519, + 336, + 535 + ], + "spans": [ + { + "bbox": [ + 104, + 519, + 128, + 535 + ], + "score": 1.0, + "content": "Here", + "type": "text" + }, + { + "bbox": [ + 128, + 519, + 146, + 533 + ], + "score": 0.81, + "content": "\\Lambda _ { h } ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 519, + 150, + 535 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 151, + 519, + 173, + 533 + ], + "score": 0.81, + "content": "\\Lambda _ { h } ^ { \\star - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 519, + 177, + 535 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 177, + 520, + 184, + 533 + ], + "score": 0.7, + "content": "\\xi", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 519, + 336, + 535 + ], + "score": 1.0, + "content": "are defined the same as Theorem 3.2.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32.5, + "bbox_fs": [ + 104, + 504, + 502, + 535 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 541, + 505, + 610 + ], + "lines": [ + { + "bbox": [ + 105, + 540, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 505, + 555 + ], + "score": 1.0, + "content": "Theorem 3.3 maintains nearly all the features of Theorem 3.2 (except higher order term is slightly", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 552, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 289, + 567 + ], + "score": 1.0, + "content": "worse) and the dominate term evolves from", + "type": "text" + }, + { + "bbox": [ + 289, + 552, + 335, + 568 + ], + "score": 0.93, + "content": "\\mathbb { E } _ { \\pi } \\left. \\phi \\right. _ { \\Lambda _ { h } ^ { - 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 552, + 348, + 567 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 349, + 552, + 398, + 568 + ], + "score": 0.93, + "content": "\\| \\mathbb { E } _ { \\pi } [ \\phi ] \\| _ { \\Lambda _ { h } ^ { - 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 552, + 506, + 567 + ], + "score": 1.0, + "content": ". Clearly, the two bounds", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 565, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 565, + 505, + 578 + ], + "score": 1.0, + "content": "differ by the magnitude of Jensen’s inequality. To provide a concrete view of how much improvement", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 576, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 104, + 576, + 505, + 590 + ], + "score": 1.0, + "content": "is made, we check the parameter dependence in the context of tabular MDPs (where we ignore the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 587, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 505, + 601 + ], + "score": 1.0, + "content": "higher order term for conciseness). In particular, we compare the results under the single-policy", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 598, + 170, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 170, + 613 + ], + "score": 1.0, + "content": "concentrability.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 36.5, + "bbox_fs": [ + 104, + 540, + 506, + 613 + ] + }, + { + "type": "list", + "bbox": [ + 106, + 613, + 506, + 639 + ], + "lines": [ + { + "bbox": [ + 106, + 612, + 507, + 626 + ], + "spans": [ + { + "bbox": [ + 106, + 612, + 477, + 626 + ], + "score": 1.0, + "content": "Assumption 3.4 (Rashidinejad et al. (2021); Xie et al. (2021b)). There exists a optimal policy", + "type": "text" + }, + { + "bbox": [ + 478, + 615, + 488, + 624 + ], + "score": 0.82, + "content": "\\pi ^ { \\star }", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 612, + 507, + 626 + ], + "score": 1.0, + "content": ", s.t.", + "type": "text" + } + ], + "index": 40, + "is_list_end_line": true + }, + { + "bbox": [ + 106, + 624, + 507, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 624, + 268, + 640 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\operatorname* { s u p } _ { h , s , a } \\bar { d } _ { h } ^ { \\pi ^ { \\star } } ( s , a ) / d _ { h } ^ { \\mu } ( s , a ) : = C ^ { \\star } < \\infty } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 624, + 299, + 640 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 299, + 626, + 310, + 636 + ], + "score": 0.84, + "content": "d ^ { \\pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 624, + 496, + 640 + ], + "score": 1.0, + "content": "is the marginal state-action probability under", + "type": "text" + }, + { + "bbox": [ + 496, + 629, + 502, + 636 + ], + "score": 0.68, + "content": "\\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 624, + 507, + 640 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 41, + "is_list_start_line": true, + "is_list_end_line": true + } + ], + "index": 40.5, + "bbox_fs": [ + 106, + 612, + 507, + 640 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 647, + 479, + 660 + ], + "lines": [ + { + "bbox": [ + 105, + 647, + 479, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 647, + 165, + 662 + ], + "score": 1.0, + "content": "In tabular RL,", + "type": "text" + }, + { + "bbox": [ + 165, + 648, + 224, + 660 + ], + "score": 0.93, + "content": "\\phi ( s , a ) = \\mathbf { 1 } _ { s , a }", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 647, + 242, + 662 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 242, + 648, + 307, + 660 + ], + "score": 0.62, + "content": "d = S \\cdot A \\left( S , A \\right.", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 647, + 479, + 662 + ], + "score": 1.0, + "content": "be the finite state, action cardinality), then", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42, + "bbox_fs": [ + 105, + 647, + 479, + 662 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 152, + 666, + 457, + 736 + ], + "lines": [ + { + "bbox": [ + 152, + 666, + 457, + 736 + ], + "spans": [ + { + "bbox": [ + 152, + 666, + 457, + 736 + ], + "score": 0.54, + "content": "\\begin{array} { r l } & { \\mathrm { T h e o r e m ~ 3 . 2 } \\sqrt { S A } \\displaystyle \\sum _ { h } ^ { H } \\sum _ { s , a } d _ { h } ^ { \\pi ^ { \\star } } ( s , a ) \\sqrt { \\frac { \\mathrm { V a r } _ { P s , a } ( r + V _ { h + 1 } ^ { \\star } ) } { K \\cdot d _ { h } ^ { \\mu } ( s , a ) } } \\leq \\sqrt { \\frac { H ^ { 3 } C ^ { \\star } S ^ { 2 } A } { K } } ; } \\\\ & { \\mathrm { T h e o r e m ~ 3 . 3 } \\sqrt { S A } \\displaystyle \\sum _ { h } ^ { H } \\sqrt { \\displaystyle \\sum _ { s , a } d _ { h } ^ { \\pi ^ { \\star } } ( s , a ) ^ { 2 } \\frac { \\mathrm { V a r } _ { P s , a } ( r + V _ { h + 1 } ^ { \\star } ) } { K \\cdot d _ { h } ^ { \\mu } ( s , a ) } } \\leq \\sqrt { \\frac { H ^ { 3 } C ^ { \\star } S A } { K } } . } \\end{array}", + "type": "interline_equation", + "image_path": "7308f6e7368f68683b5e9d15ddc915c4aa01c2cb21749620c10da7ef171bfde5.jpg" + } + ] + } + ], + "index": 44, + "virtual_lines": [ + { + "bbox": [ + 152, + 666, + 457, + 689.3333333333334 + ], + "spans": [], + "index": 43 + }, + { + "bbox": [ + 152, + 689.3333333333334, + 457, + 712.6666666666667 + ], + "spans": [], + "index": 44 + }, + { + "bbox": [ + 152, + 712.6666666666667, + 457, + 736.0000000000001 + ], + "spans": [], + "index": 45 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 505, + 123 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 195, + 95 + ], + "score": 1.0, + "content": "Theorem 3.3 enjoys a", + "type": "text" + }, + { + "bbox": [ + 195, + 83, + 203, + 92 + ], + "score": 0.8, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 81, + 505, + 95 + ], + "score": 1.0, + "content": "state improvement over Theorem 3.2 and nearly recovers the minimax rate", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 107, + 93, + 505, + 114 + ], + "spans": [ + { + "bbox": [ + 107, + 93, + 146, + 114 + ], + "score": 0.91, + "content": "\\sqrt { \\frac { H ^ { 3 } C ^ { \\star } { \\cal S } } { K } }", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 93, + 505, + 113 + ], + "score": 1.0, + "content": "(Xie et al., 2021b). The detailed derivation can be found in Appendix G. Also, to show our", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 111, + 486, + 124 + ], + "spans": [ + { + "bbox": [ + 105, + 111, + 486, + 124 + ], + "score": 1.0, + "content": "result is near-optimal, we provide the corresponding lower bound. The proof is in Appendix E.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 107, + 124, + 505, + 177 + ], + "lines": [ + { + "bbox": [ + 106, + 123, + 506, + 136 + ], + "spans": [ + { + "bbox": [ + 106, + 123, + 430, + 136 + ], + "score": 1.0, + "content": "Theorem 3.5 (Minimax lower bound). There exist a pair of universal constants", + "type": "text" + }, + { + "bbox": [ + 430, + 124, + 465, + 136 + ], + "score": 0.88, + "content": "c , c ^ { \\prime } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 123, + 506, + 136 + ], + "score": 1.0, + "content": "such that", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 134, + 506, + 148 + ], + "spans": [ + { + "bbox": [ + 105, + 134, + 176, + 148 + ], + "score": 1.0, + "content": "given dimension", + "type": "text" + }, + { + "bbox": [ + 176, + 136, + 182, + 145 + ], + "score": 0.62, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 134, + 221, + 148 + ], + "score": 1.0, + "content": ", horizon", + "type": "text" + }, + { + "bbox": [ + 221, + 136, + 231, + 145 + ], + "score": 0.72, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 134, + 300, + 148 + ], + "score": 1.0, + "content": "and sample size", + "type": "text" + }, + { + "bbox": [ + 301, + 136, + 342, + 145 + ], + "score": 0.89, + "content": "K > c ^ { \\prime } d ^ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 134, + 506, + 148 + ], + "score": 1.0, + "content": ", one can always find a family of linear", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 103, + 144, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 103, + 146, + 391, + 167 + ], + "score": 1.0, + "content": "MDP instances M such that (where Λ?h = PKk=1 φ(s h,a h)·φ(s h,a h) Varh(V ? )(sk ,ak )", + "type": "text" + }, + { + "bbox": [ + 390, + 144, + 428, + 162 + ], + "score": 1.0, + "content": "satisfies", + "type": "text" + }, + { + "bbox": [ + 428, + 149, + 459, + 162 + ], + "score": 0.92, + "content": "( \\Lambda _ { h } ^ { \\star } ) ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 144, + 506, + 162 + ], + "score": 1.0, + "content": "exists and", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + }, + { + "type": "interline_equation", + "bbox": [ + 105, + 164, + 436, + 207 + ], + "lines": [ + { + "bbox": [ + 105, + 164, + 436, + 207 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 436, + 207 + ], + "score": 0.88, + "content": "\\begin{array} { r l } & { \\mathrm { V a r } _ { h } ( V _ { h + 1 } ^ { \\star } ) ( s _ { h } ^ { k } , a _ { h } ^ { k } ) > 0 \\forall M \\in \\mathcal { M } ) } \\\\ & { \\qquad \\quad \\frac { \\operatorname* { i n f } } { \\pi } \\underset { M \\in \\mathcal { M } } { \\operatorname* { s u p } } \\mathbb { E } _ { M } \\left[ v ^ { \\star } - v ^ { \\hat { \\pi } } \\right] / \\left( \\sqrt { d } \\cdot \\displaystyle \\sum _ { h = 1 } ^ { H } \\sqrt { \\mathbb { E } _ { \\pi ^ { \\star } } \\left[ \\phi \\right] ^ { \\top } ( \\Lambda _ { h } ^ { \\star } ) ^ { - 1 } \\mathbb { E } _ { \\pi ^ { \\star } } \\left[ \\phi \\right] } \\right) \\geq c . } \\end{array}", + "type": "interline_equation", + "image_path": "b810657ad689c435f0bcbbd8d7bed61d8304acd9da1fbff77bfcc2ba7f28750c.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 105, + 164, + 436, + 178.33333333333334 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 105, + 178.33333333333334, + 436, + 192.66666666666669 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 105, + 192.66666666666669, + 436, + 207.00000000000003 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 207, + 504, + 219 + ], + "lines": [ + { + "bbox": [ + 106, + 206, + 506, + 221 + ], + "spans": [ + { + "bbox": [ + 106, + 206, + 506, + 221 + ], + "score": 1.0, + "content": "Theorem 3.5 nearly matches the main term in VAPVI-I (Theorem 3.3) and certifies it is near-optimal.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "title", + "bbox": [ + 108, + 234, + 220, + 247 + ], + "lines": [ + { + "bbox": [ + 105, + 232, + 221, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 221, + 249 + ], + "score": 1.0, + "content": "4 PROOF OVERVIEW", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 106, + 258, + 506, + 319 + ], + "lines": [ + { + "bbox": [ + 106, + 258, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 106, + 258, + 505, + 271 + ], + "score": 1.0, + "content": "Due to the space constraint, we could only provide a brief overview of the key proving ideas of the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 269, + 506, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 506, + 282 + ], + "score": 1.0, + "content": "theorems. We begin with Theorem 3.2. First, by the extended value difference lemma (Lemma H.7),", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 103, + 279, + 508, + 299 + ], + "spans": [ + { + "bbox": [ + 103, + 279, + 312, + 299 + ], + "score": 1.0, + "content": "we can convert bounding the suboptimality gap of", + "type": "text" + }, + { + "bbox": [ + 312, + 281, + 345, + 293 + ], + "score": 0.92, + "content": "\\boldsymbol { v } ^ { \\star } - \\boldsymbol { v } ^ { \\widehat { \\pi } }", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 279, + 398, + 299 + ], + "score": 1.0, + "content": "to bounding", + "type": "text" + }, + { + "bbox": [ + 399, + 281, + 502, + 295 + ], + "score": 0.89, + "content": "{ \\begin{array} { r } { \\sum _ { h = 1 } ^ { H } 2 \\cdot \\mathbb { E } _ { \\pi } \\left[ \\Gamma _ { h } { \\left( s _ { h } , a _ { h } \\right) } \\right] } \\end{array} }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 279, + 508, + 299 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 293, + 506, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 150, + 309 + ], + "score": 1.0, + "content": "given that", + "type": "text" + }, + { + "bbox": [ + 150, + 294, + 310, + 308 + ], + "score": 0.92, + "content": "| ( \\mathcal T _ { h } \\widehat { V } _ { h + 1 } - \\widehat { \\mathcal T } _ { h } \\widehat { V } _ { h + 1 } ) ( s , a ) | \\leq \\Gamma _ { h } ( s , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 310, + 293, + 339, + 309 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 339, + 297, + 365, + 308 + ], + "score": 0.92, + "content": "s , a , h", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 293, + 411, + 309 + ], + "score": 1.0, + "content": ". To bound", + "type": "text" + }, + { + "bbox": [ + 412, + 295, + 488, + 308 + ], + "score": 0.94, + "content": "\\mathcal { T } _ { h } \\widehat { V } _ { h + 1 } - \\widehat { \\mathcal { T } } _ { h } \\widehat { V } _ { h + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 293, + 506, + 309 + ], + "score": 1.0, + "content": ", by", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 306, + 319, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 306, + 319, + 320 + ], + "score": 1.0, + "content": "decomposing it reduces to bounding the key quantity", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13 + }, + { + "type": "interline_equation", + "bbox": [ + 121, + 320, + 478, + 354 + ], + "lines": [ + { + "bbox": [ + 121, + 320, + 478, + 354 + ], + "spans": [ + { + "bbox": [ + 121, + 320, + 478, + 354 + ], + "score": 0.92, + "content": "\\phi ( s , a ) ^ { \\top } \\widehat { \\Lambda } _ { h } ^ { - 1 } \\left[ \\sum _ { \\tau = 1 } ^ { K } \\phi \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\cdot \\left( r _ { h } ^ { \\tau } + \\widehat { V } _ { h + 1 } \\left( s _ { h + 1 } ^ { \\tau } \\right) - \\left( \\mathcal { T } _ { h } \\widehat { V } _ { h + 1 } \\right) \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\right) / \\widehat { \\sigma } _ { h } ^ { 2 } \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\right]", + "type": "interline_equation", + "image_path": "30c75e721e4d8cdd6fd373e0d3f390dfccfb1a8916332669093166d67337d475.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 121, + 320, + 478, + 331.3333333333333 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 121, + 331.3333333333333, + 478, + 342.66666666666663 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 121, + 342.66666666666663, + 478, + 353.99999999999994 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 354, + 506, + 506 + ], + "lines": [ + { + "bbox": [ + 104, + 354, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 104, + 355, + 347, + 375 + ], + "score": 1.0, + "content": "The term is treated in two steps. First, we bound the gap of", + "type": "text" + }, + { + "bbox": [ + 347, + 354, + 406, + 375 + ], + "score": 0.93, + "content": "\\left\\| \\sigma _ { \\widehat { V } _ { h + 1 } } ^ { 2 } - \\widehat { \\sigma } _ { h } ^ { 2 } \\right\\|", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 357, + 481, + 372 + ], + "score": 1.0, + "content": "so we can convert", + "type": "text" + }, + { + "bbox": [ + 482, + 358, + 493, + 371 + ], + "score": 0.89, + "content": "\\widehat { \\sigma } _ { h } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 357, + 505, + 372 + ], + "score": 1.0, + "content": "to", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 107, + 373, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 107, + 376, + 131, + 394 + ], + "score": 0.9, + "content": "\\sigma _ { \\widehat { V } _ { h + 1 } } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 373, + 185, + 393 + ], + "score": 1.0, + "content": ". Next, since", + "type": "text" + }, + { + "bbox": [ + 186, + 374, + 448, + 394 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\mathrm { V a r } \\left[ r _ { h } ^ { \\tau } + \\widehat { V } _ { h + 1 } \\left( s _ { h + 1 } ^ { \\tau } \\right) - \\left( \\mathcal { T } _ { h } \\widehat { V } _ { h + 1 } \\right) \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\mid s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right] \\approx \\sigma _ { \\widehat { V } _ { h + 1 } } ^ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 378, + 505, + 390 + ], + "score": 1.0, + "content": ", therefore by", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 391, + 506, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 506, + 405 + ], + "score": 1.0, + "content": "the variance-weighted scheme in (equation 7), we can leverage the recent technical development", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 403, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 505, + 416 + ], + "score": 1.0, + "content": "Bernstein inequality for self-normalized martingale (Lemma H.3) for acquiring the tight result, in", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 413, + 502, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 452, + 427 + ], + "score": 1.0, + "content": "contrast to the previous treatment of Hoeffding inequality for self-normalized martingale", + "type": "text" + }, + { + "bbox": [ + 452, + 416, + 460, + 424 + ], + "score": 0.55, + "content": "^ +", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 413, + 502, + 427 + ], + "score": 1.0, + "content": "Covering.8", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 103, + 420, + 508, + 443 + ], + "spans": [ + { + "bbox": [ + 103, + 424, + 313, + 439 + ], + "score": 1.0, + "content": "For the second part, one needs to further convert", + "type": "text" + }, + { + "bbox": [ + 314, + 425, + 339, + 441 + ], + "score": 0.93, + "content": "\\sigma _ { \\widehat { V } _ { h + 1 } } ^ { 2 ^ { - } }", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 420, + 352, + 443 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 352, + 424, + 369, + 438 + ], + "score": 0.88, + "content": "\\sigma _ { h } ^ { \\star 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 420, + 373, + 443 + ], + "score": 0.0, + "content": "", + "type": "text" + }, + { + "bbox": [ + 373, + 424, + 393, + 438 + ], + "score": 0.8, + "content": "\\cdot \\Lambda _ { h } ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 420, + 406, + 443 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 406, + 424, + 429, + 438 + ], + "score": 0.91, + "content": "\\Lambda _ { h } ^ { \\star - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 420, + 508, + 443 + ], + "score": 1.0, + "content": ") with appropriate", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 440, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 106, + 440, + 505, + 451 + ], + "score": 1.0, + "content": "concentrations. The proof of Theorem 3.3 is similar but with more complicated computations and", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 450, + 506, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 280, + 463 + ], + "score": 1.0, + "content": "relies on using the linear representation of", + "type": "text" + }, + { + "bbox": [ + 281, + 451, + 288, + 462 + ], + "score": 0.84, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 450, + 299, + 463 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 300, + 450, + 312, + 463 + ], + "score": 0.9, + "content": "\\Gamma _ { h } ^ { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 450, + 444, + 463 + ], + "score": 1.0, + "content": "(4), so that the expectation over", + "type": "text" + }, + { + "bbox": [ + 445, + 452, + 452, + 460 + ], + "score": 0.75, + "content": "\\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 450, + 506, + 463 + ], + "score": 1.0, + "content": "is inside the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 462, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 505, + 474 + ], + "score": 1.0, + "content": "square root by taking expectation over the linear representation at the beginning. The lower bound", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 471, + 506, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 506, + 487 + ], + "score": 1.0, + "content": "proof uses a simple modification of Zanette et al. (2021) which consists of the reduction from learning", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 483, + 506, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 506, + 496 + ], + "score": 1.0, + "content": "to testing with Assouad’s method, and the use of standard information inequalities (e.g. from total", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 494, + 461, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 461, + 507 + ], + "score": 1.0, + "content": "variation to KL divergence). For completeness, we provide the full proof in Appendix E.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 24.5 + }, + { + "type": "title", + "bbox": [ + 109, + 521, + 287, + 533 + ], + "lines": [ + { + "bbox": [ + 105, + 519, + 289, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 289, + 536 + ], + "score": 1.0, + "content": "5 DISCUSSION AND CONCLUSION", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 106, + 545, + 505, + 633 + ], + "lines": [ + { + "bbox": [ + 106, + 545, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 505, + 558 + ], + "score": 1.0, + "content": "This work studies offline RL with linear MDP representation and contributes Variance Aware", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 556, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 505, + 568 + ], + "score": 1.0, + "content": "Pessimistic Value Iteration (VAPVI) which adopts the conditional variance information of the value", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 567, + 505, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 505, + 580 + ], + "score": 1.0, + "content": "function. VAPVI uses the estimated variances to reweight the Bellman residuals in the least-square", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 579, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 505, + 590 + ], + "score": 1.0, + "content": "pessimistic value iteration and provides improved offline learning bounds over the existing best-known", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 588, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 505, + 601 + ], + "score": 1.0, + "content": "results. VAPVI-I further improves over VAPVI in the state-action dimension and is near-minimax", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 599, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 613 + ], + "score": 1.0, + "content": "optimal. One highlight of the theorems is that our learning bounds are expressed in terms of system", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 611, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 505, + 624 + ], + "score": 1.0, + "content": "quantities, which automatically provide natural instance-dependent characterizations that previous", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 622, + 185, + 633 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 185, + 633 + ], + "score": 1.0, + "content": "results are short of.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 35.5 + }, + { + "type": "text", + "bbox": [ + 107, + 639, + 505, + 705 + ], + "lines": [ + { + "bbox": [ + 106, + 638, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 505, + 651 + ], + "score": 1.0, + "content": "On the other hand, while VAPVI/VAPVI-I close the existing gap from previous literature (Jin et al.,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 649, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 505, + 662 + ], + "score": 1.0, + "content": "2021b; Xie et al., 2021a), the optimal guarantee is in the minimax sense. Although our upper bounds", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 661, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 661, + 506, + 673 + ], + "score": 1.0, + "content": "possess instance-dependent characterizations, the lower bound only holds true for a class of hard", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 672, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 106, + 672, + 505, + 683 + ], + "score": 1.0, + "content": "instances. In this sense, whether “instance-dependent optimality” can be achieved remains elusive in", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 682, + 506, + 695 + ], + "spans": [ + { + "bbox": [ + 106, + 682, + 506, + 695 + ], + "score": 1.0, + "content": "the current linear MDP setting (such a discussion is recently initiated in MAB problems (Xiao et al.,", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 693, + 256, + 705 + ], + "spans": [ + { + "bbox": [ + 106, + 693, + 256, + 705 + ], + "score": 1.0, + "content": "2021)). We leave this as future work.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 42.5 + } + ], + "page_idx": 8, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 711, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 117, + 708, + 506, + 725 + ], + "spans": [ + { + "bbox": [ + 117, + 708, + 506, + 725 + ], + "score": 1.0, + "content": "8Variance-reweighting in (7) is important, since applying Bernstein inequality for self-normalized martingale", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 720, + 388, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 388, + 732 + ], + "score": 1.0, + "content": "(Lemma H.3) without variance-reweighting cannot provide any improvement.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 107, + 26, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "9", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 505, + 123 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 195, + 95 + ], + "score": 1.0, + "content": "Theorem 3.3 enjoys a", + "type": "text" + }, + { + "bbox": [ + 195, + 83, + 203, + 92 + ], + "score": 0.8, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 81, + 505, + 95 + ], + "score": 1.0, + "content": "state improvement over Theorem 3.2 and nearly recovers the minimax rate", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 107, + 93, + 505, + 114 + ], + "spans": [ + { + "bbox": [ + 107, + 93, + 146, + 114 + ], + "score": 0.91, + "content": "\\sqrt { \\frac { H ^ { 3 } C ^ { \\star } { \\cal S } } { K } }", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 93, + 505, + 113 + ], + "score": 1.0, + "content": "(Xie et al., 2021b). The detailed derivation can be found in Appendix G. Also, to show our", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 111, + 486, + 124 + ], + "spans": [ + { + "bbox": [ + 105, + 111, + 486, + 124 + ], + "score": 1.0, + "content": "result is near-optimal, we provide the corresponding lower bound. The proof is in Appendix E.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1, + "bbox_fs": [ + 105, + 81, + 505, + 124 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 124, + 505, + 177 + ], + "lines": [ + { + "bbox": [ + 106, + 123, + 506, + 136 + ], + "spans": [ + { + "bbox": [ + 106, + 123, + 430, + 136 + ], + "score": 1.0, + "content": "Theorem 3.5 (Minimax lower bound). There exist a pair of universal constants", + "type": "text" + }, + { + "bbox": [ + 430, + 124, + 465, + 136 + ], + "score": 0.88, + "content": "c , c ^ { \\prime } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 123, + 506, + 136 + ], + "score": 1.0, + "content": "such that", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 134, + 506, + 148 + ], + "spans": [ + { + "bbox": [ + 105, + 134, + 176, + 148 + ], + "score": 1.0, + "content": "given dimension", + "type": "text" + }, + { + "bbox": [ + 176, + 136, + 182, + 145 + ], + "score": 0.62, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 134, + 221, + 148 + ], + "score": 1.0, + "content": ", horizon", + "type": "text" + }, + { + "bbox": [ + 221, + 136, + 231, + 145 + ], + "score": 0.72, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 134, + 300, + 148 + ], + "score": 1.0, + "content": "and sample size", + "type": "text" + }, + { + "bbox": [ + 301, + 136, + 342, + 145 + ], + "score": 0.89, + "content": "K > c ^ { \\prime } d ^ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 134, + 506, + 148 + ], + "score": 1.0, + "content": ", one can always find a family of linear", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 103, + 144, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 103, + 146, + 391, + 167 + ], + "score": 1.0, + "content": "MDP instances M such that (where Λ?h = PKk=1 φ(s h,a h)·φ(s h,a h) Varh(V ? )(sk ,ak )", + "type": "text" + }, + { + "bbox": [ + 390, + 144, + 428, + 162 + ], + "score": 1.0, + "content": "satisfies", + "type": "text" + }, + { + "bbox": [ + 428, + 149, + 459, + 162 + ], + "score": 0.92, + "content": "( \\Lambda _ { h } ^ { \\star } ) ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 144, + 506, + 162 + ], + "score": 1.0, + "content": "exists and", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4, + "bbox_fs": [ + 103, + 123, + 506, + 167 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 105, + 164, + 436, + 207 + ], + "lines": [ + { + "bbox": [ + 105, + 164, + 436, + 207 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 436, + 207 + ], + "score": 0.88, + "content": "\\begin{array} { r l } & { \\mathrm { V a r } _ { h } ( V _ { h + 1 } ^ { \\star } ) ( s _ { h } ^ { k } , a _ { h } ^ { k } ) > 0 \\forall M \\in \\mathcal { M } ) } \\\\ & { \\qquad \\quad \\frac { \\operatorname* { i n f } } { \\pi } \\underset { M \\in \\mathcal { M } } { \\operatorname* { s u p } } \\mathbb { E } _ { M } \\left[ v ^ { \\star } - v ^ { \\hat { \\pi } } \\right] / \\left( \\sqrt { d } \\cdot \\displaystyle \\sum _ { h = 1 } ^ { H } \\sqrt { \\mathbb { E } _ { \\pi ^ { \\star } } \\left[ \\phi \\right] ^ { \\top } ( \\Lambda _ { h } ^ { \\star } ) ^ { - 1 } \\mathbb { E } _ { \\pi ^ { \\star } } \\left[ \\phi \\right] } \\right) \\geq c . } \\end{array}", + "type": "interline_equation", + "image_path": "b810657ad689c435f0bcbbd8d7bed61d8304acd9da1fbff77bfcc2ba7f28750c.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 105, + 164, + 436, + 178.33333333333334 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 105, + 178.33333333333334, + 436, + 192.66666666666669 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 105, + 192.66666666666669, + 436, + 207.00000000000003 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 207, + 504, + 219 + ], + "lines": [ + { + "bbox": [ + 106, + 206, + 506, + 221 + ], + "spans": [ + { + "bbox": [ + 106, + 206, + 506, + 221 + ], + "score": 1.0, + "content": "Theorem 3.5 nearly matches the main term in VAPVI-I (Theorem 3.3) and certifies it is near-optimal.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9, + "bbox_fs": [ + 106, + 206, + 506, + 221 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 234, + 220, + 247 + ], + "lines": [ + { + "bbox": [ + 105, + 232, + 221, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 221, + 249 + ], + "score": 1.0, + "content": "4 PROOF OVERVIEW", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 106, + 258, + 506, + 319 + ], + "lines": [ + { + "bbox": [ + 106, + 258, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 106, + 258, + 505, + 271 + ], + "score": 1.0, + "content": "Due to the space constraint, we could only provide a brief overview of the key proving ideas of the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 269, + 506, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 506, + 282 + ], + "score": 1.0, + "content": "theorems. We begin with Theorem 3.2. First, by the extended value difference lemma (Lemma H.7),", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 103, + 279, + 508, + 299 + ], + "spans": [ + { + "bbox": [ + 103, + 279, + 312, + 299 + ], + "score": 1.0, + "content": "we can convert bounding the suboptimality gap of", + "type": "text" + }, + { + "bbox": [ + 312, + 281, + 345, + 293 + ], + "score": 0.92, + "content": "\\boldsymbol { v } ^ { \\star } - \\boldsymbol { v } ^ { \\widehat { \\pi } }", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 279, + 398, + 299 + ], + "score": 1.0, + "content": "to bounding", + "type": "text" + }, + { + "bbox": [ + 399, + 281, + 502, + 295 + ], + "score": 0.89, + "content": "{ \\begin{array} { r } { \\sum _ { h = 1 } ^ { H } 2 \\cdot \\mathbb { E } _ { \\pi } \\left[ \\Gamma _ { h } { \\left( s _ { h } , a _ { h } \\right) } \\right] } \\end{array} }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 279, + 508, + 299 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 293, + 506, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 150, + 309 + ], + "score": 1.0, + "content": "given that", + "type": "text" + }, + { + "bbox": [ + 150, + 294, + 310, + 308 + ], + "score": 0.92, + "content": "| ( \\mathcal T _ { h } \\widehat { V } _ { h + 1 } - \\widehat { \\mathcal T } _ { h } \\widehat { V } _ { h + 1 } ) ( s , a ) | \\leq \\Gamma _ { h } ( s , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 310, + 293, + 339, + 309 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 339, + 297, + 365, + 308 + ], + "score": 0.92, + "content": "s , a , h", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 293, + 411, + 309 + ], + "score": 1.0, + "content": ". To bound", + "type": "text" + }, + { + "bbox": [ + 412, + 295, + 488, + 308 + ], + "score": 0.94, + "content": "\\mathcal { T } _ { h } \\widehat { V } _ { h + 1 } - \\widehat { \\mathcal { T } } _ { h } \\widehat { V } _ { h + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 293, + 506, + 309 + ], + "score": 1.0, + "content": ", by", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 306, + 319, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 306, + 319, + 320 + ], + "score": 1.0, + "content": "decomposing it reduces to bounding the key quantity", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13, + "bbox_fs": [ + 103, + 258, + 508, + 320 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 121, + 320, + 478, + 354 + ], + "lines": [ + { + "bbox": [ + 121, + 320, + 478, + 354 + ], + "spans": [ + { + "bbox": [ + 121, + 320, + 478, + 354 + ], + "score": 0.92, + "content": "\\phi ( s , a ) ^ { \\top } \\widehat { \\Lambda } _ { h } ^ { - 1 } \\left[ \\sum _ { \\tau = 1 } ^ { K } \\phi \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\cdot \\left( r _ { h } ^ { \\tau } + \\widehat { V } _ { h + 1 } \\left( s _ { h + 1 } ^ { \\tau } \\right) - \\left( \\mathcal { T } _ { h } \\widehat { V } _ { h + 1 } \\right) \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\right) / \\widehat { \\sigma } _ { h } ^ { 2 } \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\right]", + "type": "interline_equation", + "image_path": "30c75e721e4d8cdd6fd373e0d3f390dfccfb1a8916332669093166d67337d475.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 121, + 320, + 478, + 331.3333333333333 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 121, + 331.3333333333333, + 478, + 342.66666666666663 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 121, + 342.66666666666663, + 478, + 353.99999999999994 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 354, + 506, + 506 + ], + "lines": [ + { + "bbox": [ + 104, + 354, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 104, + 355, + 347, + 375 + ], + "score": 1.0, + "content": "The term is treated in two steps. First, we bound the gap of", + "type": "text" + }, + { + "bbox": [ + 347, + 354, + 406, + 375 + ], + "score": 0.93, + "content": "\\left\\| \\sigma _ { \\widehat { V } _ { h + 1 } } ^ { 2 } - \\widehat { \\sigma } _ { h } ^ { 2 } \\right\\|", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 357, + 481, + 372 + ], + "score": 1.0, + "content": "so we can convert", + "type": "text" + }, + { + "bbox": [ + 482, + 358, + 493, + 371 + ], + "score": 0.89, + "content": "\\widehat { \\sigma } _ { h } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 357, + 505, + 372 + ], + "score": 1.0, + "content": "to", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 107, + 373, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 107, + 376, + 131, + 394 + ], + "score": 0.9, + "content": "\\sigma _ { \\widehat { V } _ { h + 1 } } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 373, + 185, + 393 + ], + "score": 1.0, + "content": ". Next, since", + "type": "text" + }, + { + "bbox": [ + 186, + 374, + 448, + 394 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\mathrm { V a r } \\left[ r _ { h } ^ { \\tau } + \\widehat { V } _ { h + 1 } \\left( s _ { h + 1 } ^ { \\tau } \\right) - \\left( \\mathcal { T } _ { h } \\widehat { V } _ { h + 1 } \\right) \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\mid s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right] \\approx \\sigma _ { \\widehat { V } _ { h + 1 } } ^ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 378, + 505, + 390 + ], + "score": 1.0, + "content": ", therefore by", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 391, + 506, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 506, + 405 + ], + "score": 1.0, + "content": "the variance-weighted scheme in (equation 7), we can leverage the recent technical development", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 403, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 505, + 416 + ], + "score": 1.0, + "content": "Bernstein inequality for self-normalized martingale (Lemma H.3) for acquiring the tight result, in", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 413, + 502, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 452, + 427 + ], + "score": 1.0, + "content": "contrast to the previous treatment of Hoeffding inequality for self-normalized martingale", + "type": "text" + }, + { + "bbox": [ + 452, + 416, + 460, + 424 + ], + "score": 0.55, + "content": "^ +", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 413, + 502, + 427 + ], + "score": 1.0, + "content": "Covering.8", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 103, + 420, + 508, + 443 + ], + "spans": [ + { + "bbox": [ + 103, + 424, + 313, + 439 + ], + "score": 1.0, + "content": "For the second part, one needs to further convert", + "type": "text" + }, + { + "bbox": [ + 314, + 425, + 339, + 441 + ], + "score": 0.93, + "content": "\\sigma _ { \\widehat { V } _ { h + 1 } } ^ { 2 ^ { - } }", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 420, + 352, + 443 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 352, + 424, + 369, + 438 + ], + "score": 0.88, + "content": "\\sigma _ { h } ^ { \\star 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 420, + 373, + 443 + ], + "score": 0.0, + "content": "", + "type": "text" + }, + { + "bbox": [ + 373, + 424, + 393, + 438 + ], + "score": 0.8, + "content": "\\cdot \\Lambda _ { h } ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 420, + 406, + 443 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 406, + 424, + 429, + 438 + ], + "score": 0.91, + "content": "\\Lambda _ { h } ^ { \\star - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 420, + 508, + 443 + ], + "score": 1.0, + "content": ") with appropriate", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 440, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 106, + 440, + 505, + 451 + ], + "score": 1.0, + "content": "concentrations. The proof of Theorem 3.3 is similar but with more complicated computations and", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 450, + 506, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 280, + 463 + ], + "score": 1.0, + "content": "relies on using the linear representation of", + "type": "text" + }, + { + "bbox": [ + 281, + 451, + 288, + 462 + ], + "score": 0.84, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 450, + 299, + 463 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 300, + 450, + 312, + 463 + ], + "score": 0.9, + "content": "\\Gamma _ { h } ^ { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 450, + 444, + 463 + ], + "score": 1.0, + "content": "(4), so that the expectation over", + "type": "text" + }, + { + "bbox": [ + 445, + 452, + 452, + 460 + ], + "score": 0.75, + "content": "\\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 450, + 506, + 463 + ], + "score": 1.0, + "content": "is inside the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 462, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 505, + 474 + ], + "score": 1.0, + "content": "square root by taking expectation over the linear representation at the beginning. The lower bound", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 471, + 506, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 506, + 487 + ], + "score": 1.0, + "content": "proof uses a simple modification of Zanette et al. (2021) which consists of the reduction from learning", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 483, + 506, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 506, + 496 + ], + "score": 1.0, + "content": "to testing with Assouad’s method, and the use of standard information inequalities (e.g. from total", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 494, + 461, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 461, + 507 + ], + "score": 1.0, + "content": "variation to KL divergence). For completeness, we provide the full proof in Appendix E.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 24.5, + "bbox_fs": [ + 103, + 354, + 508, + 507 + ] + }, + { + "type": "title", + "bbox": [ + 109, + 521, + 287, + 533 + ], + "lines": [ + { + "bbox": [ + 105, + 519, + 289, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 289, + 536 + ], + "score": 1.0, + "content": "5 DISCUSSION AND CONCLUSION", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 106, + 545, + 505, + 633 + ], + "lines": [ + { + "bbox": [ + 106, + 545, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 505, + 558 + ], + "score": 1.0, + "content": "This work studies offline RL with linear MDP representation and contributes Variance Aware", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 556, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 505, + 568 + ], + "score": 1.0, + "content": "Pessimistic Value Iteration (VAPVI) which adopts the conditional variance information of the value", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 567, + 505, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 505, + 580 + ], + "score": 1.0, + "content": "function. VAPVI uses the estimated variances to reweight the Bellman residuals in the least-square", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 579, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 505, + 590 + ], + "score": 1.0, + "content": "pessimistic value iteration and provides improved offline learning bounds over the existing best-known", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 588, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 505, + 601 + ], + "score": 1.0, + "content": "results. VAPVI-I further improves over VAPVI in the state-action dimension and is near-minimax", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 599, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 613 + ], + "score": 1.0, + "content": "optimal. One highlight of the theorems is that our learning bounds are expressed in terms of system", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 611, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 505, + 624 + ], + "score": 1.0, + "content": "quantities, which automatically provide natural instance-dependent characterizations that previous", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 622, + 185, + 633 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 185, + 633 + ], + "score": 1.0, + "content": "results are short of.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 35.5, + "bbox_fs": [ + 105, + 545, + 505, + 633 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 639, + 505, + 705 + ], + "lines": [ + { + "bbox": [ + 106, + 638, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 505, + 651 + ], + "score": 1.0, + "content": "On the other hand, while VAPVI/VAPVI-I close the existing gap from previous literature (Jin et al.,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 649, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 505, + 662 + ], + "score": 1.0, + "content": "2021b; Xie et al., 2021a), the optimal guarantee is in the minimax sense. Although our upper bounds", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 661, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 661, + 506, + 673 + ], + "score": 1.0, + "content": "possess instance-dependent characterizations, the lower bound only holds true for a class of hard", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 672, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 106, + 672, + 505, + 683 + ], + "score": 1.0, + "content": "instances. In this sense, whether “instance-dependent optimality” can be achieved remains elusive in", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 682, + 506, + 695 + ], + "spans": [ + { + "bbox": [ + 106, + 682, + 506, + 695 + ], + "score": 1.0, + "content": "the current linear MDP setting (such a discussion is recently initiated in MAB problems (Xiao et al.,", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 693, + 256, + 705 + ], + "spans": [ + { + "bbox": [ + 106, + 693, + 256, + 705 + ], + "score": 1.0, + "content": "2021)). We leave this as future work.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 42.5, + "bbox_fs": [ + 105, + 638, + 506, + 705 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 83, + 200, + 93 + ], + "lines": [ + { + "bbox": [ + 107, + 84, + 200, + 94 + ], + "spans": [ + { + "bbox": [ + 107, + 84, + 200, + 94 + ], + "score": 1.0, + "content": "ACKNOWLEDGMENTS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 101, + 505, + 168 + ], + "lines": [ + { + "bbox": [ + 105, + 100, + 506, + 115 + ], + "spans": [ + { + "bbox": [ + 105, + 100, + 506, + 115 + ], + "score": 1.0, + "content": "The authors would like to thank Quanquan Gu for explaining Min et al. (2021) and introducing a", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 113, + 505, + 126 + ], + "spans": [ + { + "bbox": [ + 106, + 113, + 505, + 126 + ], + "score": 1.0, + "content": "couple of related literatures. Ming Yin would like to thank Zhuoran Yang for the helpful suggestions", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 124, + 505, + 136 + ], + "spans": [ + { + "bbox": [ + 106, + 124, + 505, + 136 + ], + "score": 1.0, + "content": "and Dan Qiao for a careful proofreading. Mengdi Wang gratefully acknowledges funding from Office", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 134, + 506, + 147 + ], + "spans": [ + { + "bbox": [ + 105, + 134, + 506, + 147 + ], + "score": 1.0, + "content": "of Naval Research (ONR) N00014-21-1-2288, Air Force Office of Scientific Research (AFOSR)", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 145, + 505, + 158 + ], + "spans": [ + { + "bbox": [ + 105, + 145, + 505, + 158 + ], + "score": 1.0, + "content": "FA9550-19-1-0203, and NSF 19-589, CMMI-1653435. 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Other", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 138 + ], + "score": 1.0, + "content": "works impose weaker assumptions (e.g. partial coverage (Liu et al., 2020; Kidambi et al., 2020;", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 136, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 483, + 150 + ], + "score": 1.0, + "content": "Uehara and Sun, 2021)), and their finite sample analysis are generally suboptimal in terms of", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 483, + 138, + 493, + 147 + ], + "score": 0.8, + "content": "H", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 493, + 136, + 506, + 150 + ], + "score": 1.0, + "content": "or", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 147, + 236, + 162 + ], + "spans": [ + { + "bbox": [ + 106, + 147, + 191, + 162 + ], + "score": 1.0, + "content": "the effective horizon", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 192, + 148, + 233, + 160 + ], + "score": 0.92, + "content": "( 1 - \\gamma ) ^ { - 1 }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 233, + 147, + 236, + 162 + ], + "score": 1.0, + "content": ".", + "type": "text", + "cross_page": true + } + ], + "index": 6 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 687, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 160 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "improved learning results. However, owing to the aim for tackling general function approximation,", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "those learning bounds are expressed in terms of the stringent concentrability coefficients (therefore,", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 505, + 116 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 505, + 116 + ], + "score": 1.0, + "content": "are less adaptive to individual instances) and are usually only information-theoretical, due to the", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 127 + ], + "score": 1.0, + "content": "computational intractability of the optimization procedure over the general function classes. Other", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 138 + ], + "score": 1.0, + "content": "works impose weaker assumptions (e.g. partial coverage (Liu et al., 2020; Kidambi et al., 2020;", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 136, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 483, + 150 + ], + "score": 1.0, + "content": "Uehara and Sun, 2021)), and their finite sample analysis are generally suboptimal in terms of", + "type": "text" + }, + { + "bbox": [ + 483, + 138, + 493, + 147 + ], + "score": 0.8, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 136, + 506, + 150 + ], + "score": 1.0, + "content": "or", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 147, + 236, + 162 + ], + "spans": [ + { + "bbox": [ + 106, + 147, + 191, + 162 + ], + "score": 1.0, + "content": "the effective horizon", + "type": "text" + }, + { + "bbox": [ + 192, + 148, + 233, + 160 + ], + "score": 0.92, + "content": "( 1 - \\gamma ) ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 147, + 236, + 162 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 107, + 165, + 505, + 264 + ], + "lines": [ + { + "bbox": [ + 106, + 165, + 505, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 505, + 177 + ], + "score": 1.0, + "content": "Offline RL with tabular models. For tabular MDPs, tight learning bounds can be achieved under", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 175, + 505, + 188 + ], + "spans": [ + { + "bbox": [ + 106, + 175, + 505, + 188 + ], + "score": 1.0, + "content": "several data-coverage assumptions. For the class of problems with uniform data-visitation measure", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 186, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 120, + 198 + ], + "score": 0.88, + "content": "d _ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 120, + 186, + 346, + 200 + ], + "score": 1.0, + "content": ", the near-optimal sample complexity bound has the rate", + "type": "text" + }, + { + "bbox": [ + 347, + 187, + 402, + 199 + ], + "score": 0.93, + "content": "O ( H ^ { 3 } / d _ { m } \\epsilon ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 186, + 505, + 200 + ], + "score": 1.0, + "content": "for time-inhomogeneous", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 197, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 106, + 197, + 220, + 210 + ], + "score": 1.0, + "content": "MDPs (Yin et al., 2021) and", + "type": "text" + }, + { + "bbox": [ + 221, + 198, + 272, + 210 + ], + "score": 0.93, + "content": "O ( H ^ { 2 } / d _ { m } \\dot { \\epsilon } )", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 197, + 505, + 210 + ], + "score": 1.0, + "content": "for time-homogeneous MDPs (Yin and Wang, 2021a; Ren", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 208, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 104, + 208, + 394, + 222 + ], + "score": 1.0, + "content": "et al., 2021). Under the single concentrability assumption, the tight rate", + "type": "text" + }, + { + "bbox": [ + 395, + 208, + 457, + 221 + ], + "score": 0.92, + "content": "{ \\cal O } ( H ^ { 3 } S C ^ { \\star } / \\epsilon ^ { \\top } )", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 208, + 506, + 222 + ], + "score": 1.0, + "content": "is obtained", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 219, + 506, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 219, + 506, + 232 + ], + "score": 1.0, + "content": "by Xie et al. (2021b). In particular, the recent study Yin and Wang (2021b) introduces the intrinsic", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 231, + 506, + 243 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 506, + 243 + ], + "score": 1.0, + "content": "offline learning bound that is not only instance-dependent but also subsumes previous optimal results.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 241, + 506, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 506, + 255 + ], + "score": 1.0, + "content": "More recently, Shi et al. (2022) uses model-free approach to achieve minimax rate with a larger", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 107, + 253, + 141, + 267 + ], + "spans": [ + { + "bbox": [ + 107, + 255, + 112, + 263 + ], + "score": 0.49, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 112, + 253, + 141, + 267 + ], + "score": 1.0, + "content": "-range.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 106, + 269, + 505, + 511 + ], + "lines": [ + { + "bbox": [ + 105, + 268, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 505, + 282 + ], + "score": 1.0, + "content": "Offline RL with linear model representations. Recently, there are more focus on studying the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 280, + 506, + 294 + ], + "spans": [ + { + "bbox": [ + 104, + 280, + 506, + 294 + ], + "score": 1.0, + "content": "provable efficient offline RL under the linear model representations. Jin et al. (2021b) first shows", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 290, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 505, + 304 + ], + "score": 1.0, + "content": "offline RL with linear MDP is provably efficient by the pessimistic value iteration (PEVI), which", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 302, + 506, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 506, + 315 + ], + "score": 1.0, + "content": "is an offline counterpart of LSVI-UCB in Jin et al. (2020). Their analysis deviates from their lower", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 314, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 314, + 192, + 325 + ], + "score": 1.0, + "content": "bound by a factor of", + "type": "text" + }, + { + "bbox": [ + 192, + 314, + 216, + 324 + ], + "score": 0.88, + "content": "d \\cdot H", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 314, + 505, + 325 + ], + "score": 1.0, + "content": "(check their Theorem 4.4 and 4.6). Later, Xie et al. (2021a) considers", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 324, + 505, + 337 + ], + "spans": [ + { + "bbox": [ + 106, + 324, + 505, + 337 + ], + "score": 1.0, + "content": "function approximation under the Bellman-consistent assumptions, and, when realized to linear MDP", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 334, + 507, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 422, + 349 + ], + "score": 1.0, + "content": "setting, improve the sample complexity guarantee of Jin et al. (2021b) by a order", + "type": "text" + }, + { + "bbox": [ + 423, + 335, + 444, + 347 + ], + "score": 0.9, + "content": "O ( d )", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 334, + 507, + 349 + ], + "score": 1.0, + "content": "(Theorem 3.2).", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 345, + 506, + 359 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 453, + 359 + ], + "score": 1.0, + "content": "However, their improvement only holds for finite action space (due to the dependence", + "type": "text" + }, + { + "bbox": [ + 454, + 347, + 486, + 358 + ], + "score": 0.87, + "content": "\\log \\left| { \\mathcal { A } } \\right| )", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 345, + 506, + 359 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 357, + 506, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 506, + 370 + ], + "score": 1.0, + "content": "by the direct reduction (from Theorem 3.1) their result does not imply a computationally tractable", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 369, + 505, + 380 + ], + "spans": [ + { + "bbox": [ + 106, + 369, + 505, + 380 + ], + "score": 1.0, + "content": "algorithm. In addition, there is no improvement on the horizon dependence. Concurrently, Zanette", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 378, + 506, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 506, + 392 + ], + "score": 1.0, + "content": "et al. (2021) considers the Linear Bellman Complete model (which originates from its online version", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 389, + 506, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 506, + 403 + ], + "score": 1.0, + "content": "Zanette et al. (2020)) and designs the actor-critic style algorithm that achieves tight result under", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 400, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 505, + 414 + ], + "score": 1.0, + "content": "the assumption that the value function is bounded by 1. While their algorithm is efficient (which is", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 411, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 505, + 425 + ], + "score": 1.0, + "content": "based on solving a sequence of second-order cone programs), the resulting learning bound requires", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 423, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 505, + 436 + ], + "score": 1.0, + "content": "the action space to be finite due to the mirror descent/natural policy gradient updates in the Actor", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 433, + 506, + 447 + ], + "spans": [ + { + "bbox": [ + 104, + 433, + 506, + 447 + ], + "score": 1.0, + "content": "procedure (Agarwal et al., 2021). Besides, assuming the value function to be less than 1 simplifies", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 445, + 506, + 457 + ], + "spans": [ + { + "bbox": [ + 106, + 445, + 258, + 457 + ], + "score": 1.0, + "content": "the challenges in dealing with horizon", + "type": "text" + }, + { + "bbox": [ + 259, + 446, + 269, + 455 + ], + "score": 0.72, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 445, + 402, + 457 + ], + "score": 1.0, + "content": "since when rescale their result to", + "type": "text" + }, + { + "bbox": [ + 402, + 445, + 426, + 457 + ], + "score": 0.92, + "content": "[ 0 , H ]", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 445, + 468, + 457 + ], + "score": 1.0, + "content": ", there is a", + "type": "text" + }, + { + "bbox": [ + 469, + 446, + 479, + 455 + ], + "score": 0.78, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 445, + 506, + 457 + ], + "score": 1.0, + "content": "factor", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 456, + 506, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 506, + 469 + ], + "score": 1.0, + "content": "blow-up, which makes no improvement in the horizon dependence comparing to Jin et al. (2021b).", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 467, + 506, + 479 + ], + "spans": [ + { + "bbox": [ + 106, + 467, + 506, + 479 + ], + "score": 1.0, + "content": "On the other hand, Wang et al. (2021a); Zanette (2021) study the statistical hardness of offline RL", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 476, + 506, + 492 + ], + "spans": [ + { + "bbox": [ + 104, + 476, + 506, + 492 + ], + "score": 1.0, + "content": "with linear representations by proofing the exponential lower bounds. As a result, none of the existing", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 488, + 506, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 506, + 501 + ], + "score": 1.0, + "content": "algorithms can achieve the statistical limit for the well-structured linear MDP model with the general", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 499, + 363, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 363, + 513 + ], + "score": 1.0, + "content": "(infinite or continuous) state-action spaces in the offline regime.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 26.5 + }, + { + "type": "title", + "bbox": [ + 108, + 529, + 251, + 542 + ], + "lines": [ + { + "bbox": [ + 106, + 528, + 253, + 544 + ], + "spans": [ + { + "bbox": [ + 106, + 528, + 253, + 544 + ], + "score": 1.0, + "content": "C PROOFS IN SECTION 3.2", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 108, + 555, + 504, + 578 + ], + "lines": [ + { + "bbox": [ + 105, + 553, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 553, + 235, + 569 + ], + "score": 1.0, + "content": "Instead of proofing the result for", + "type": "text" + }, + { + "bbox": [ + 236, + 555, + 269, + 567 + ], + "score": 0.92, + "content": "v ^ { \\star } - v ^ { \\widehat { \\pi } }", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 553, + 426, + 569 + ], + "score": 1.0, + "content": ", in most parts of the proof we deal with", + "type": "text" + }, + { + "bbox": [ + 427, + 555, + 466, + 568 + ], + "score": 0.94, + "content": "V _ { 1 } ^ { \\star } - V _ { 1 } ^ { \\widehat \\pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 553, + 506, + 569 + ], + "score": 1.0, + "content": ", which is", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 567, + 164, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 164, + 579 + ], + "score": 1.0, + "content": "more general.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39.5 + }, + { + "type": "title", + "bbox": [ + 107, + 594, + 225, + 605 + ], + "lines": [ + { + "bbox": [ + 106, + 593, + 226, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 593, + 226, + 606 + ], + "score": 1.0, + "content": "C.1 SOME PREPARATIONS", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 106, + 613, + 506, + 641 + ], + "lines": [ + { + "bbox": [ + 105, + 613, + 504, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 249, + 629 + ], + "score": 1.0, + "content": "Define the Bellman update error", + "type": "text" + }, + { + "bbox": [ + 249, + 613, + 416, + 628 + ], + "score": 0.94, + "content": "\\zeta _ { h } ( s , a ) \\ : = \\ ( { \\mathcal T } _ { h } \\widehat V _ { h + 1 } ) ( s , a ) \\ - \\ - \\ - \\widetilde Q _ { h } ( s , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 613, + 465, + 629 + ], + "score": 1.0, + "content": "and recall", + "type": "text" + }, + { + "bbox": [ + 465, + 615, + 504, + 627 + ], + "score": 0.89, + "content": "\\widehat { \\pi } _ { h } ( s ) \\ =", + "type": "inline_equation" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 626, + 423, + 643 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 122, + 643 + ], + "score": 1.0, + "content": "arg", + "type": "text" + }, + { + "bbox": [ + 123, + 626, + 238, + 641 + ], + "score": 0.93, + "content": "\\operatorname* { m a x } _ { \\pi _ { h } } \\langle \\widehat { Q } _ { h } ( s , \\cdot ) , \\pi _ { h } ( \\cdot \\mid s ) \\rangle _ { \\mathcal { A } }", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 626, + 423, + 643 + ], + "score": 1.0, + "content": ", then by the direct application of Lemma H.8", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 42.5 + }, + { + "type": "interline_equation", + "bbox": [ + 144, + 648, + 467, + 683 + ], + "lines": [ + { + "bbox": [ + 144, + 648, + 467, + 683 + ], + "spans": [ + { + "bbox": [ + 144, + 648, + 467, + 683 + ], + "score": 0.93, + "content": "V _ { 1 } ^ { \\pi } ( s ) - V _ { 1 } ^ { \\widehat { \\pi } } ( s ) \\leq \\sum _ { h = 1 } ^ { H } \\mathbb { E } _ { \\pi } \\left[ \\zeta _ { h } ( s _ { h } , a _ { h } ) \\ | \\ s _ { 1 } = s \\right] - \\sum _ { h = 1 } ^ { H } \\mathbb { E } _ { \\widehat { \\pi } } \\left[ \\zeta _ { h } ( s _ { h } , a _ { h } ) \\ | \\ s _ { 1 } = s \\right] .", + "type": "interline_equation", + "image_path": "f036dd51f54ff83e1fbc2531865d476277df7179bdb966b7d011606d688224d9.jpg" + } + ] + } + ], + "index": 45, + "virtual_lines": [ + { + "bbox": [ + 144, + 648, + 467, + 659.6666666666666 + ], + "spans": [], + "index": 44 + }, + { + "bbox": [ + 144, + 659.6666666666666, + 467, + 671.3333333333333 + ], + "spans": [], + "index": 45 + }, + { + "bbox": [ + 144, + 671.3333333333333, + 467, + 682.9999999999999 + ], + "spans": [], + "index": 46 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 689, + 505, + 702 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 506, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 703 + ], + "score": 1.0, + "content": "The next lemma shows it is sufficient to bound the pessimistic penalty, which is the key in the proof.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 47 + }, + { + "type": "text", + "bbox": [ + 106, + 706, + 505, + 733 + ], + "lines": [ + { + "bbox": [ + 105, + 704, + 506, + 721 + ], + "spans": [ + { + "bbox": [ + 105, + 704, + 262, + 721 + ], + "score": 1.0, + "content": "Lemma C.1. Suppose with probability", + "type": "text" + }, + { + "bbox": [ + 262, + 708, + 284, + 718 + ], + "score": 0.72, + "content": "1 - \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 704, + 346, + 721 + ], + "score": 1.0, + "content": ", it holds for all", + "type": "text" + }, + { + "bbox": [ + 346, + 707, + 436, + 720 + ], + "score": 0.91, + "content": "h , s , a \\in [ H ] \\times S \\times A", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 704, + 455, + 721 + ], + "score": 1.0, + "content": "that", + "type": "text" + }, + { + "bbox": [ + 455, + 705, + 506, + 720 + ], + "score": 0.9, + "content": "| ( \\widehat { T _ { h } } \\widehat { V _ { h + 1 } } -", + "type": "inline_equation" + } + ], + "index": 48 + }, + { + "bbox": [ + 107, + 718, + 506, + 735 + ], + "spans": [ + { + "bbox": [ + 107, + 718, + 218, + 733 + ], + "score": 0.91, + "content": "\\widehat { \\mathcal { T } } _ { h } \\widehat { V } _ { h + 1 } ) ( s , a ) | \\ \\leq \\ \\Gamma _ { h } ( s , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 718, + 285, + 735 + ], + "score": 1.0, + "content": ", then it implies", + "type": "text" + }, + { + "bbox": [ + 285, + 720, + 389, + 732 + ], + "score": 0.65, + "content": "\\forall s , a , h \\in \\mathcal { S } \\times \\mathcal { A } \\times [ H ]", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 718, + 393, + 735 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 394, + 720, + 502, + 732 + ], + "score": 0.81, + "content": "0 \\leq \\zeta _ { h } ( s , a ) \\leq 2 \\Gamma _ { h } ( s , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 718, + 506, + 735 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 48.5 + } + ], + "page_idx": 15, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 765 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 765 + ], + "score": 1.0, + "content": "16", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 160 + ], + "lines": [], + "index": 3, + "bbox_fs": [ + 105, + 82, + 506, + 162 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 165, + 505, + 264 + ], + "lines": [ + { + "bbox": [ + 106, + 165, + 505, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 505, + 177 + ], + "score": 1.0, + "content": "Offline RL with tabular models. For tabular MDPs, tight learning bounds can be achieved under", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 175, + 505, + 188 + ], + "spans": [ + { + "bbox": [ + 106, + 175, + 505, + 188 + ], + "score": 1.0, + "content": "several data-coverage assumptions. For the class of problems with uniform data-visitation measure", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 186, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 120, + 198 + ], + "score": 0.88, + "content": "d _ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 120, + 186, + 346, + 200 + ], + "score": 1.0, + "content": ", the near-optimal sample complexity bound has the rate", + "type": "text" + }, + { + "bbox": [ + 347, + 187, + 402, + 199 + ], + "score": 0.93, + "content": "O ( H ^ { 3 } / d _ { m } \\epsilon ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 186, + 505, + 200 + ], + "score": 1.0, + "content": "for time-inhomogeneous", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 197, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 106, + 197, + 220, + 210 + ], + "score": 1.0, + "content": "MDPs (Yin et al., 2021) and", + "type": "text" + }, + { + "bbox": [ + 221, + 198, + 272, + 210 + ], + "score": 0.93, + "content": "O ( H ^ { 2 } / d _ { m } \\dot { \\epsilon } )", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 197, + 505, + 210 + ], + "score": 1.0, + "content": "for time-homogeneous MDPs (Yin and Wang, 2021a; Ren", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 208, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 104, + 208, + 394, + 222 + ], + "score": 1.0, + "content": "et al., 2021). Under the single concentrability assumption, the tight rate", + "type": "text" + }, + { + "bbox": [ + 395, + 208, + 457, + 221 + ], + "score": 0.92, + "content": "{ \\cal O } ( H ^ { 3 } S C ^ { \\star } / \\epsilon ^ { \\top } )", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 208, + 506, + 222 + ], + "score": 1.0, + "content": "is obtained", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 219, + 506, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 219, + 506, + 232 + ], + "score": 1.0, + "content": "by Xie et al. (2021b). In particular, the recent study Yin and Wang (2021b) introduces the intrinsic", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 231, + 506, + 243 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 506, + 243 + ], + "score": 1.0, + "content": "offline learning bound that is not only instance-dependent but also subsumes previous optimal results.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 241, + 506, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 506, + 255 + ], + "score": 1.0, + "content": "More recently, Shi et al. 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Recently, there are more focus on studying the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 280, + 506, + 294 + ], + "spans": [ + { + "bbox": [ + 104, + 280, + 506, + 294 + ], + "score": 1.0, + "content": "provable efficient offline RL under the linear model representations. Jin et al. (2021b) first shows", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 290, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 505, + 304 + ], + "score": 1.0, + "content": "offline RL with linear MDP is provably efficient by the pessimistic value iteration (PEVI), which", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 302, + 506, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 506, + 315 + ], + "score": 1.0, + "content": "is an offline counterpart of LSVI-UCB in Jin et al. (2020). Their analysis deviates from their lower", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 314, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 314, + 192, + 325 + ], + "score": 1.0, + "content": "bound by a factor of", + "type": "text" + }, + { + "bbox": [ + 192, + 314, + 216, + 324 + ], + "score": 0.88, + "content": "d \\cdot H", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 314, + 505, + 325 + ], + "score": 1.0, + "content": "(check their Theorem 4.4 and 4.6). Later, Xie et al. (2021a) considers", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 324, + 505, + 337 + ], + "spans": [ + { + "bbox": [ + 106, + 324, + 505, + 337 + ], + "score": 1.0, + "content": "function approximation under the Bellman-consistent assumptions, and, when realized to linear MDP", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 334, + 507, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 422, + 349 + ], + "score": 1.0, + "content": "setting, improve the sample complexity guarantee of Jin et al. (2021b) by a order", + "type": "text" + }, + { + "bbox": [ + 423, + 335, + 444, + 347 + ], + "score": 0.9, + "content": "O ( d )", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 334, + 507, + 349 + ], + "score": 1.0, + "content": "(Theorem 3.2).", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 345, + 506, + 359 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 453, + 359 + ], + "score": 1.0, + "content": "However, their improvement only holds for finite action space (due to the dependence", + "type": "text" + }, + { + "bbox": [ + 454, + 347, + 486, + 358 + ], + "score": 0.87, + "content": "\\log \\left| { \\mathcal { A } } \\right| )", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 345, + 506, + 359 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 357, + 506, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 506, + 370 + ], + "score": 1.0, + "content": "by the direct reduction (from Theorem 3.1) their result does not imply a computationally tractable", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 369, + 505, + 380 + ], + "spans": [ + { + "bbox": [ + 106, + 369, + 505, + 380 + ], + "score": 1.0, + "content": "algorithm. In addition, there is no improvement on the horizon dependence. Concurrently, Zanette", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 378, + 506, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 506, + 392 + ], + "score": 1.0, + "content": "et al. (2021) considers the Linear Bellman Complete model (which originates from its online version", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 389, + 506, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 506, + 403 + ], + "score": 1.0, + "content": "Zanette et al. (2020)) and designs the actor-critic style algorithm that achieves tight result under", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 400, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 505, + 414 + ], + "score": 1.0, + "content": "the assumption that the value function is bounded by 1. While their algorithm is efficient (which is", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 411, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 505, + 425 + ], + "score": 1.0, + "content": "based on solving a sequence of second-order cone programs), the resulting learning bound requires", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 423, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 505, + 436 + ], + "score": 1.0, + "content": "the action space to be finite due to the mirror descent/natural policy gradient updates in the Actor", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 433, + 506, + 447 + ], + "spans": [ + { + "bbox": [ + 104, + 433, + 506, + 447 + ], + "score": 1.0, + "content": "procedure (Agarwal et al., 2021). Besides, assuming the value function to be less than 1 simplifies", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 445, + 506, + 457 + ], + "spans": [ + { + "bbox": [ + 106, + 445, + 258, + 457 + ], + "score": 1.0, + "content": "the challenges in dealing with horizon", + "type": "text" + }, + { + "bbox": [ + 259, + 446, + 269, + 455 + ], + "score": 0.72, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 445, + 402, + 457 + ], + "score": 1.0, + "content": "since when rescale their result to", + "type": "text" + }, + { + "bbox": [ + 402, + 445, + 426, + 457 + ], + "score": 0.92, + "content": "[ 0 , H ]", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 445, + 468, + 457 + ], + "score": 1.0, + "content": ", there is a", + "type": "text" + }, + { + "bbox": [ + 469, + 446, + 479, + 455 + ], + "score": 0.78, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 445, + 506, + 457 + ], + "score": 1.0, + "content": "factor", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 456, + 506, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 506, + 469 + ], + "score": 1.0, + "content": "blow-up, which makes no improvement in the horizon dependence comparing to Jin et al. (2021b).", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 467, + 506, + 479 + ], + "spans": [ + { + "bbox": [ + 106, + 467, + 506, + 479 + ], + "score": 1.0, + "content": "On the other hand, Wang et al. (2021a); Zanette (2021) study the statistical hardness of offline RL", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 476, + 506, + 492 + ], + "spans": [ + { + "bbox": [ + 104, + 476, + 506, + 492 + ], + "score": 1.0, + "content": "with linear representations by proofing the exponential lower bounds. As a result, none of the existing", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 488, + 506, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 506, + 501 + ], + "score": 1.0, + "content": "algorithms can achieve the statistical limit for the well-structured linear MDP model with the general", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 499, + 363, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 363, + 513 + ], + "score": 1.0, + "content": "(infinite or continuous) state-action spaces in the offline regime.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 26.5, + "bbox_fs": [ + 104, + 268, + 507, + 513 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 529, + 251, + 542 + ], + "lines": [ + { + "bbox": [ + 106, + 528, + 253, + 544 + ], + "spans": [ + { + "bbox": [ + 106, + 528, + 253, + 544 + ], + "score": 1.0, + "content": "C PROOFS IN SECTION 3.2", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 108, + 555, + 504, + 578 + ], + "lines": [ + { + "bbox": [ + 105, + 553, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 553, + 235, + 569 + ], + "score": 1.0, + "content": "Instead of proofing the result for", + "type": "text" + }, + { + "bbox": [ + 236, + 555, + 269, + 567 + ], + "score": 0.92, + "content": "v ^ { \\star } - v ^ { \\widehat { \\pi } }", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 553, + 426, + 569 + ], + "score": 1.0, + "content": ", in most parts of the proof we deal with", + "type": "text" + }, + { + "bbox": [ + 427, + 555, + 466, + 568 + ], + "score": 0.94, + "content": "V _ { 1 } ^ { \\star } - V _ { 1 } ^ { \\widehat \\pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 553, + 506, + 569 + ], + "score": 1.0, + "content": ", which is", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 567, + 164, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 164, + 579 + ], + "score": 1.0, + "content": "more general.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39.5, + "bbox_fs": [ + 105, + 553, + 506, + 579 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 594, + 225, + 605 + ], + "lines": [ + { + "bbox": [ + 106, + 593, + 226, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 593, + 226, + 606 + ], + "score": 1.0, + "content": "C.1 SOME PREPARATIONS", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 106, + 613, + 506, + 641 + ], + "lines": [ + { + "bbox": [ + 105, + 613, + 504, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 249, + 629 + ], + "score": 1.0, + "content": "Define the Bellman update error", + "type": "text" + }, + { + "bbox": [ + 249, + 613, + 416, + 628 + ], + "score": 0.94, + "content": "\\zeta _ { h } ( s , a ) \\ : = \\ ( { \\mathcal T } _ { h } \\widehat V _ { h + 1 } ) ( s , a ) \\ - \\ - \\ - \\widetilde Q _ { h } ( s , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 613, + 465, + 629 + ], + "score": 1.0, + "content": "and recall", + "type": "text" + }, + { + "bbox": [ + 465, + 615, + 504, + 627 + ], + "score": 0.89, + "content": "\\widehat { \\pi } _ { h } ( s ) \\ =", + "type": "inline_equation" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 626, + 423, + 643 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 122, + 643 + ], + "score": 1.0, + "content": "arg", + "type": "text" + }, + { + "bbox": [ + 123, + 626, + 238, + 641 + ], + "score": 0.93, + "content": "\\operatorname* { m a x } _ { \\pi _ { h } } \\langle \\widehat { Q } _ { h } ( s , \\cdot ) , \\pi _ { h } ( \\cdot \\mid s ) \\rangle _ { \\mathcal { A } }", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 626, + 423, + 643 + ], + "score": 1.0, + "content": ", then by the direct application of Lemma H.8", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 42.5, + "bbox_fs": [ + 105, + 613, + 504, + 643 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 144, + 648, + 467, + 683 + ], + "lines": [ + { + "bbox": [ + 144, + 648, + 467, + 683 + ], + "spans": [ + { + "bbox": [ + 144, + 648, + 467, + 683 + ], + "score": 0.93, + "content": "V _ { 1 } ^ { \\pi } ( s ) - V _ { 1 } ^ { \\widehat { \\pi } } ( s ) \\leq \\sum _ { h = 1 } ^ { H } \\mathbb { E } _ { \\pi } \\left[ \\zeta _ { h } ( s _ { h } , a _ { h } ) \\ | \\ s _ { 1 } = s \\right] - \\sum _ { h = 1 } ^ { H } \\mathbb { E } _ { \\widehat { \\pi } } \\left[ \\zeta _ { h } ( s _ { h } , a _ { h } ) \\ | \\ s _ { 1 } = s \\right] .", + "type": "interline_equation", + "image_path": "f036dd51f54ff83e1fbc2531865d476277df7179bdb966b7d011606d688224d9.jpg" + } + ] + } + ], + "index": 45, + "virtual_lines": [ + { + "bbox": [ + 144, + 648, + 467, + 659.6666666666666 + ], + "spans": [], + "index": 44 + }, + { + "bbox": [ + 144, + 659.6666666666666, + 467, + 671.3333333333333 + ], + "spans": [], + "index": 45 + }, + { + "bbox": [ + 144, + 671.3333333333333, + 467, + 682.9999999999999 + ], + "spans": [], + "index": 46 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 689, + 505, + 702 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 506, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 703 + ], + "score": 1.0, + "content": "The next lemma shows it is sufficient to bound the pessimistic penalty, which is the key in the proof.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 47, + "bbox_fs": [ + 105, + 687, + 506, + 703 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 706, + 505, + 733 + ], + "lines": [ + { + "bbox": [ + 105, + 704, + 506, + 721 + ], + "spans": [ + { + "bbox": [ + 105, + 704, + 262, + 721 + ], + "score": 1.0, + "content": "Lemma C.1. Suppose with probability", + "type": "text" + }, + { + "bbox": [ + 262, + 708, + 284, + 718 + ], + "score": 0.72, + "content": "1 - \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 704, + 346, + 721 + ], + "score": 1.0, + "content": ", it holds for all", + "type": "text" + }, + { + "bbox": [ + 346, + 707, + 436, + 720 + ], + "score": 0.91, + "content": "h , s , a \\in [ H ] \\times S \\times A", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 704, + 455, + 721 + ], + "score": 1.0, + "content": "that", + "type": "text" + }, + { + "bbox": [ + 455, + 705, + 506, + 720 + ], + "score": 0.9, + "content": "| ( \\widehat { T _ { h } } \\widehat { V _ { h + 1 } } -", + "type": "inline_equation" + } + ], + "index": 48 + }, + { + "bbox": [ + 107, + 718, + 506, + 735 + ], + "spans": [ + { + "bbox": [ + 107, + 718, + 218, + 733 + ], + "score": 0.91, + "content": "\\widehat { \\mathcal { T } } _ { h } \\widehat { V } _ { h + 1 } ) ( s , a ) | \\ \\leq \\ \\Gamma _ { h } ( s , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 718, + 285, + 735 + ], + "score": 1.0, + "content": ", then it implies", + "type": "text" + }, + { + "bbox": [ + 285, + 720, + 389, + 732 + ], + "score": 0.65, + "content": "\\forall s , a , h \\in \\mathcal { S } \\times \\mathcal { A } \\times [ H ]", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 718, + 393, + 735 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 394, + 720, + 502, + 732 + ], + "score": 0.81, + "content": "0 \\leq \\zeta _ { h } ( s , a ) \\leq 2 \\Gamma _ { h } ( s , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 718, + 506, + 735 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 48.5, + "bbox_fs": [ + 105, + 704, + 506, + 735 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 81, + 416, + 95 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 417, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 251, + 96 + ], + "score": 1.0, + "content": "Furthermore, it holds for any policy", + "type": "text" + }, + { + "bbox": [ + 252, + 85, + 259, + 92 + ], + "score": 0.65, + "content": "\\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 82, + 390, + 96 + ], + "score": 1.0, + "content": "simultaneously, with probability", + "type": "text" + }, + { + "bbox": [ + 391, + 83, + 413, + 93 + ], + "score": 0.74, + "content": "1 - \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 82, + 417, + 96 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 199, + 104, + 411, + 139 + ], + "lines": [ + { + "bbox": [ + 199, + 104, + 411, + 139 + ], + "spans": [ + { + "bbox": [ + 199, + 104, + 411, + 139 + ], + "score": 0.92, + "content": "V _ { 1 } ^ { \\pi } ( s ) - V _ { 1 } ^ { \\widehat { \\pi } } ( s ) \\leq \\sum _ { h = 1 } ^ { H } 2 \\cdot \\mathbb { E } _ { \\pi } \\left[ \\Gamma _ { h } ( s _ { h } , a _ { h } ) \\mid s _ { 1 } = s \\right] .", + "type": "interline_equation", + "image_path": "18e91ffb4e85d3e8f561a4a1a041d7b87c759fc6d168a61eb7fe3f9db99507f4.jpg" + } + ] + } + ], + "index": 1.5, + "virtual_lines": [ + { + "bbox": [ + 199, + 104, + 411, + 121.5 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 199, + 121.5, + 411, + 139.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 168, + 505, + 194 + ], + "lines": [ + { + "bbox": [ + 104, + 166, + 505, + 186 + ], + "spans": [ + { + "bbox": [ + 104, + 166, + 285, + 186 + ], + "score": 1.0, + "content": "Proof of Lemma C.1. We first show given", + "type": "text" + }, + { + "bbox": [ + 286, + 168, + 455, + 183 + ], + "score": 0.91, + "content": "| ( \\mathcal { T } _ { h } \\widehat { V } _ { h + 1 } - \\widehat { \\mathcal { T } } _ { h } \\widehat { V } _ { h + 1 } ) ( s , a ) | \\ \\leq \\ \\Gamma _ { h } ( s , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 166, + 483, + 186 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 483, + 171, + 505, + 183 + ], + "score": 0.82, + "content": "0 ~ \\leq", + "type": "inline_equation" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 180, + 297, + 195 + ], + "spans": [ + { + "bbox": [ + 106, + 182, + 292, + 194 + ], + "score": 0.83, + "content": "\\zeta _ { h } ( s , a ) \\le 2 \\Gamma _ { h } ( s , a ) , \\forall s , a , h \\in \\mathcal { S } \\times \\bar { \\mathcal { A } } \\times [ H ]", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 180, + 297, + 195 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 105, + 198, + 348, + 211 + ], + "lines": [ + { + "bbox": [ + 106, + 198, + 348, + 212 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 192, + 212 + ], + "score": 1.0, + "content": "Step1: we first show", + "type": "text" + }, + { + "bbox": [ + 192, + 198, + 345, + 211 + ], + "score": 0.9, + "content": "0 \\leq \\zeta _ { h } ( s , a ) , \\forall s , a , h \\in S \\times \\mathcal { A } \\times [ H ] .", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 198, + 348, + 212 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 106, + 216, + 506, + 255 + ], + "lines": [ + { + "bbox": [ + 104, + 215, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 104, + 215, + 154, + 232 + ], + "score": 1.0, + "content": "Indeed, if", + "type": "text" + }, + { + "bbox": [ + 154, + 217, + 219, + 230 + ], + "score": 0.88, + "content": "\\bar { Q } _ { h } ( s , a ) ~ \\leq ~ 0", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 215, + 308, + 232 + ], + "score": 1.0, + "content": ", then by definition", + "type": "text" + }, + { + "bbox": [ + 308, + 216, + 373, + 230 + ], + "score": 0.9, + "content": "\\widehat { Q } _ { h } ( s , a ) ~ = ~ 0", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 215, + 452, + 232 + ], + "score": 1.0, + "content": "and in this case", + "type": "text" + }, + { + "bbox": [ + 452, + 218, + 505, + 230 + ], + "score": 0.87, + "content": "\\zeta _ { h } ( s , a ) : =", + "type": "inline_equation" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 228, + 505, + 245 + ], + "spans": [ + { + "bbox": [ + 105, + 229, + 317, + 244 + ], + "score": 0.83, + "content": "( { \\mathcal { T } } _ { h } { \\widehat { V } } _ { h + 1 } ) ( s , a ) - { \\widehat { Q } } _ { h } ( s , a ) = ( { \\mathcal { T } } _ { h } { \\widehat { V } } _ { h + 1 } ) ( s , a ) \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 228, + 332, + 245 + ], + "score": 1.0, + "content": "; 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Note", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 101, + 421, + 510, + 448 + ], + "spans": [ + { + "bbox": [ + 101, + 421, + 165, + 448 + ], + "score": 1.0, + "content": "conditional on", + "type": "text" + }, + { + "bbox": [ + 165, + 430, + 172, + 441 + ], + "score": 0.64, + "content": "\\mathfrak { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 421, + 250, + 448 + ], + "score": 1.0, + "content": ", then by equation 8,", + "type": "text" + }, + { + "bbox": [ + 250, + 428, + 467, + 443 + ], + "score": 0.9, + "content": "\\begin{array} { r } { , V _ { 1 } ^ { \\pi } ( s ) - V _ { 1 } ^ { \\widehat { \\pi } } ( s ) \\leq \\sum _ { h = 1 } ^ { H } 2 \\cdot \\mathbb { E } _ { \\pi } [ \\Gamma _ { h } ( s _ { h } , a _ { h } ) \\mid s _ { 1 } = s ] } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 421, + 510, + 448 + ], + "score": 1.0, + "content": "holds for", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 440, + 262, + 455 + ], + "spans": [ + { + "bbox": [ + 106, + 440, + 150, + 455 + ], + "score": 1.0, + "content": "any policy", + "type": "text" + }, + { + "bbox": [ + 150, + 443, + 158, + 451 + ], + "score": 0.77, + "content": "\\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 440, + 262, + 455 + ], + "score": 1.0, + "content": "almost surely. Therefore,", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19, + "bbox_fs": [ + 101, + 414, + 510, + 455 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 119, + 462, + 491, + 605 + ], + "lines": [ + { + "bbox": [ + 119, + 462, + 491, + 605 + ], + "spans": [ + { + "bbox": [ + 119, + 462, + 491, + 605 + ], + "score": 0.95, + "content": "\\begin{array} { r l } & { \\mathbb { P } \\left[ \\forall \\pi , \\ V _ { 1 } ^ { \\pi } ( s ) - V _ { 1 } ^ { \\widehat { \\pi } } ( s ) \\leq \\displaystyle \\sum _ { h = 1 } ^ { H } 2 \\cdot \\mathbb { E } _ { \\pi } \\big [ \\Gamma _ { h } ( s _ { h } , a _ { h } ) \\mid s _ { 1 } = s \\big ] . \\right] } \\\\ & { = \\mathbb { P } \\Bigg [ \\forall \\pi , \\ V _ { 1 } ^ { \\pi } ( s ) - V _ { 1 } ^ { \\widehat { \\pi } } ( s ) \\leq \\displaystyle \\sum _ { h = 1 } ^ { H } 2 \\cdot \\mathbb { E } _ { \\pi } \\big [ \\Gamma _ { h } ( s _ { h } , a _ { h } ) \\mid s _ { 1 } = s \\bigg ] \\Bigg | \\mathfrak { F } \\Bigg ] \\cdot \\mathbb { P } [ \\mathfrak { F } ] } \\\\ & { + \\mathbb { P } \\left[ \\forall \\pi , \\ V _ { 1 } ^ { \\pi } ( s ) - V _ { 1 } ^ { \\widehat { \\pi } } ( s ) \\leq \\displaystyle \\sum _ { h = 1 } ^ { H } 2 \\cdot \\mathbb { R } _ { \\pi } \\big [ \\Gamma _ { h } ( s _ { h } , a _ { h } ) \\mid s _ { 1 } = s \\big ] \\Bigg | \\mathfrak { F } ^ { \\overline { { \\kappa } } } \\right] \\cdot \\mathbb { P } [ \\mathfrak { F } ^ { c } ] } \\\\ & { \\geq \\mathbb { P } \\Bigg [ \\forall \\pi , \\ V _ { 1 } ^ { \\pi } ( s ) - V _ { 1 } ^ { \\widehat { \\pi } } ( s ) \\leq \\displaystyle \\sum _ { h = 1 } ^ { H } 2 \\cdot \\mathbb { E } _ { \\pi } \\big [ \\Gamma _ { h } ( s _ { h } , a _ { h } ) \\mid s _ { 1 } = s \\big ] \\Bigg | \\mathfrak { F } \\Bigg ] \\cdot \\mathbb { P } [ \\mathfrak { F } ] \\geq 1 \\cdot \\mathbb { P } [ \\mathfrak { F } ] \\geq 1 - \\delta , } \\end{array}", + "type": "interline_equation", + "image_path": "08a12ee17209e642ceeb368aecea0c8b993be64f0c3e03b6f240f1c5548412be.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 119, + 462, + 491, + 509.6666666666667 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 119, + 509.6666666666667, + 491, + 557.3333333333334 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 119, + 557.3333333333334, + 491, + 605.0 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 628, + 207, + 640 + ], + "lines": [ + { + "bbox": [ + 106, + 627, + 207, + 641 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 207, + 641 + ], + "score": 1.0, + "content": "which finishes the proof.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24, + "bbox_fs": [ + 106, + 627, + 207, + 641 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 674, + 329, + 694 + ], + "lines": [ + { + "bbox": [ + 103, + 671, + 328, + 695 + ], + "spans": [ + { + "bbox": [ + 103, + 671, + 182, + 695 + ], + "score": 1.0, + "content": "C.2 BOUNDING", + "type": "text" + }, + { + "bbox": [ + 183, + 674, + 328, + 695 + ], + "score": 0.81, + "content": "\\Big | ( { \\mathcal T } _ { h } \\widehat { V } _ { h + 1 } ) ( s , a ) - ( \\widehat { { \\mathcal T } } _ { h } \\widehat { V } _ { h + 1 } ) ( s , a ) \\Big | .", + "type": "inline_equation" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 106, + 705, + 504, + 734 + ], + "lines": [ + { + "bbox": [ + 105, + 704, + 506, + 720 + ], + "spans": [ + { + "bbox": [ + 105, + 704, + 247, + 720 + ], + "score": 1.0, + "content": "By Lemma C.1, it remains to bound", + "type": "text" + }, + { + "bbox": [ + 247, + 705, + 385, + 720 + ], + "score": 0.92, + "content": "| ( { \\mathcal T } _ { h } \\widehat { V } _ { h + 1 } ) ( s , a ) - ( \\widehat { \\mathcal T } _ { h } \\widehat { V } _ { h + 1 } ) ( s , a ) |", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 704, + 424, + 720 + ], + "score": 1.0, + "content": ". 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\\left( \\widehat { \\mathcal { T } } _ { h } \\widehat { V } _ { h + 1 } \\right) ( s , a ) = \\phi ( s , a ) ^ { \\top } ( w _ { h } - \\widehat { w } _ { h } ) } \\\\ & { = \\phi ( s , a ) ^ { \\top } w _ { h } - \\phi ( s , a ) ^ { \\top } \\widehat { \\Lambda } _ { h } ^ { - 1 } \\left( \\displaystyle \\sum _ { \\tau = 1 } ^ { K } \\phi \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\cdot \\left( r _ { h } ^ { \\tau } + \\widehat { V } _ { h + 1 } \\left( s _ { h + 1 } ^ { \\tau } \\right) \\right) / \\widehat { \\sigma } _ { h } ^ { 2 } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) \\right) } \\\\ & { = \\phi ( s , a ) ^ { \\top } w _ { h } - \\phi ( s , a ) ^ { \\top } \\widehat { \\Lambda } _ { h } ^ { - 1 } \\left( \\displaystyle \\sum _ { \\tau = 1 } ^ { K } \\phi \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\cdot \\left( \\mathcal { T } _ { h } \\widehat { V } _ { h + 1 } \\right) \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) / \\widehat { \\sigma } _ { h } ^ { 2 } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) \\right) } \\\\ & { { \\overset { { , } } { , } } } \\end{array}", + "type": "interline_equation", + "image_path": "474425d8f3846016016b019bd799c067bde88f60244a4b9b14f5dce2ae45fa1d.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 103, + 104, + 444, + 135.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 103, + 135.0, + 444, + 166.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 103, + 166.0, + 444, + 197.0 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "interline_equation", + "bbox": [ + 133, + 213, + 510, + 246 + ], + "lines": [ + { + "bbox": [ + 133, + 213, + 510, + 246 + ], + "spans": [ + { + "bbox": [ + 133, + 213, + 510, + 246 + ], + "score": 0.77, + "content": "+ \\phi ( s , a ) ^ { \\top } \\widehat { \\Lambda } _ { h } ^ { - 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Suppose", + "type": "text" + }, + { + "bbox": [ + 324, + 295, + 503, + 308 + ], + "score": 0.87, + "content": "\\begin{array} { r } { K \\ge \\operatorname* { m a x } \\left\\{ 5 1 2 H ^ { 4 } / \\kappa ^ { 2 } \\log \\left( \\frac { 2 d } { \\delta } \\right) , 4 \\lambda H ^ { 2 } / \\kappa \\right\\} } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 292, + 507, + 309 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 305, + 338, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 305, + 192, + 321 + ], + "score": 1.0, + "content": "then with probability", + "type": "text" + }, + { + "bbox": [ + 192, + 307, + 216, + 318 + ], + "score": 0.68, + "content": "1 - \\delta _ { : }", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 305, + 246, + 321 + ], + "score": 1.0, + "content": ", for all", + "type": "text" + }, + { + "bbox": [ + 246, + 307, + 338, + 319 + ], + "score": 0.9, + "content": "s , a , h \\in \\mathcal S \\times \\mathcal A \\times [ H ]", + "type": "inline_equation" + } + ], + "index": 9 + } + ], + "index": 8.5 + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 329, + 504, + 363 + ], + "lines": [ + { + "bbox": [ + 111, + 329, + 504, + 363 + ], + "spans": [ + { + "bbox": [ + 111, + 329, + 504, + 363 + ], + "score": 0.91, + "content": "\\phi ( s , a ) ^ { \\top } w _ { h } - \\phi ( s , a ) ^ { \\top } \\widehat { \\Lambda } _ { h } ^ { - 1 } \\left( \\sum _ { \\tau = 1 } ^ { K } \\phi \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\cdot \\left( \\mathcal { T } _ { h } \\widehat { V } _ { h + 1 } \\right) \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) / \\widehat { \\sigma } ^ { 2 } \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\right) \\Bigg | \\leq \\frac { 2 \\lambda H ^ { 3 } \\sqrt { d } / \\kappa } { K } .", + "type": "interline_equation", + "image_path": "5df864ea3b54fac8520c9eb7b7537a004c4de3dc7587c274078d3cac9f38c97d.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 111, + 329, + 504, + 340.3333333333333 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 111, + 340.3333333333333, + 504, + 351.66666666666663 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 111, + 351.66666666666663, + 504, + 362.99999999999994 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 391, + 505, + 417 + ], + "lines": [ + { + "bbox": [ + 105, + 390, + 505, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 165, + 407 + ], + "score": 1.0, + "content": "Proof. 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\\phi ( s , a ) ^ { \\top } \\widehat \\Lambda _ { h } ^ { - 1 } \\left( \\displaystyle \\sum _ { \\tau = 1 } ^ { K } \\phi \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\cdot \\left( \\mathcal { T } _ { h } \\widehat { V } _ { h + 1 } \\right) \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) / \\widehat \\sigma ^ { 2 } \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\right) } \\\\ & { = \\phi ( s , a ) ^ { \\top } w _ { h } - \\phi ( s , a ) ^ { \\top } \\widehat \\Lambda _ { h } ^ { - 1 } \\left( \\displaystyle \\sum _ { \\tau = 1 } ^ { K } \\phi \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\cdot \\phi ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) ^ { \\top } w _ { h } / \\widehat \\sigma ^ { 2 } \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\right) } \\\\ & { = \\phi ( s , a ) ^ { \\top } w _ { h } - \\phi ( s , a ) ^ { \\top } \\widehat \\Lambda _ { h } ^ { - 1 } \\left( \\widehat { \\Lambda } _ { h } - 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\\left( \\widehat { \\mathcal { T } } _ { h } \\widehat { V } _ { h + 1 } \\right) ( s , a ) = \\phi ( s , a ) ^ { \\top } ( w _ { h } - \\widehat { w } _ { h } ) } \\\\ & { = \\phi ( s , a ) ^ { \\top } w _ { h } - \\phi ( s , a ) ^ { \\top } \\widehat { \\Lambda } _ { h } ^ { - 1 } \\left( \\displaystyle \\sum _ { \\tau = 1 } ^ { K } \\phi \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\cdot \\left( r _ { h } ^ { \\tau } + \\widehat { V } _ { h + 1 } \\left( s _ { h + 1 } ^ { \\tau } \\right) \\right) / \\widehat { \\sigma } _ { h } ^ { 2 } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) \\right) } \\\\ & { = \\phi ( s , a ) ^ { \\top } w _ { h } - \\phi ( s , a ) ^ { \\top } \\widehat { \\Lambda } _ { h } ^ { - 1 } \\left( \\displaystyle \\sum _ { \\tau = 1 } ^ { K } \\phi \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\cdot \\left( \\mathcal { T } _ { h } \\widehat { V } _ { h + 1 } \\right) \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) / \\widehat { \\sigma } _ { h } ^ { 2 } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) \\right) } \\\\ & { { \\overset { { , } } { , } } } \\end{array}", + "type": "interline_equation", + "image_path": "474425d8f3846016016b019bd799c067bde88f60244a4b9b14f5dce2ae45fa1d.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 103, + 104, + 444, + 135.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 103, + 135.0, + 444, + 166.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 103, + 166.0, + 444, + 197.0 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "interline_equation", + "bbox": [ + 133, + 213, + 510, + 246 + ], + "lines": [ + { + "bbox": [ + 133, + 213, + 510, + 246 + ], + "spans": [ + { + "bbox": [ + 133, + 213, + 510, + 246 + ], + "score": 0.77, + "content": "+ \\phi ( s , a ) ^ { \\top } \\widehat { \\Lambda } _ { h } ^ { - 1 } \\left( \\sum _ { \\tau = 1 } ^ { K } \\phi \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\cdot \\left( r _ { h } ^ { \\tau } + \\widehat { V } _ { h + 1 } \\left( s _ { h + 1 } ^ { \\tau } \\right) - \\left( \\tau _ { h } \\widehat { V } _ { h + 1 } \\right) \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\right) / \\widehat { \\sigma } _ { h } ^ { 2 } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) \\right) .", + "type": "interline_equation", + "image_path": "d985a0834b83e526509a28b18b216835fd3ac31b93c0da8d50e3d8bd3485b0ef.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 133, + 213, + 510, + 224.0 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 133, + 224.0, + 510, + 235.0 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 133, + 235.0, + 510, + 246.0 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 276, + 288, + 289 + ], + "lines": [ + { + "bbox": [ + 105, + 276, + 288, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 288, + 291 + ], + "score": 1.0, + "content": "The term (i) is dealt by the following lemma.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 276, + 288, + 291 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 293, + 506, + 320 + ], + "lines": [ + { + "bbox": [ + 105, + 292, + 507, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 195, + 309 + ], + "score": 1.0, + "content": "Lemma C.2. Recall", + "type": "text" + }, + { + "bbox": [ + 195, + 297, + 203, + 305 + ], + "score": 0.67, + "content": "\\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 292, + 324, + 309 + ], + "score": 1.0, + "content": "in Assumption 2.2. Suppose", + "type": "text" + }, + { + "bbox": [ + 324, + 295, + 503, + 308 + ], + "score": 0.87, + "content": "\\begin{array} { r } { K \\ge \\operatorname* { m a x } \\left\\{ 5 1 2 H ^ { 4 } / \\kappa ^ { 2 } \\log \\left( \\frac { 2 d } { \\delta } \\right) , 4 \\lambda H ^ { 2 } / \\kappa \\right\\} } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 292, + 507, + 309 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 305, + 338, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 305, + 192, + 321 + ], + "score": 1.0, + "content": "then with probability", + "type": "text" + }, + { + "bbox": [ + 192, + 307, + 216, + 318 + ], + "score": 0.68, + "content": "1 - \\delta _ { : }", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 305, + 246, + 321 + ], + "score": 1.0, + "content": ", for all", + "type": "text" + }, + { + "bbox": [ + 246, + 307, + 338, + 319 + ], + "score": 0.9, + "content": "s , a , h \\in \\mathcal S \\times \\mathcal A \\times [ H ]", + "type": "inline_equation" + } + ], + "index": 9 + } + ], + "index": 8.5, + "bbox_fs": [ + 105, + 292, + 507, + 321 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 329, + 504, + 363 + ], + "lines": [ + { + "bbox": [ + 111, + 329, + 504, + 363 + ], + "spans": [ + { + "bbox": [ + 111, + 329, + 504, + 363 + ], + "score": 0.91, + "content": "\\phi ( s , a ) ^ { \\top } w _ { h } - \\phi ( s , a ) ^ { \\top } \\widehat { \\Lambda } _ { h } ^ { - 1 } \\left( \\sum _ { \\tau = 1 } ^ { K } \\phi \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\cdot \\left( \\mathcal { T } _ { h } \\widehat { V } _ { h + 1 } \\right) \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) / \\widehat { \\sigma } ^ { 2 } \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\right) \\Bigg | \\leq \\frac { 2 \\lambda H ^ { 3 } \\sqrt { d } / \\kappa } { K } .", + "type": "interline_equation", + "image_path": "5df864ea3b54fac8520c9eb7b7537a004c4de3dc7587c274078d3cac9f38c97d.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 111, + 329, + 504, + 340.3333333333333 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 111, + 340.3333333333333, + 504, + 351.66666666666663 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 111, + 351.66666666666663, + 504, + 362.99999999999994 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 391, + 505, + 417 + ], + "lines": [ + { + "bbox": [ + 105, + 390, + 505, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 165, + 407 + ], + "score": 1.0, + "content": "Proof. Recall", + "type": "text" + }, + { + "bbox": [ + 165, + 392, + 238, + 405 + ], + "score": 0.92, + "content": "\\mathcal { T } _ { h } \\widehat { V } _ { h + 1 } = \\phi ^ { \\top } w _ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 390, + 449, + 407 + ], + "score": 1.0, + "content": "and apply Lemma H.6, we obtain with probability", + "type": "text" + }, + { + "bbox": [ + 450, + 395, + 473, + 404 + ], + "score": 0.86, + "content": "1 - \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 390, + 505, + 407 + ], + "score": 1.0, + "content": ", for all", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 107, + 403, + 203, + 417 + ], + "spans": [ + { + "bbox": [ + 107, + 405, + 198, + 416 + ], + "score": 0.9, + "content": "s , a , h \\in \\mathcal S \\times \\mathcal A \\times [ H ]", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 403, + 203, + 417 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 390, + 505, + 417 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 139, + 426, + 471, + 572 + ], + "lines": [ + { + "bbox": [ + 139, + 426, + 471, + 572 + ], + "spans": [ + { + "bbox": [ + 139, + 426, + 471, + 572 + ], + "score": 0.95, + "content": "\\begin{array} { r l } & { \\phi ( s , a ) ^ { \\top } w _ { h } - 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\\lambda I \\right) w _ { h } = \\lambda \\cdot \\phi ( s , a ) ^ { \\top } \\widehat \\Lambda _ { h } ^ { - 1 } w _ { h } } \\\\ & { \\leq \\lambda \\| \\phi ( s , a ) \\| _ { \\widehat \\Lambda _ { h } ^ { - 1 } } \\cdot \\| w _ { h } \\| _ { \\widehat \\Lambda _ { h } ^ { - 1 } } \\leq \\displaystyle \\frac { \\lambda } { K } \\left\\| \\phi ( s , a ) \\right\\| _ { \\{ \\widehat \\Lambda _ { h } ^ { p } } - 1 } \\cdot \\left\\| w _ { h } \\right\\| _ { ( \\widehat { \\Lambda } _ { h } ^ { p } ) ^ { - 1 } } \\\\ & { \\leq \\displaystyle \\frac { \\lambda } { K } 1 \\cdot \\sqrt { \\left\\| ( \\widehat { \\Lambda } _ { h } ^ { p } ) ^ { - 1 } \\right\\| } \\cdot 2 H \\sqrt { d } \\cdot \\sqrt { \\left\\| ( \\widehat { \\Lambda } _ { h } ^ { p } ) ^ { - 1 } \\right\\| } } \\end{array}", + "type": "interline_equation", + "image_path": "a4063bd519014fa410d64e777145eabac33280af056ea974c2adfb3b9665671b.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 139, + 426, + 471, + 474.6666666666667 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 139, + 474.6666666666667, + 471, + 523.3333333333334 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 139, + 523.3333333333334, + 471, + 572.0 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 595, + 506, + 667 + ], + "lines": [ + { + "bbox": [ + 105, + 595, + 506, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 133, + 611 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 596, + 299, + 611 + ], + "score": 0.91, + "content": "\\tilde { \\Lambda } _ { h } ^ { p } : = \\mathbb { E } _ { \\mu , h } \\left[ \\widehat { \\sigma } _ { h } ( s , a ) ^ { - 2 } \\phi ( s , a ) \\phi ( s , a ) ^ { \\top } \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 595, + 506, + 611 + ], + "score": 1.0, + "content": "and the second inequality is by Lemma H.6 (with", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 107, + 610, + 505, + 625 + ], + "spans": [ + { + "bbox": [ + 107, + 611, + 159, + 623 + ], + "score": 0.9, + "content": "\\phi ^ { \\prime } ~ = ~ \\phi / \\widehat { \\sigma } _ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 610, + 181, + 625 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 182, + 612, + 310, + 624 + ], + "score": 0.86, + "content": "\\| \\phi / { \\widehat { \\sigma } } _ { h } \\| \\ \\leq \\ \\| \\phi \\| \\ \\leq \\ 1 \\ : = \\ C )", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 610, + 441, + 625 + ], + "score": 1.0, + "content": "and the third inequality uses", + "type": "text" + }, + { + "bbox": [ + 441, + 610, + 505, + 623 + ], + "score": 0.88, + "content": "{ \\sqrt { a ^ { \\top } \\cdot A \\cdot a } } \\leq", + "type": "inline_equation" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 622, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 624, + 258, + 638 + ], + "score": 0.89, + "content": "\\sqrt { \\| a \\| _ { 2 } \\| A \\| _ { 2 } \\| a \\| _ { 2 } } = \\| a \\| _ { 2 } \\sqrt { \\| A \\| _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 622, + 284, + 639 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 285, + 627, + 292, + 635 + ], + "score": 0.73, + "content": "a", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 622, + 350, + 639 + ], + "score": 1.0, + "content": "to be either", + "type": "text" + }, + { + "bbox": [ + 351, + 625, + 358, + 636 + ], + "score": 0.83, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 622, + 375, + 639 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 375, + 626, + 388, + 636 + ], + "score": 0.85, + "content": "w _ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 622, + 448, + 639 + ], + "score": 1.0, + "content": ". 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\\left( { \\mathcal T } _ { h } { \\widehat V } _ { h + 1 } \\right) \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\right) / { \\widehat { \\sigma } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) } , } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 507, + 79, + 511, + 100 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 97, + 255, + 111 + ], + "spans": [ + { + "bbox": [ + 106, + 97, + 255, + 111 + ], + "score": 1.0, + "content": "bthen by Cauchy inequality it follows", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "interline_equation", + "bbox": [ + 117, + 125, + 495, + 195 + ], + "lines": [ + { + "bbox": [ + 117, + 125, + 495, + 195 + ], + "spans": [ + { + "bbox": [ + 117, + 125, + 495, + 195 + ], + "score": 0.92, + "content": "\\begin{array} { l } { \\displaystyle \\left| \\phi ( s , a ) ^ { \\top } \\widehat { \\Lambda } _ { h } ^ { - 1 } \\left( \\displaystyle \\sum _ { \\tau = 1 } ^ { K } \\phi \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\cdot \\left( r _ { h } ^ { \\tau } + \\widehat { V } _ { h + 1 } \\left( s _ { h + 1 } ^ { \\tau } \\right) - \\left( \\mathcal { T } _ { h } \\widehat { V } _ { h + 1 } \\right) \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\right) / \\widehat { \\sigma } _ { h } ^ { 2 } \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\right) \\right| } \\\\ { \\leq \\sqrt { \\phi ( s , a ) ^ { \\top } \\widehat { \\Lambda } _ { h } ^ { - 1 } \\phi ( s , a ) } \\cdot \\displaystyle \\| \\displaystyle \\sum _ { \\tau = 1 } ^ { K } x _ { \\tau } \\eta _ { \\tau } \\big \\| _ { \\widehat { \\Lambda } _ { h } ^ { - 1 } } } \\\\ { \\leq \\sqrt { \\phi ( s , a ) ^ { \\top } \\widehat { \\Lambda } _ { h } ^ { - 1 } \\phi ( s , a ) } \\cdot \\displaystyle \\| \\displaystyle \\sum _ { \\tau = 1 } ^ { K } x _ { \\tau } \\eta _ { \\tau } \\big \\| _ { \\widehat { \\Lambda } _ { h } ^ { - 1 } } } \\end{array}", + "type": "interline_equation", + "image_path": "2c2033056f9f694d9fef6cf473b0fde71c3ac8c8e0454e443c264a78bf904ca7.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 117, + 125, + 495, + 148.33333333333334 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 117, + 148.33333333333334, + 495, + 171.66666666666669 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 117, + 171.66666666666669, + 495, + 195.00000000000003 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 226, + 328, + 245 + ], + "lines": [ + { + "bbox": [ + 104, + 226, + 326, + 247 + ], + "spans": [ + { + "bbox": [ + 104, + 227, + 241, + 247 + ], + "score": 1.0, + "content": "C.2.1 ANALYZING THE TERM", + "type": "text" + }, + { + "bbox": [ + 241, + 226, + 326, + 246 + ], + "score": 0.92, + "content": "\\sqrt { \\phi ( s , a ) \\widehat { \\Lambda } _ { h } ^ { - 1 } \\phi ( s , a ) }", + "type": "inline_equation" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 105, + 257, + 506, + 308 + ], + "lines": [ + { + "bbox": [ + 103, + 254, + 507, + 276 + ], + "spans": [ + { + "bbox": [ + 103, + 254, + 264, + 276 + ], + "score": 1.0, + "content": "Recall (in Theorem 3.2) the estimated", + "type": "text" + }, + { + "bbox": [ + 264, + 258, + 487, + 273 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\widehat { \\Lambda } _ { h } = \\sum _ { \\tau = 1 } ^ { K } \\phi \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\phi \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) ^ { \\top } / \\widehat { \\sigma } ^ { 2 } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) + \\lambda \\cdot I } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 254, + 507, + 276 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 264, + 509, + 292 + ], + "spans": [ + { + "bbox": [ + 106, + 272, + 334, + 290 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\Lambda _ { h } = \\sum _ { \\tau = 1 } ^ { K } \\phi ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) ^ { \\top } \\phi ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) / \\sigma _ { \\widehat { V } _ { h + 1 } } ^ { 2 } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) + \\lambda I } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 264, + 509, + 292 + ], + "score": 1.0, + "content": "h h h h b h h . Then we have the following lemma to", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 290, + 261, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 173, + 308 + ], + "score": 1.0, + "content": "control the term", + "type": "text" + }, + { + "bbox": [ + 173, + 290, + 257, + 309 + ], + "score": 0.9, + "content": "\\sqrt { \\phi ( s , a ) \\widehat { \\Lambda } _ { h } ^ { - 1 } \\phi ( s , a ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 292, + 261, + 308 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 106, + 316, + 506, + 358 + ], + "lines": [ + { + "bbox": [ + 105, + 316, + 506, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 261, + 331 + ], + "score": 1.0, + "content": "Lemma C.3. Denote the quantities", + "type": "text" + }, + { + "bbox": [ + 261, + 317, + 484, + 330 + ], + "score": 0.88, + "content": "C _ { 1 } \\ = \\ \\operatorname* { m a x } \\{ 2 \\lambda , 1 2 8 \\log ( 2 d / \\delta ) , 1 2 8 H ^ { 4 } \\log ( 2 d / \\delta ) / \\kappa ^ { 2 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 316, + 506, + 331 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 107, + 327, + 508, + 349 + ], + "spans": [ + { + "bbox": [ + 107, + 329, + 369, + 345 + ], + "score": 0.86, + "content": "\\begin{array} { r } { C _ { 2 } = \\operatorname* { m a x } \\{ \\frac { \\lambda ^ { 2 } } { \\kappa \\log ( ( \\lambda + K ) H / \\lambda \\delta ) } , 9 6 ^ { 2 } H ^ { 1 2 } d \\log ( ( \\lambda + K ) H / \\lambda \\delta ) / \\kappa ^ { 5 } \\} } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 327, + 508, + 349 + ], + "score": 1.0, + "content": ". Suppose the number of episode", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 107, + 343, + 349, + 358 + ], + "spans": [ + { + "bbox": [ + 107, + 345, + 117, + 355 + ], + "score": 0.74, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 343, + 152, + 358 + ], + "score": 1.0, + "content": "satisfies", + "type": "text" + }, + { + "bbox": [ + 152, + 345, + 231, + 357 + ], + "score": 0.9, + "content": "K > \\operatorname* { m a x } \\{ C _ { 1 } , C _ { 2 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 343, + 321, + 358 + ], + "score": 1.0, + "content": ", then with probability", + "type": "text" + }, + { + "bbox": [ + 321, + 345, + 344, + 356 + ], + "score": 0.75, + "content": "1 - \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 343, + 349, + 358 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10 + }, + { + "type": "interline_equation", + "bbox": [ + 173, + 372, + 437, + 394 + ], + "lines": [ + { + "bbox": [ + 173, + 372, + 437, + 394 + ], + "spans": [ + { + "bbox": [ + 173, + 372, + 437, + 394 + ], + "score": 0.86, + "content": "\\sqrt { \\phi ( s , a ) \\widehat \\Lambda _ { h } ^ { - 1 } \\phi ( s , a ) } \\le 2 \\sqrt { \\phi ( s , a ) \\Lambda _ { h } ^ { - 1 } \\phi ( s , a ) } , \\quad \\forall s , a \\in \\mathcal { S } \\times \\mathcal { A } .", + "type": "interline_equation", + "image_path": "72da16f29276fb22dbc52b9a54d7ac4808c736b5a525b13c9405ef837e5870a2.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 173, + 372, + 437, + 394 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 438, + 458, + 459 + ], + "lines": [ + { + "bbox": [ + 104, + 438, + 459, + 459 + ], + "spans": [ + { + "bbox": [ + 104, + 438, + 250, + 459 + ], + "score": 1.0, + "content": "Proof of Lemma C.3. By definition", + "type": "text" + }, + { + "bbox": [ + 251, + 438, + 401, + 459 + ], + "score": 0.93, + "content": "\\sqrt { \\phi ( s , a ) \\widehat { \\Lambda } _ { h } ^ { - 1 } \\phi ( s , a ) } = \\| \\phi ( s , a ) \\| _ { \\widehat { \\Lambda } _ { h } ^ { - 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 438, + 459, + 459 + ], + "score": 1.0, + "content": ". Then denote", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "interline_equation", + "bbox": [ + 246, + 473, + 363, + 497 + ], + "lines": [ + { + "bbox": [ + 246, + 473, + 363, + 497 + ], + "spans": [ + { + "bbox": [ + 246, + 473, + 363, + 497 + ], + "score": 0.93, + "content": "\\widehat { \\Lambda } _ { h } ^ { \\prime } = \\frac { 1 } { K } \\widehat { \\Lambda } _ { h } , \\quad \\Lambda _ { h } ^ { \\prime } = \\frac { 1 } { K } \\Lambda _ { h } ,", + "type": "interline_equation", + "image_path": "7315700146a1a0ac0f660a8939b706a62a638ec0fcfad1e700ceef66f404a05c.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 246, + 473, + 363, + 497 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 512, + 505, + 541 + ], + "lines": [ + { + "bbox": [ + 103, + 507, + 508, + 532 + ], + "spans": [ + { + "bbox": [ + 103, + 507, + 134, + 532 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 135, + 513, + 366, + 531 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\Lambda _ { h } \\ = \\ \\sum _ { \\tau = 1 } ^ { K } \\phi ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) ^ { \\top } \\phi ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) / \\sigma _ { \\widehat { V } _ { h + 1 } } ^ { 2 } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) + \\lambda I . } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 507, + 476, + 532 + ], + "score": 1.0, + "content": ". Under the condition of", + "type": "text" + }, + { + "bbox": [ + 477, + 515, + 487, + 524 + ], + "score": 0.84, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 507, + 508, + 532 + ], + "score": 1.0, + "content": ", by", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 527, + 248, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 225, + 541 + ], + "score": 1.0, + "content": "Lemma C.7, with probability", + "type": "text" + }, + { + "bbox": [ + 225, + 529, + 248, + 540 + ], + "score": 0.7, + "content": "1 - \\delta", + "type": "inline_equation" + } + ], + "index": 16 + } + ], + "index": 15.5 + }, + { + "type": "interline_equation", + "bbox": [ + 144, + 555, + 468, + 661 + ], + "lines": [ + { + "bbox": [ + 144, + 555, + 468, + 661 + ], + "spans": [ + { + "bbox": [ + 144, + 555, + 468, + 661 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { \\left\\| \\widehat { \\Lambda } _ { h } ^ { \\prime } - \\Lambda _ { h } ^ { \\prime } \\right\\| \\leq \\underset { s , a } { \\operatorname* { s u p } } \\left\\| \\frac { \\phi ( s , a ) \\phi ( s , a ) ^ { \\top } } { \\widehat { \\sigma } _ { h } ^ { 2 } ( s , a ) } - \\frac { \\phi ( s , a ) \\phi ( s , a ) ^ { \\top } } { \\widehat { \\sigma } _ { \\widehat { V } _ { h + 1 } } ^ { 2 } ( s , a ) } \\right\\| } \\\\ & { \\leq \\underset { s , a } { \\operatorname* { s u p } } \\left| \\frac { \\widehat { \\sigma } _ { h } ^ { 2 } ( s , a ) - \\sigma _ { \\widehat { V } _ { h + 1 } } ^ { 2 } ( s , a ) } { \\widehat { \\sigma } _ { h } ^ { 2 } ( s , a ) \\sigma _ { \\widehat { V } _ { h + 1 } } ^ { 2 } ( s , a ) } \\right| \\cdot \\left\\| \\phi ( s , a ) \\right\\| ^ { 2 } \\leq \\underset { s , a } { \\operatorname* { s u p } } \\left| \\frac { \\widehat { \\sigma } _ { h } ^ { 2 } ( s , a ) - \\sigma _ { \\widehat { V } _ { h + 1 } } ^ { 2 } ( s , a ) } { 1 } \\right| \\cdot 1 } \\\\ & { \\leq 1 2 \\sqrt { \\frac { H ^ { 4 } d } { \\kappa K } \\log \\left( \\frac { \\left( \\lambda + K \\right) H } { \\lambda \\delta } \\right) } + 1 2 \\lambda \\frac { H ^ { 2 } \\sqrt { d } } { \\kappa K } . } \\end{array}", + "type": "interline_equation", + "image_path": "0b55496ca8b38f5a50b3a1b2e1c6c4587a1d6adf5eedf5c4809544b52531d8c4.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 144, + 555, + 468, + 590.3333333333334 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 144, + 590.3333333333334, + 468, + 625.6666666666667 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 144, + 625.6666666666667, + 468, + 661.0000000000001 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 674, + 460, + 690 + ], + "lines": [ + { + "bbox": [ + 103, + 672, + 461, + 694 + ], + "spans": [ + { + "bbox": [ + 103, + 672, + 214, + 694 + ], + "score": 1.0, + "content": "Next by Lemma H.5 (with", + "type": "text" + }, + { + "bbox": [ + 214, + 677, + 221, + 687 + ], + "score": 0.84, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 672, + 245, + 694 + ], + "score": 1.0, + "content": "to be", + "type": "text" + }, + { + "bbox": [ + 245, + 676, + 281, + 691 + ], + "score": 0.93, + "content": "\\phi / \\sigma _ { \\widehat { V } _ { h + 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 672, + 299, + 694 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 299, + 676, + 327, + 686 + ], + "score": 0.85, + "content": "C = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 672, + 432, + 694 + ], + "score": 1.0, + "content": "), it holds with probability", + "type": "text" + }, + { + "bbox": [ + 432, + 676, + 455, + 686 + ], + "score": 0.82, + "content": "1 - \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 672, + 461, + 694 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "interline_equation", + "bbox": [ + 146, + 706, + 464, + 736 + ], + "lines": [ + { + "bbox": [ + 146, + 706, + 464, + 736 + ], + "spans": [ + { + "bbox": [ + 146, + 706, + 464, + 736 + ], + "score": 0.9, + "content": "\\left\\| \\Lambda _ { h } ^ { \\prime } - \\left( \\mathbb { E } _ { \\mu , h } [ \\phi ( s , a ) \\phi ( s , a ) ^ { \\top } / \\sigma _ { \\hat { V } _ { h + 1 } } ^ { 2 } ( s , a ) ] + \\frac { \\lambda } { K } I _ { d } \\right) \\right\\| \\leq \\frac { 4 \\sqrt { 2 } } { \\sqrt { K } } \\left( \\log \\frac { 2 d } { \\delta } \\right) ^ { 1 / 2 } .", + "type": "interline_equation", + "image_path": "930944bad7cbc297d0ab3d035137d3c0317083086eee697073e9e0f0c6c2a0de.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 146, + 706, + 464, + 736 + ], + "spans": [], + "index": 21 + } + ] + } + ], + "page_idx": 18, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 26, + 294, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 39 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 39 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 761 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 765 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 765 + ], + "score": 1.0, + "content": "19", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 79, + 510, + 111 + ], + "lines": [ + { + "bbox": [ + 105, + 79, + 511, + 101 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 216, + 97 + ], + "score": 1.0, + "content": "For term (ii), denote: xτ =", + "type": "text" + }, + { + "bbox": [ + 189, + 81, + 507, + 101 + ], + "score": 0.9, + "content": "\\begin{array} { r } { x _ { \\tau } = \\frac { \\phi ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) } { \\widehat { \\sigma } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) } , \\quad \\eta _ { \\tau } = \\left( r _ { h } ^ { \\tau } + { \\widehat V } _ { h + 1 } \\left( s _ { h + 1 } ^ { \\tau } \\right) - \\left( { \\mathcal T } _ { h } { \\widehat V } _ { h + 1 } \\right) \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\right) / { \\widehat { \\sigma } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) } , } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 507, + 79, + 511, + 100 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 97, + 255, + 111 + ], + "spans": [ + { + "bbox": [ + 106, + 97, + 255, + 111 + ], + "score": 1.0, + "content": "bthen by Cauchy inequality it follows", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 105, + 79, + 511, + 111 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 117, + 125, + 495, + 195 + ], + "lines": [ + { + "bbox": [ + 117, + 125, + 495, + 195 + ], + "spans": [ + { + "bbox": [ + 117, + 125, + 495, + 195 + ], + "score": 0.92, + "content": "\\begin{array} { l } { \\displaystyle \\left| \\phi ( s , a ) ^ { \\top } \\widehat { \\Lambda } _ { h } ^ { - 1 } \\left( \\displaystyle \\sum _ { \\tau = 1 } ^ { K } \\phi \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\cdot \\left( r _ { h } ^ { \\tau } + \\widehat { V } _ { h + 1 } \\left( s _ { h + 1 } ^ { \\tau } \\right) - \\left( \\mathcal { T } _ { h } \\widehat { V } _ { h + 1 } \\right) \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\right) / \\widehat { \\sigma } _ { h } ^ { 2 } \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\right) \\right| } \\\\ { \\leq \\sqrt { \\phi ( s , a ) ^ { \\top } \\widehat { \\Lambda } _ { h } ^ { - 1 } \\phi ( s , a ) } \\cdot \\displaystyle \\| \\displaystyle \\sum _ { \\tau = 1 } ^ { K } x _ { \\tau } \\eta _ { \\tau } \\big \\| _ { \\widehat { \\Lambda } _ { h } ^ { - 1 } } } \\\\ { \\leq \\sqrt { \\phi ( s , a ) ^ { \\top } \\widehat { \\Lambda } _ { h } ^ { - 1 } \\phi ( s , a ) } \\cdot \\displaystyle \\| \\displaystyle \\sum _ { \\tau = 1 } ^ { K } x _ { \\tau } \\eta _ { \\tau } \\big \\| _ { \\widehat { \\Lambda } _ { h } ^ { - 1 } } } \\end{array}", + "type": "interline_equation", + "image_path": "2c2033056f9f694d9fef6cf473b0fde71c3ac8c8e0454e443c264a78bf904ca7.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 117, + 125, + 495, + 148.33333333333334 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 117, + 148.33333333333334, + 495, + 171.66666666666669 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 117, + 171.66666666666669, + 495, + 195.00000000000003 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 226, + 328, + 245 + ], + "lines": [ + { + "bbox": [ + 104, + 226, + 326, + 247 + ], + "spans": [ + { + "bbox": [ + 104, + 227, + 241, + 247 + ], + "score": 1.0, + "content": "C.2.1 ANALYZING THE TERM", + "type": "text" + }, + { + "bbox": [ + 241, + 226, + 326, + 246 + ], + "score": 0.92, + "content": "\\sqrt { \\phi ( s , a ) \\widehat { \\Lambda } _ { h } ^ { - 1 } \\phi ( s , a ) }", + "type": "inline_equation" + } + ], + "index": 5 + } + ], + "index": 5, + "bbox_fs": [ + 104, + 226, + 326, + 247 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 257, + 506, + 308 + ], + "lines": [ + { + "bbox": [ + 103, + 254, + 507, + 276 + ], + "spans": [ + { + "bbox": [ + 103, + 254, + 264, + 276 + ], + "score": 1.0, + "content": "Recall (in Theorem 3.2) the estimated", + "type": "text" + }, + { + "bbox": [ + 264, + 258, + 487, + 273 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\widehat { \\Lambda } _ { h } = \\sum _ { \\tau = 1 } ^ { K } \\phi \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\phi \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) ^ { \\top } / \\widehat { \\sigma } ^ { 2 } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) + \\lambda \\cdot I } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 254, + 507, + 276 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 264, + 509, + 292 + ], + "spans": [ + { + "bbox": [ + 106, + 272, + 334, + 290 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\Lambda _ { h } = \\sum _ { \\tau = 1 } ^ { K } \\phi ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) ^ { \\top } \\phi ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) / \\sigma _ { \\widehat { V } _ { h + 1 } } ^ { 2 } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) + \\lambda I } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 264, + 509, + 292 + ], + "score": 1.0, + "content": "h h h h b h h . Then we have the following lemma to", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 290, + 261, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 173, + 308 + ], + "score": 1.0, + "content": "control the term", + "type": "text" + }, + { + "bbox": [ + 173, + 290, + 257, + 309 + ], + "score": 0.9, + "content": "\\sqrt { \\phi ( s , a ) \\widehat { \\Lambda } _ { h } ^ { - 1 } \\phi ( s , a ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 292, + 261, + 308 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7, + "bbox_fs": [ + 103, + 254, + 509, + 309 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 316, + 506, + 358 + ], + "lines": [ + { + "bbox": [ + 105, + 316, + 506, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 261, + 331 + ], + "score": 1.0, + "content": "Lemma C.3. Denote the quantities", + "type": "text" + }, + { + "bbox": [ + 261, + 317, + 484, + 330 + ], + "score": 0.88, + "content": "C _ { 1 } \\ = \\ \\operatorname* { m a x } \\{ 2 \\lambda , 1 2 8 \\log ( 2 d / \\delta ) , 1 2 8 H ^ { 4 } \\log ( 2 d / \\delta ) / \\kappa ^ { 2 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 316, + 506, + 331 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 107, + 327, + 508, + 349 + ], + "spans": [ + { + "bbox": [ + 107, + 329, + 369, + 345 + ], + "score": 0.86, + "content": "\\begin{array} { r } { C _ { 2 } = \\operatorname* { m a x } \\{ \\frac { \\lambda ^ { 2 } } { \\kappa \\log ( ( \\lambda + K ) H / \\lambda \\delta ) } , 9 6 ^ { 2 } H ^ { 1 2 } d \\log ( ( \\lambda + K ) H / \\lambda \\delta ) / \\kappa ^ { 5 } \\} } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 327, + 508, + 349 + ], + "score": 1.0, + "content": ". Suppose the number of episode", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 107, + 343, + 349, + 358 + ], + "spans": [ + { + "bbox": [ + 107, + 345, + 117, + 355 + ], + "score": 0.74, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 343, + 152, + 358 + ], + "score": 1.0, + "content": "satisfies", + "type": "text" + }, + { + "bbox": [ + 152, + 345, + 231, + 357 + ], + "score": 0.9, + "content": "K > \\operatorname* { m a x } \\{ C _ { 1 } , C _ { 2 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 343, + 321, + 358 + ], + "score": 1.0, + "content": ", then with probability", + "type": "text" + }, + { + "bbox": [ + 321, + 345, + 344, + 356 + ], + "score": 0.75, + "content": "1 - \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 343, + 349, + 358 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10, + "bbox_fs": [ + 105, + 316, + 508, + 358 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 173, + 372, + 437, + 394 + ], + "lines": [ + { + "bbox": [ + 173, + 372, + 437, + 394 + ], + "spans": [ + { + "bbox": [ + 173, + 372, + 437, + 394 + ], + "score": 0.86, + "content": "\\sqrt { \\phi ( s , a ) \\widehat \\Lambda _ { h } ^ { - 1 } \\phi ( s , a ) } \\le 2 \\sqrt { \\phi ( s , a ) \\Lambda _ { h } ^ { - 1 } \\phi ( s , a ) } , \\quad \\forall s , a \\in \\mathcal { S } \\times \\mathcal { A } .", + "type": "interline_equation", + "image_path": "72da16f29276fb22dbc52b9a54d7ac4808c736b5a525b13c9405ef837e5870a2.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 173, + 372, + 437, + 394 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 438, + 458, + 459 + ], + "lines": [ + { + "bbox": [ + 104, + 438, + 459, + 459 + ], + "spans": [ + { + "bbox": [ + 104, + 438, + 250, + 459 + ], + "score": 1.0, + "content": "Proof of Lemma C.3. By definition", + "type": "text" + }, + { + "bbox": [ + 251, + 438, + 401, + 459 + ], + "score": 0.93, + "content": "\\sqrt { \\phi ( s , a ) \\widehat { \\Lambda } _ { h } ^ { - 1 } \\phi ( s , a ) } = \\| \\phi ( s , a ) \\| _ { \\widehat { \\Lambda } _ { h } ^ { - 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 438, + 459, + 459 + ], + "score": 1.0, + "content": ". Then denote", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13, + "bbox_fs": [ + 104, + 438, + 459, + 459 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 246, + 473, + 363, + 497 + ], + "lines": [ + { + "bbox": [ + 246, + 473, + 363, + 497 + ], + "spans": [ + { + "bbox": [ + 246, + 473, + 363, + 497 + ], + "score": 0.93, + "content": "\\widehat { \\Lambda } _ { h } ^ { \\prime } = \\frac { 1 } { K } \\widehat { \\Lambda } _ { h } , \\quad \\Lambda _ { h } ^ { \\prime } = \\frac { 1 } { K } \\Lambda _ { h } ,", + "type": "interline_equation", + "image_path": "7315700146a1a0ac0f660a8939b706a62a638ec0fcfad1e700ceef66f404a05c.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 246, + 473, + 363, + 497 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 512, + 505, + 541 + ], + "lines": [ + { + "bbox": [ + 103, + 507, + 508, + 532 + ], + "spans": [ + { + "bbox": [ + 103, + 507, + 134, + 532 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 135, + 513, + 366, + 531 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\Lambda _ { h } \\ = \\ \\sum _ { \\tau = 1 } ^ { K } \\phi ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) ^ { \\top } \\phi ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) / \\sigma _ { \\widehat { V } _ { h + 1 } } ^ { 2 } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) + \\lambda I . } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 507, + 476, + 532 + ], + "score": 1.0, + "content": ". Under the condition of", + "type": "text" + }, + { + "bbox": [ + 477, + 515, + 487, + 524 + ], + "score": 0.84, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 507, + 508, + 532 + ], + "score": 1.0, + "content": ", by", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 527, + 248, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 225, + 541 + ], + "score": 1.0, + "content": "Lemma C.7, with probability", + "type": "text" + }, + { + "bbox": [ + 225, + 529, + 248, + 540 + ], + "score": 0.7, + "content": "1 - \\delta", + "type": "inline_equation" + } + ], + "index": 16 + } + ], + "index": 15.5, + "bbox_fs": [ + 103, + 507, + 508, + 541 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 144, + 555, + 468, + 661 + ], + "lines": [ + { + "bbox": [ + 144, + 555, + 468, + 661 + ], + "spans": [ + { + "bbox": [ + 144, + 555, + 468, + 661 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { \\left\\| \\widehat { \\Lambda } _ { h } ^ { \\prime } - \\Lambda _ { h } ^ { \\prime } \\right\\| \\leq \\underset { s , a } { \\operatorname* { s u p } } \\left\\| \\frac { \\phi ( s , a ) \\phi ( s , a ) ^ { \\top } } { \\widehat { \\sigma } _ { h } ^ { 2 } ( s , a ) } - \\frac { \\phi ( s , a ) \\phi ( s , a ) ^ { \\top } } { \\widehat { \\sigma } _ { \\widehat { V } _ { h + 1 } } ^ { 2 } ( s , a ) } \\right\\| } \\\\ & { \\leq \\underset { s , a } { \\operatorname* { s u p } } \\left| \\frac { \\widehat { \\sigma } _ { h } ^ { 2 } ( s , a ) - \\sigma _ { \\widehat { V } _ { h + 1 } } ^ { 2 } ( s , a ) } { \\widehat { \\sigma } _ { h } ^ { 2 } ( s , a ) \\sigma _ { \\widehat { V } _ { h + 1 } } ^ { 2 } ( s , a ) } \\right| \\cdot \\left\\| \\phi ( s , a ) \\right\\| ^ { 2 } \\leq \\underset { s , a } { \\operatorname* { s u p } } \\left| \\frac { \\widehat { \\sigma } _ { h } ^ { 2 } ( s , a ) - \\sigma _ { \\widehat { V } _ { h + 1 } } ^ { 2 } ( s , a ) } { 1 } \\right| \\cdot 1 } \\\\ & { \\leq 1 2 \\sqrt { \\frac { H ^ { 4 } d } { \\kappa K } \\log \\left( \\frac { \\left( \\lambda + K \\right) H } { \\lambda \\delta } \\right) } + 1 2 \\lambda \\frac { H ^ { 2 } \\sqrt { d } } { \\kappa K } . } \\end{array}", + "type": "interline_equation", + "image_path": "0b55496ca8b38f5a50b3a1b2e1c6c4587a1d6adf5eedf5c4809544b52531d8c4.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 144, + 555, + 468, + 590.3333333333334 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 144, + 590.3333333333334, + 468, + 625.6666666666667 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 144, + 625.6666666666667, + 468, + 661.0000000000001 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 674, + 460, + 690 + ], + "lines": [ + { + "bbox": [ + 103, + 672, + 461, + 694 + ], + "spans": [ + { + "bbox": [ + 103, + 672, + 214, + 694 + ], + "score": 1.0, + "content": "Next by Lemma H.5 (with", + "type": "text" + }, + { + "bbox": [ + 214, + 677, + 221, + 687 + ], + "score": 0.84, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 672, + 245, + 694 + ], + "score": 1.0, + "content": "to be", + "type": "text" + }, + { + "bbox": [ + 245, + 676, + 281, + 691 + ], + "score": 0.93, + "content": "\\phi / \\sigma _ { \\widehat { V } _ { h + 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 672, + 299, + 694 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 299, + 676, + 327, + 686 + ], + "score": 0.85, + "content": "C = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 672, + 432, + 694 + ], + "score": 1.0, + "content": "), it holds with probability", + "type": "text" + }, + { + "bbox": [ + 432, + 676, + 455, + 686 + ], + "score": 0.82, + "content": "1 - \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 672, + 461, + 694 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20, + "bbox_fs": [ + 103, + 672, + 461, + 694 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 146, + 706, + 464, + 736 + ], + "lines": [ + { + "bbox": [ + 146, + 706, + 464, + 736 + ], + "spans": [ + { + "bbox": [ + 146, + 706, + 464, + 736 + ], + "score": 0.9, + "content": "\\left\\| \\Lambda _ { h } ^ { \\prime } - \\left( \\mathbb { E } _ { \\mu , h } [ \\phi ( s , a ) \\phi ( s , a ) ^ { \\top } / \\sigma _ { \\hat { V } _ { h + 1 } } ^ { 2 } ( s , a ) ] + \\frac { \\lambda } { K } I _ { d } \\right) \\right\\| \\leq \\frac { 4 \\sqrt { 2 } } { \\sqrt { K } } \\left( \\log \\frac { 2 d } { \\delta } \\right) ^ { 1 / 2 } .", + "type": "interline_equation", + "image_path": "930944bad7cbc297d0ab3d035137d3c0317083086eee697073e9e0f0c6c2a0de.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 146, + 706, + 464, + 736 + ], + "spans": [], + "index": 21 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 81, + 504, + 106 + ], + "lines": [ + { + "bbox": [ + 106, + 80, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 221, + 95 + ], + "score": 1.0, + "content": "Therefore by Weyl’s", + "type": "text" + }, + { + "bbox": [ + 232, + 83, + 274, + 94 + ], + "score": 1.0, + "content": "spectrum", + "type": "text" + }, + { + "bbox": [ + 287, + 82, + 325, + 94 + ], + "score": 1.0, + "content": "theorem", + "type": "text" + }, + { + "bbox": [ + 336, + 80, + 388, + 95 + ], + "score": 1.0, + "content": "and the", + "type": "text" + }, + { + "bbox": [ + 398, + 81, + 455, + 95 + ], + "score": 1.0, + "content": "condition", + "type": "text" + }, + { + "bbox": [ + 456, + 82, + 466, + 93 + ], + "score": 0.25, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 84, + 505, + 94 + ], + "score": 1.0, + "content": ">", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 92, + 376, + 107 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 299, + 106 + ], + "score": 0.83, + "content": "\\operatorname* { m a x } \\{ 2 \\lambda , 1 2 8 \\log ( 2 d / \\delta ) , 1 2 8 H ^ { 4 } \\log ( 2 d / \\delta ) / \\kappa ^ { 2 } \\} .", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 92, + 376, + 107 + ], + "score": 1.0, + "content": ", the above implies", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "interline_equation", + "bbox": [ + 113, + 114, + 503, + 298 + ], + "lines": [ + { + "bbox": [ + 113, + 114, + 503, + 298 + ], + "spans": [ + { + "bbox": [ + 113, + 114, + 503, + 298 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { \\| \\Lambda _ { t } ^ { \\prime } \\| = \\lambda _ { \\operatorname* { m a x } } ( N _ { h } ) \\leq \\lambda _ { \\operatorname* { m a x } } \\Big ( \\mathrm { E } _ { \\mu h } \\big [ \\phi ( s , a ) \\phi ( s , a ) \\big ] \\Big ) + \\frac { \\lambda } { K } + \\frac { 4 \\sqrt { 2 } } { \\sqrt { K } } \\Big ( \\log \\frac { 2 d } { \\delta } \\Big ) ^ { 1 / 2 } } \\\\ & { \\qquad - \\Big \\| \\mathrm { E } _ { \\mu h } [ \\delta ( s , a ) \\phi ( s , a ) ^ { \\top } / \\sigma _ { b \\mathrm { t } \\mathrm { \\scriptsize \\ ' \\ ' } } ^ { 2 } ( s , a ) \\mathbb { S } , a ] \\Big \\| _ { 2 } + \\frac { \\lambda } { K } + \\frac { 4 \\sqrt { 2 } } { \\sqrt { K } } \\Big ( \\log \\frac { 2 d } { \\delta } \\Big ) ^ { 1 / 2 } } \\\\ & { \\qquad \\leq \\| \\phi ( s , a ) \\| ^ { 2 } + \\frac { \\lambda } { K } + \\frac { 4 \\sqrt { 2 } } { \\sqrt { K } } \\Big ( \\log \\frac { 2 d } { \\delta } \\Big ) ^ { 1 / 2 } \\leq 1 + \\frac { \\lambda } { K } + \\frac { 4 \\sqrt { 2 } } { \\sqrt { K } } \\Big ( \\log \\frac { 2 d } { \\delta } \\Big ) ^ { 1 / 2 } \\leq 2 , } \\\\ & { \\qquad \\quad \\times _ { \\operatorname* { m i n } } \\big ( \\Delta _ { h } ^ { \\prime } \\big ) \\leq \\lambda _ { \\operatorname* { m i n } } \\Big ( \\mathrm { E } _ { \\mu h } \\big ) \\big ( \\tilde { \\phi } ( s , a ) \\phi ( s , a ) ^ { \\top } / \\sigma _ { b \\mathrm { t } \\mathrm { \\scriptsize \\ ' } + 1 } ^ { 2 } ( s , a ) \\big ] \\Big ) + \\frac { \\lambda } { K } - \\frac { 4 \\sqrt { 2 } } { \\sqrt { K } } \\Big ( \\log \\frac { 2 d } { \\delta } \\Big ) ^ { 1 / 2 } } \\\\ & { \\qquad \\quad \\geq \\lambda _ { \\operatorname* { m i n } } \\Big ( \\mathrm { E } _ { \\mu h } \\big [ \\phi ( s , a ) \\phi ( s , a ) ^ { \\top } / \\sigma _ { b \\mathrm { t } \\mathrm { \\scriptsize \\ ' } + 1 } ^ { 2 } ( s , a ) \\big ] \\Big ) - \\frac { 4 \\sqrt { 2 } } { \\sqrt { K } } \\Big ( \\log \\frac { 2 d } { \\delta } \\Big ) ^ { 1 / 2 } } \\\\ & \\qquad \\quad \\geq \\frac { \\kappa } { H ^ { 2 } } - \\frac { 4 \\sqrt { 2 } } { \\sqrt { K } } \\Big ( \\log \\frac \\end{array}", + "type": "interline_equation", + "image_path": "0eaef0a745df068251121ccb4815b6c111cb574fd4d9b188402e6b893f998f80.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 113, + 114, + 503, + 175.33333333333334 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 113, + 175.33333333333334, + 503, + 236.66666666666669 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 113, + 236.66666666666669, + 503, + 298.0 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 304, + 506, + 339 + ], + "lines": [ + { + "bbox": [ + 105, + 304, + 506, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 201, + 321 + ], + "score": 1.0, + "content": "Hence with probability", + "type": "text" + }, + { + "bbox": [ + 202, + 306, + 270, + 319 + ], + "score": 0.88, + "content": "1 - \\delta , \\| \\Lambda _ { h } ^ { \\prime } \\| \\leq 2", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 304, + 288, + 321 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 289, + 305, + 426, + 320 + ], + "score": 0.94, + "content": "\\left\\| \\Lambda _ { h } ^ { \\prime - 1 } \\right\\| = 1 / \\lambda _ { \\operatorname* { m i n } } ( \\Lambda _ { h } ^ { \\prime } ) \\le 2 H ^ { 2 } / \\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 304, + 506, + 321 + ], + "score": 1.0, + "content": ". Similarly, one can", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 103, + 316, + 297, + 339 + ], + "spans": [ + { + "bbox": [ + 103, + 316, + 130, + 339 + ], + "score": 1.0, + "content": "show", + "type": "text" + }, + { + "bbox": [ + 131, + 318, + 205, + 339 + ], + "score": 0.93, + "content": "\\left\\| \\widehat { \\Lambda } _ { h } ^ { \\prime - 1 } \\right\\| \\leq 2 H ^ { 2 } / \\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 316, + 297, + 339 + ], + "score": 1.0, + "content": "with high probability.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5 + }, + { + "type": "text", + "bbox": [ + 105, + 343, + 504, + 358 + ], + "lines": [ + { + "bbox": [ + 104, + 342, + 504, + 360 + ], + "spans": [ + { + "bbox": [ + 104, + 342, + 210, + 360 + ], + "score": 1.0, + "content": "Now apply Lemma H.4 to", + "type": "text" + }, + { + "bbox": [ + 210, + 343, + 223, + 358 + ], + "score": 0.91, + "content": "\\widehat { \\Lambda } _ { h } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 342, + 241, + 360 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 241, + 345, + 254, + 358 + ], + "score": 0.89, + "content": "\\Lambda _ { h } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 342, + 437, + 360 + ], + "score": 1.0, + "content": "and a union bound, we obtain with probability", + "type": "text" + }, + { + "bbox": [ + 438, + 347, + 459, + 356 + ], + "score": 0.86, + "content": "1 - \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 342, + 488, + 360 + ], + "score": 1.0, + "content": ", for all", + "type": "text" + }, + { + "bbox": [ + 489, + 350, + 504, + 357 + ], + "score": 0.84, + "content": "s , a", + "type": "inline_equation" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "interline_equation", + "bbox": [ + 114, + 365, + 503, + 526 + ], + "lines": [ + { + "bbox": [ + 114, + 365, + 503, + 526 + ], + "spans": [ + { + "bbox": [ + 114, + 365, + 503, + 526 + ], + "score": 0.94, + "content": "\\begin{array} { r l } { \\left\\| \\phi ( s , a ) \\right\\| _ { \\widetilde { \\Lambda } _ { h } ^ { - 1 } } \\leq \\left[ 1 + \\sqrt { \\left\\| \\Lambda _ { h } ^ { \\prime - 1 } \\right\\| \\left\\| \\Lambda _ { h } ^ { \\prime } \\right\\| } \\cdot \\left\\| \\hat { \\Lambda } _ { h } ^ { \\prime - 1 } \\right\\| \\cdot \\left\\| \\hat { \\Lambda } _ { h } ^ { \\prime } - \\Lambda _ { h } ^ { \\prime } \\right\\| \\right] \\cdot \\left\\| \\phi ( s , a ) \\right\\| _ { \\Lambda _ { h } ^ { \\prime - 1 } } } & { } \\\\ { \\leq \\left[ 1 + \\sqrt { \\frac { 2 H ^ { 2 } } { \\kappa } } \\cdot 1 \\cdot \\frac { 2 H ^ { 2 } } { \\kappa } \\cdot \\left\\| \\hat { \\Lambda } _ { h } ^ { \\prime } - \\Lambda _ { h } ^ { \\prime } \\right\\| \\right] \\cdot \\left\\| \\phi ( s , a ) \\right\\| _ { \\Lambda _ { h } ^ { \\prime - 1 } } } & { } \\\\ { \\leq \\left[ 1 + \\sqrt { \\frac { 4 \\Delta H ^ { 4 } } { \\kappa ^ { 2 } } \\left( \\sqrt { \\frac { H ^ { 4 } d } { \\kappa K } } \\log \\left( \\frac { ( \\lambda + K ) H } { \\lambda \\delta } \\right) + \\lambda \\frac { H ^ { 2 } \\sqrt { d } } { \\kappa K } \\right) } \\right] \\cdot \\left\\| \\phi ( s , a ) \\right\\| _ { \\Lambda _ { h } ^ { \\prime - 1 } } } & { } \\\\ { \\leq \\left[ 1 + \\sqrt { \\frac { 9 6 H ^ { 4 } } { \\kappa ^ { 2 } } \\sqrt { \\frac { H ^ { 4 } d } { \\kappa K } } \\log \\left( \\frac { ( \\lambda + K ) H } { \\lambda \\delta } \\right) } \\right] \\cdot \\left\\| \\phi ( s , a ) \\right\\| _ { \\Lambda _ { h } ^ { \\prime - 1 } } \\leq 2 \\left\\| \\phi ( s , a ) \\right\\| _ { \\Lambda _ { h } ^ { \\prime - 1 } } } & { } \\end{array}", + "type": "interline_equation", + "image_path": "d5967e38fba8c718d18ae360a13bbf55703eb7ad38d01b91497cb4c7f686fce2.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 114, + 365, + 503, + 418.6666666666667 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 114, + 418.6666666666667, + 503, + 472.33333333333337 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 114, + 472.33333333333337, + 503, + 526.0 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 546, + 506, + 594 + ], + "lines": [ + { + "bbox": [ + 105, + 544, + 505, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 505, + 560 + ], + "score": 1.0, + "content": "where the third inequality uses equation 11 and the last and the second last inequality use", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 107, + 555, + 509, + 579 + ], + "spans": [ + { + "bbox": [ + 107, + 558, + 369, + 573 + ], + "score": 0.86, + "content": "\\begin{array} { r } { K > \\operatorname* { m a x } \\lbrace \\frac { \\lambda ^ { 2 } } { \\kappa \\log ( ( \\lambda + K ) H / \\lambda \\delta ) } , 9 6 ^ { 2 } H ^ { 1 2 } d \\log ( ( \\lambda + K ) H / \\lambda \\delta ) / \\kappa ^ { 5 } \\rbrace . } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 555, + 509, + 579 + ], + "score": 1.0, + "content": ". Note the above is equivalent to", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 107, + 575, + 445, + 595 + ], + "spans": [ + { + "bbox": [ + 107, + 575, + 293, + 595 + ], + "score": 0.92, + "content": "\\sqrt { \\phi ( s , a ) \\widehat \\Lambda _ { h } ^ { - 1 } \\phi ( s , a ) } \\le 2 \\sqrt { \\phi ( s , a ) \\Lambda _ { h } ^ { - 1 } \\phi ( s , a ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 576, + 356, + 595 + ], + "score": 1.0, + "content": "by multiplying", + "type": "text" + }, + { + "bbox": [ + 356, + 577, + 385, + 591 + ], + "score": 0.92, + "content": "1 / \\sqrt { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 576, + 445, + 595 + ], + "score": 1.0, + "content": "on both sides.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 106, + 630, + 316, + 645 + ], + "lines": [ + { + "bbox": [ + 102, + 623, + 316, + 651 + ], + "spans": [ + { + "bbox": [ + 102, + 623, + 241, + 651 + ], + "score": 1.0, + "content": "C.2.2 ANALYZING THE TERM", + "type": "text" + }, + { + "bbox": [ + 241, + 629, + 316, + 645 + ], + "score": 0.91, + "content": "\\begin{array} { r } { | | \\sum _ { \\tau = 1 } ^ { K } x _ { \\tau } \\eta _ { \\tau } | | _ { \\widehat { \\Lambda } ^ { - 1 } } } \\end{array}", + "type": "inline_equation" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 106, + 652, + 512, + 697 + ], + "lines": [ + { + "bbox": [ + 105, + 651, + 513, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 651, + 188, + 675 + ], + "score": 1.0, + "content": "Lemma C.4. Recall", + "type": "text" + }, + { + "bbox": [ + 189, + 654, + 246, + 672 + ], + "score": 0.93, + "content": "\\begin{array} { r } { x _ { \\tau } = \\frac { \\phi ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) } { \\widehat { \\sigma } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 653, + 510, + 674 + ], + "score": 0.45, + "content": "\\begin{array} { r } { \\frac { a _ { h } ^ { \\tau } } { a _ { h } ^ { \\tau } ) } a n d \\eta _ { \\tau } = \\left( r _ { h } ^ { \\tau } + \\widehat { V } _ { h + 1 } \\left( s _ { h + 1 } ^ { \\tau } \\right) - \\left( \\mathcal { T } _ { h } \\widehat { V } _ { h + 1 } \\right) ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) \\right) / \\widehat { \\sigma } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) . } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 510, + 651, + 513, + 675 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 672, + 402, + 698 + ], + "spans": [ + { + "bbox": [ + 104, + 676, + 122, + 696 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 122, + 673, + 349, + 698 + ], + "score": 0.92, + "content": "\\begin{array} { r } { C _ { H , d , \\kappa , K } : = 3 6 \\sqrt { \\frac { H ^ { 4 } d ^ { 3 } } { \\kappa } \\log \\left( \\frac { ( \\lambda + K ) 2 K d H ^ { 2 } } { \\lambda \\delta } \\right) } + 1 2 \\lambda \\frac { H ^ { 2 } \\sqrt { d } } { \\kappa } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 672, + 402, + 696 + ], + "score": 1.0, + "content": "and denote", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15.5 + }, + { + "type": "interline_equation", + "bbox": [ + 181, + 706, + 428, + 735 + ], + "lines": [ + { + "bbox": [ + 181, + 706, + 428, + 735 + ], + "spans": [ + { + "bbox": [ + 181, + 706, + 428, + 735 + ], + "score": 0.91, + "content": "\\xi : = \\operatorname* { s u p } _ { V \\in \\left[ 0 , H \\right] , s ^ { \\prime } \\sim P _ { h } \\left( s , a \\right) , h \\in \\left[ H \\right] } \\left| \\frac { r _ { h } + V \\left( s ^ { \\prime } \\right) - \\left( \\mathcal T _ { h } V \\right) \\left( s , a \\right) } { \\sigma _ { V } \\left( s , a \\right) } \\right| .", + "type": "interline_equation", + "image_path": "f46644a87b6e47aba6cd17b7a42b936a0b8de966101822058b87a01b8e6a98f3.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 181, + 706, + 428, + 735 + ], + "spans": [], + "index": 17 + } + ] + } + ], + "page_idx": 19, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 293, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 494, + 599, + 505, + 609 + ], + "lines": [ + { + "bbox": [ + 495, + 600, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 495, + 600, + 505, + 610 + ], + "score": 0.998, + "content": "□", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 312, + 761 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "score": 1.0, + "content": "20", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 81, + 504, + 106 + ], + "lines": [ + { + "bbox": [ + 106, + 80, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 221, + 95 + ], + "score": 1.0, + "content": "Therefore by Weyl’s", + "type": "text" + }, + { + "bbox": [ + 232, + 83, + 274, + 94 + ], + "score": 1.0, + "content": "spectrum", + "type": "text" + }, + { + "bbox": [ + 287, + 82, + 325, + 94 + ], + "score": 1.0, + "content": "theorem", + "type": "text" + }, + { + "bbox": [ + 336, + 80, + 388, + 95 + ], + "score": 1.0, + "content": "and the", + "type": "text" + }, + { + "bbox": [ + 398, + 81, + 455, + 95 + ], + "score": 1.0, + "content": "condition", + "type": "text" + }, + { + "bbox": [ + 456, + 82, + 466, + 93 + ], + "score": 0.25, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 84, + 505, + 94 + ], + "score": 1.0, + "content": ">", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 92, + 376, + 107 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 299, + 106 + ], + "score": 0.83, + "content": "\\operatorname* { m a x } \\{ 2 \\lambda , 1 2 8 \\log ( 2 d / \\delta ) , 1 2 8 H ^ { 4 } \\log ( 2 d / \\delta ) / \\kappa ^ { 2 } \\} .", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 92, + 376, + 107 + ], + "score": 1.0, + "content": ", the above implies", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 106, + 80, + 505, + 107 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 113, + 114, + 503, + 298 + ], + "lines": [ + { + "bbox": [ + 113, + 114, + 503, + 298 + ], + "spans": [ + { + "bbox": [ + 113, + 114, + 503, + 298 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { \\| \\Lambda _ { t } ^ { \\prime } \\| = \\lambda _ { \\operatorname* { m a x } } ( N _ { h } ) \\leq \\lambda _ { \\operatorname* { m a x } } \\Big ( \\mathrm { E } _ { \\mu h } \\big [ \\phi ( s , a ) \\phi ( s , a ) \\big ] \\Big ) + \\frac { \\lambda } { K } + \\frac { 4 \\sqrt { 2 } } { \\sqrt { K } } \\Big ( \\log \\frac { 2 d } { \\delta } \\Big ) ^ { 1 / 2 } } \\\\ & { \\qquad - \\Big \\| \\mathrm { E } _ { \\mu h } [ \\delta ( s , a ) \\phi ( s , a ) ^ { \\top } / \\sigma _ { b \\mathrm { t } \\mathrm { \\scriptsize \\ ' \\ ' } } ^ { 2 } ( s , a ) \\mathbb { S } , a ] \\Big \\| _ { 2 } + \\frac { \\lambda } { K } + \\frac { 4 \\sqrt { 2 } } { \\sqrt { K } } \\Big ( \\log \\frac { 2 d } { \\delta } \\Big ) ^ { 1 / 2 } } \\\\ & { \\qquad \\leq \\| \\phi ( s , a ) \\| ^ { 2 } + \\frac { \\lambda } { K } + \\frac { 4 \\sqrt { 2 } } { \\sqrt { K } } \\Big ( \\log \\frac { 2 d } { \\delta } \\Big ) ^ { 1 / 2 } \\leq 1 + \\frac { \\lambda } { K } + \\frac { 4 \\sqrt { 2 } } { \\sqrt { K } } \\Big ( \\log \\frac { 2 d } { \\delta } \\Big ) ^ { 1 / 2 } \\leq 2 , } \\\\ & { \\qquad \\quad \\times _ { \\operatorname* { m i n } } \\big ( \\Delta _ { h } ^ { \\prime } \\big ) \\leq \\lambda _ { \\operatorname* { m i n } } \\Big ( \\mathrm { E } _ { \\mu h } \\big ) \\big ( \\tilde { \\phi } ( s , a ) \\phi ( s , a ) ^ { \\top } / \\sigma _ { b \\mathrm { t } \\mathrm { \\scriptsize \\ ' } + 1 } ^ { 2 } ( s , a ) \\big ] \\Big ) + \\frac { \\lambda } { K } - \\frac { 4 \\sqrt { 2 } } { \\sqrt { K } } \\Big ( \\log \\frac { 2 d } { \\delta } \\Big ) ^ { 1 / 2 } } \\\\ & { \\qquad \\quad \\geq \\lambda _ { \\operatorname* { m i n } } \\Big ( \\mathrm { E } _ { \\mu h } \\big [ \\phi ( s , a ) \\phi ( s , a ) ^ { \\top } / \\sigma _ { b \\mathrm { t } \\mathrm { \\scriptsize \\ ' } + 1 } ^ { 2 } ( s , a ) \\big ] \\Big ) - \\frac { 4 \\sqrt { 2 } } { \\sqrt { K } } \\Big ( \\log \\frac { 2 d } { \\delta } \\Big ) ^ { 1 / 2 } } \\\\ & \\qquad \\quad \\geq \\frac { \\kappa } { H ^ { 2 } } - \\frac { 4 \\sqrt { 2 } } { \\sqrt { K } } \\Big ( \\log \\frac \\end{array}", + "type": "interline_equation", + "image_path": "0eaef0a745df068251121ccb4815b6c111cb574fd4d9b188402e6b893f998f80.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 113, + 114, + 503, + 175.33333333333334 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 113, + 175.33333333333334, + 503, + 236.66666666666669 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 113, + 236.66666666666669, + 503, + 298.0 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 304, + 506, + 339 + ], + "lines": [ + { + "bbox": [ + 105, + 304, + 506, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 201, + 321 + ], + "score": 1.0, + "content": "Hence with probability", + "type": "text" + }, + { + "bbox": [ + 202, + 306, + 270, + 319 + ], + "score": 0.88, + "content": "1 - \\delta , \\| \\Lambda _ { h } ^ { \\prime } \\| \\leq 2", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 304, + 288, + 321 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 289, + 305, + 426, + 320 + ], + "score": 0.94, + "content": "\\left\\| \\Lambda _ { h } ^ { \\prime - 1 } \\right\\| = 1 / \\lambda _ { \\operatorname* { m i n } } ( \\Lambda _ { h } ^ { \\prime } ) \\le 2 H ^ { 2 } / \\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 304, + 506, + 321 + ], + "score": 1.0, + "content": ". Similarly, one can", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 103, + 316, + 297, + 339 + ], + "spans": [ + { + "bbox": [ + 103, + 316, + 130, + 339 + ], + "score": 1.0, + "content": "show", + "type": "text" + }, + { + "bbox": [ + 131, + 318, + 205, + 339 + ], + "score": 0.93, + "content": "\\left\\| \\widehat { \\Lambda } _ { h } ^ { \\prime - 1 } \\right\\| \\leq 2 H ^ { 2 } / \\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 316, + 297, + 339 + ], + "score": 1.0, + "content": "with high probability.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5, + "bbox_fs": [ + 103, + 304, + 506, + 339 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 343, + 504, + 358 + ], + "lines": [ + { + "bbox": [ + 104, + 342, + 504, + 360 + ], + "spans": [ + { + "bbox": [ + 104, + 342, + 210, + 360 + ], + "score": 1.0, + "content": "Now apply Lemma H.4 to", + "type": "text" + }, + { + "bbox": [ + 210, + 343, + 223, + 358 + ], + "score": 0.91, + "content": "\\widehat { \\Lambda } _ { h } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 342, + 241, + 360 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 241, + 345, + 254, + 358 + ], + "score": 0.89, + "content": "\\Lambda _ { h } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 342, + 437, + 360 + ], + "score": 1.0, + "content": "and a union bound, we obtain with probability", + "type": "text" + }, + { + "bbox": [ + 438, + 347, + 459, + 356 + ], + "score": 0.86, + "content": "1 - \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 342, + 488, + 360 + ], + "score": 1.0, + "content": ", for all", + "type": "text" + }, + { + "bbox": [ + 489, + 350, + 504, + 357 + ], + "score": 0.84, + "content": "s , a", + "type": "inline_equation" + } + ], + "index": 7 + } + ], + "index": 7, + "bbox_fs": [ + 104, + 342, + 504, + 360 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 114, + 365, + 503, + 526 + ], + "lines": [ + { + "bbox": [ + 114, + 365, + 503, + 526 + ], + "spans": [ + { + "bbox": [ + 114, + 365, + 503, + 526 + ], + "score": 0.94, + "content": "\\begin{array} { r l } { \\left\\| \\phi ( s , a ) \\right\\| _ { \\widetilde { \\Lambda } _ { h } ^ { - 1 } } \\leq \\left[ 1 + \\sqrt { \\left\\| \\Lambda _ { h } ^ { \\prime - 1 } \\right\\| \\left\\| \\Lambda _ { h } ^ { \\prime } \\right\\| } \\cdot \\left\\| \\hat { \\Lambda } _ { h } ^ { \\prime - 1 } \\right\\| \\cdot \\left\\| \\hat { \\Lambda } _ { h } ^ { \\prime } - \\Lambda _ { h } ^ { \\prime } \\right\\| \\right] \\cdot \\left\\| \\phi ( s , a ) \\right\\| _ { \\Lambda _ { h } ^ { \\prime - 1 } } } & { } \\\\ { \\leq \\left[ 1 + \\sqrt { \\frac { 2 H ^ { 2 } } { \\kappa } } \\cdot 1 \\cdot \\frac { 2 H ^ { 2 } } { \\kappa } \\cdot \\left\\| \\hat { \\Lambda } _ { h } ^ { \\prime } - \\Lambda _ { h } ^ { \\prime } \\right\\| \\right] \\cdot \\left\\| \\phi ( s , a ) \\right\\| _ { \\Lambda _ { h } ^ { \\prime - 1 } } } & { } \\\\ { \\leq \\left[ 1 + \\sqrt { \\frac { 4 \\Delta H ^ { 4 } } { \\kappa ^ { 2 } } \\left( \\sqrt { \\frac { H ^ { 4 } d } { \\kappa K } } \\log \\left( \\frac { ( \\lambda + K ) H } { \\lambda \\delta } \\right) + \\lambda \\frac { H ^ { 2 } \\sqrt { d } } { \\kappa K } \\right) } \\right] \\cdot \\left\\| \\phi ( s , a ) \\right\\| _ { \\Lambda _ { h } ^ { \\prime - 1 } } } & { } \\\\ { \\leq \\left[ 1 + \\sqrt { \\frac { 9 6 H ^ { 4 } } { \\kappa ^ { 2 } } \\sqrt { \\frac { H ^ { 4 } d } { \\kappa K } } \\log \\left( \\frac { ( \\lambda + K ) H } { \\lambda \\delta } \\right) } \\right] \\cdot \\left\\| \\phi ( s , a ) \\right\\| _ { \\Lambda _ { h } ^ { \\prime - 1 } } \\leq 2 \\left\\| \\phi ( s , a ) \\right\\| _ { \\Lambda _ { h } ^ { \\prime - 1 } } } & { } \\end{array}", + "type": "interline_equation", + "image_path": "d5967e38fba8c718d18ae360a13bbf55703eb7ad38d01b91497cb4c7f686fce2.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 114, + 365, + 503, + 418.6666666666667 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 114, + 418.6666666666667, + 503, + 472.33333333333337 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 114, + 472.33333333333337, + 503, + 526.0 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 546, + 506, + 594 + ], + "lines": [ + { + "bbox": [ + 105, + 544, + 505, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 505, + 560 + ], + "score": 1.0, + "content": "where the third inequality uses equation 11 and the last and the second last inequality use", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 107, + 555, + 509, + 579 + ], + "spans": [ + { + "bbox": [ + 107, + 558, + 369, + 573 + ], + "score": 0.86, + "content": "\\begin{array} { r } { K > \\operatorname* { m a x } \\lbrace \\frac { \\lambda ^ { 2 } } { \\kappa \\log ( ( \\lambda + K ) H / \\lambda \\delta ) } , 9 6 ^ { 2 } H ^ { 1 2 } d \\log ( ( \\lambda + K ) H / \\lambda \\delta ) / \\kappa ^ { 5 } \\rbrace . } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 555, + 509, + 579 + ], + "score": 1.0, + "content": ". Note the above is equivalent to", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 107, + 575, + 445, + 595 + ], + "spans": [ + { + "bbox": [ + 107, + 575, + 293, + 595 + ], + "score": 0.92, + "content": "\\sqrt { \\phi ( s , a ) \\widehat \\Lambda _ { h } ^ { - 1 } \\phi ( s , a ) } \\le 2 \\sqrt { \\phi ( s , a ) \\Lambda _ { h } ^ { - 1 } \\phi ( s , a ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 576, + 356, + 595 + ], + "score": 1.0, + "content": "by multiplying", + "type": "text" + }, + { + "bbox": [ + 356, + 577, + 385, + 591 + ], + "score": 0.92, + "content": "1 / \\sqrt { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 576, + 445, + 595 + ], + "score": 1.0, + "content": "on both sides.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12, + "bbox_fs": [ + 105, + 544, + 509, + 595 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 630, + 316, + 645 + ], + "lines": [ + { + "bbox": [ + 102, + 623, + 316, + 651 + ], + "spans": [ + { + "bbox": [ + 102, + 623, + 241, + 651 + ], + "score": 1.0, + "content": "C.2.2 ANALYZING THE TERM", + "type": "text" + }, + { + "bbox": [ + 241, + 629, + 316, + 645 + ], + "score": 0.91, + "content": "\\begin{array} { r } { | | \\sum _ { \\tau = 1 } ^ { K } x _ { \\tau } \\eta _ { \\tau } | | _ { \\widehat { \\Lambda } ^ { - 1 } } } \\end{array}", + "type": "inline_equation" + } + ], + "index": 14 + } + ], + "index": 14, + "bbox_fs": [ + 102, + 623, + 316, + 651 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 652, + 512, + 697 + ], + "lines": [ + { + "bbox": [ + 105, + 651, + 513, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 651, + 188, + 675 + ], + "score": 1.0, + "content": "Lemma C.4. Recall", + "type": "text" + }, + { + "bbox": [ + 189, + 654, + 246, + 672 + ], + "score": 0.93, + "content": "\\begin{array} { r } { x _ { \\tau } = \\frac { \\phi ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) } { \\widehat { \\sigma } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 653, + 510, + 674 + ], + "score": 0.45, + "content": "\\begin{array} { r } { \\frac { a _ { h } ^ { \\tau } } { a _ { h } ^ { \\tau } ) } a n d \\eta _ { \\tau } = \\left( r _ { h } ^ { \\tau } + \\widehat { V } _ { h + 1 } \\left( s _ { h + 1 } ^ { \\tau } \\right) - \\left( \\mathcal { T } _ { h } \\widehat { V } _ { h + 1 } \\right) ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) \\right) / \\widehat { \\sigma } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) . } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 510, + 651, + 513, + 675 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 672, + 402, + 698 + ], + "spans": [ + { + "bbox": [ + 104, + 676, + 122, + 696 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 122, + 673, + 349, + 698 + ], + "score": 0.92, + "content": "\\begin{array} { r } { C _ { H , d , \\kappa , K } : = 3 6 \\sqrt { \\frac { H ^ { 4 } d ^ { 3 } } { \\kappa } \\log \\left( \\frac { ( \\lambda + K ) 2 K d H ^ { 2 } } { \\lambda \\delta } \\right) } + 1 2 \\lambda \\frac { H ^ { 2 } \\sqrt { d } } { \\kappa } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 672, + 402, + 696 + ], + "score": 1.0, + "content": "and denote", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15.5, + "bbox_fs": [ + 104, + 651, + 513, + 698 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 181, + 706, + 428, + 735 + ], + "lines": [ + { + "bbox": [ + 181, + 706, + 428, + 735 + ], + "spans": [ + { + "bbox": [ + 181, + 706, + 428, + 735 + ], + "score": 0.91, + "content": "\\xi : = \\operatorname* { s u p } _ { V \\in \\left[ 0 , H \\right] , s ^ { \\prime } \\sim P _ { h } \\left( s , a \\right) , h \\in \\left[ H \\right] } \\left| \\frac { r _ { h } + V \\left( s ^ { \\prime } \\right) - \\left( \\mathcal T _ { h } V \\right) \\left( s , a \\right) } { \\sigma _ { V } \\left( s , a \\right) } \\right| .", + "type": "interline_equation", + "image_path": "f46644a87b6e47aba6cd17b7a42b936a0b8de966101822058b87a01b8e6a98f3.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 181, + 706, + 428, + 735 + ], + "spans": [], + "index": 17 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 80, + 385, + 96 + ], + "lines": [ + { + "bbox": [ + 102, + 78, + 387, + 99 + ], + "spans": [ + { + "bbox": [ + 102, + 78, + 115, + 99 + ], + "score": 1.0, + "content": "If", + "type": "text" + }, + { + "bbox": [ + 115, + 81, + 181, + 96 + ], + "score": 0.92, + "content": "K \\ge 4 C _ { H , d , \\kappa , K } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 78, + 199, + 99 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 200, + 80, + 268, + 95 + ], + "score": 0.92, + "content": "K \\ge \\widetilde { \\cal O } ( H ^ { 6 } d / \\kappa ) .", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 78, + 357, + 99 + ], + "score": 1.0, + "content": ", then with probability", + "type": "text" + }, + { + "bbox": [ + 358, + 83, + 381, + 93 + ], + "score": 0.7, + "content": "1 - \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 78, + 387, + 99 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 115, + 103, + 493, + 138 + ], + "lines": [ + { + "bbox": [ + 115, + 103, + 493, + 138 + ], + "spans": [ + { + "bbox": [ + 115, + 103, + 493, + 138 + ], + "score": 0.92, + "content": "\\left\\| \\sum _ { \\tau = 1 } ^ { K } x _ { \\tau } \\eta _ { \\tau } \\right\\| _ { \\widehat { \\Lambda } ^ { - 1 } } \\leq 1 6 \\sqrt { d \\log \\left( 1 + \\frac { K } { \\lambda d } \\right) } \\cdot \\log \\left( \\frac { 4 K ^ { 2 } } { \\delta } \\right) + 4 \\xi \\log \\left( \\frac { 4 K ^ { 2 } } { \\delta } \\right) \\leq \\widetilde { O } \\operatorname* { m a x } \\left\\{ \\sqrt { d } , \\xi \\right\\} ,", + "type": "interline_equation", + "image_path": "87a788e78e4a4624516bff7ea43a860cc5631671d1b210be0cd2e6b2fa86694d.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 115, + 103, + 493, + 114.66666666666667 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 115, + 114.66666666666667, + 493, + 126.33333333333334 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 115, + 126.33333333333334, + 493, + 138.0 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 147, + 308, + 160 + ], + "lines": [ + { + "bbox": [ + 105, + 145, + 309, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 145, + 133, + 162 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 146, + 142, + 158 + ], + "score": 0.84, + "content": "\\widetilde O", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 145, + 309, + 162 + ], + "score": 1.0, + "content": "absorbs the constants and Polylog terms.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 107, + 177, + 505, + 203 + ], + "lines": [ + { + "bbox": [ + 105, + 177, + 505, + 192 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 305, + 192 + ], + "score": 1.0, + "content": "Proof of Lemma C.4. By construction, we have", + "type": "text" + }, + { + "bbox": [ + 306, + 178, + 394, + 191 + ], + "score": 0.92, + "content": "\\| x _ { \\tau } \\| \\le \\| \\phi / \\widehat { \\sigma } \\| \\le 1", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 177, + 505, + 192 + ], + "score": 1.0, + "content": "and by Lemma C.7, with", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 190, + 190, + 203 + ], + "spans": [ + { + "bbox": [ + 105, + 190, + 153, + 203 + ], + "score": 1.0, + "content": "probability", + "type": "text" + }, + { + "bbox": [ + 153, + 190, + 186, + 202 + ], + "score": 0.92, + "content": "1 - \\delta / 3", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 190, + 190, + 203 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5 + }, + { + "type": "interline_equation", + "bbox": [ + 123, + 208, + 487, + 252 + ], + "lines": [ + { + "bbox": [ + 123, + 208, + 487, + 252 + ], + "spans": [ + { + "bbox": [ + 123, + 208, + 487, + 252 + ], + "score": 0.94, + "content": "\\left\\| \\sigma _ { \\widehat { V } _ { h + 1 } } - \\widehat { \\sigma } _ { h } \\right\\| _ { \\infty } = \\operatorname* { s u p } _ { s , a } \\left| \\frac { \\sigma _ { \\widehat { V } _ { h + 1 } } ^ { 2 } ( s , a ) - \\widehat { \\sigma } _ { h } ^ { 2 } ( s , a ) } { \\sigma _ { \\widehat { V } _ { h + 1 } } ( s , a ) + \\widehat { \\sigma } _ { h } ( s , a ) } \\right| \\leq \\frac { 1 } { 2 } \\left\\| \\sigma _ { \\widehat { V } _ { h + 1 } } ^ { 2 } - \\widehat { \\sigma } _ { h } ^ { 2 } \\right\\| _ { \\infty } \\leq C _ { H , d , \\kappa , K } \\sqrt { \\frac { 1 } { K } }", + "type": "interline_equation", + "image_path": "1a05682e6bf307b8706bc1a9a1b48122d0ebfe3bee7380fbfaf98bbd7a772305.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 123, + 208, + 487, + 222.66666666666666 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 123, + 222.66666666666666, + 487, + 237.33333333333331 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 123, + 237.33333333333331, + 487, + 251.99999999999997 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 259, + 455, + 279 + ], + "lines": [ + { + "bbox": [ + 104, + 258, + 457, + 281 + ], + "spans": [ + { + "bbox": [ + 104, + 258, + 174, + 281 + ], + "score": 1.0, + "content": "Therefore, when", + "type": "text" + }, + { + "bbox": [ + 174, + 258, + 410, + 280 + ], + "score": 0.93, + "content": "K \\geq 4 C _ { H , d , \\kappa , K } ^ { 2 } , C _ { H , d , \\kappa , K } \\sqrt { \\frac { 1 } { K } } \\leq 1 / 2 \\leq \\sigma _ { \\widehat { V } _ { h + 1 } } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 258, + 457, + 281 + ], + "score": 1.0, + "content": "and hence", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 286, + 505, + 357 + ], + "lines": [ + { + "bbox": [ + 111, + 286, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 111, + 286, + 505, + 357 + ], + "score": 0.92, + "content": "\\begin{array} { r l } & { \\eta _ { \\tau } | \\leq \\left| \\frac { r _ { h } ^ { \\tau } + \\widehat { V } _ { h + 1 } \\left( s _ { h + 1 } ^ { \\tau } \\right) - \\left( \\mathcal { T } _ { h } \\widehat { V } _ { h + 1 } \\right) \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) } { \\sigma _ { \\widehat { V } _ { h + 1 } } \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) - \\frac { C _ { H , d , \\kappa , K } } { K ^ { 1 / 2 } } } \\right| \\leq 2 \\left| \\frac { r _ { h } ^ { \\tau } + \\widehat { V } _ { h + 1 } \\left( s _ { h + 1 } ^ { \\tau } \\right) - \\left( \\mathcal { T } _ { h } \\widehat { V } _ { h + 1 } \\right) \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) } { \\sigma _ { \\widehat { V } _ { h + 1 } } \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) } \\right| } \\\\ & { \\quad \\leq 2 \\underset { V \\in [ 0 , H ] , s ^ { \\prime } \\sim P _ { h } ( s , a ) } { \\operatorname* { s u p } } \\left| \\frac { r + V \\left( s ^ { \\prime } \\right) - \\left( \\mathcal { T } _ { h } V \\right) \\left( s , a \\right) } { \\sigma _ { V } \\left( s , a \\right) } \\right| : = \\xi . } \\end{array}", + "type": "interline_equation", + "image_path": "649ba0c516232a49c240d91990639d87ff0b65dea268745a3f6f1f097c46ea3d.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 111, + 286, + 505, + 309.6666666666667 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 111, + 309.6666666666667, + 505, + 333.33333333333337 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 111, + 333.33333333333337, + 505, + 357.00000000000006 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 378, + 506, + 402 + ], + "lines": [ + { + "bbox": [ + 104, + 376, + 507, + 392 + ], + "spans": [ + { + "bbox": [ + 104, + 376, + 206, + 392 + ], + "score": 1.0, + "content": "Next, for a fixed function", + "type": "text" + }, + { + "bbox": [ + 206, + 379, + 215, + 389 + ], + "score": 0.79, + "content": "V", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 376, + 338, + 392 + ], + "score": 1.0, + "content": ", we define the Bellman error as", + "type": "text" + }, + { + "bbox": [ + 339, + 379, + 503, + 391 + ], + "score": 0.92, + "content": "B _ { h } ( V ) ( s , a ) = r _ { h } + V ( s ^ { \\prime } ) - ( T _ { h } V ) ( s , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 376, + 507, + 392 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 388, + 128, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 128, + 402 + ], + "score": 1.0, + "content": "then", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5 + }, + { + "type": "interline_equation", + "bbox": [ + 162, + 409, + 514, + 659 + ], + "lines": [ + { + "bbox": [ + 162, + 409, + 514, + 659 + ], + "spans": [ + { + "bbox": [ + 162, + 409, + 514, + 659 + ], + "score": 0.78, + "content": "\\begin{array} { r l } & { - \\frac { \\nabla \\cdot \\mathbf { S } _ { \\perp + } [ \\hat { \\mathbf { x } } _ { \\perp } + \\hat { \\mathbf { x } } _ { \\perp } - ( \\hat { \\mathbf { x } } _ { \\perp } - [ ( \\hat { \\mathbf { x } } _ { \\perp } \\hat { \\mathbf { e } } _ { \\perp + } ^ { - \\mathbf { A } } ) ] ) \\cdot \\mathbf { \\bar { x } } _ { \\perp } \\hat { \\mathbf { e } } _ { \\perp } ^ { - \\mathbf { A } } ] \\cdot [ \\hat { \\mathbf { x } } _ { \\perp - } ] } { \\hat { \\mathcal { Q } } _ { \\perp + } [ \\hat { \\mathbf { x } } _ { \\perp } + \\hat { \\mathbf { x } } _ { \\perp } ] } } \\\\ & { - \\frac { \\nabla \\cdot \\mathbf { S } _ { \\perp + } [ \\hat { \\mathbf { x } } _ { \\perp } + \\hat { \\mathbf { x } } _ { \\perp } - \\hat { \\mathbf { x } } _ { \\perp } ] } { 2 } \\cdot \\frac { \\partial \\hat { \\mathbf { x } } _ { \\perp + } [ \\hat { \\mathbf { x } } _ { \\perp } + \\hat { \\mathbf { x } } _ { \\perp } ] \\cdot ( \\hat { \\mathbf { x } } _ { \\perp } - \\hat { \\mathbf { x } } _ { \\perp } ) } { \\partial \\hat { \\mathbf { x } } _ { \\perp } [ \\hat { \\mathbf { x } } _ { \\perp } + \\hat { \\mathbf { x } } _ { \\perp } ] } [ \\hat { \\mathbf { x } } _ { \\perp } \\hat { \\mathbf { e } } _ { \\perp + } ^ { \\mathbf { A } } ] \\hat { \\mathcal { Q } } _ { \\perp - } \\Bigg ] } \\\\ & \\phantom { = } \\frac { \\nabla \\cdot \\mathbf { S } _ { \\perp + } [ \\hat { \\mathbf { x } } _ { \\perp + } ^ { \\mathbf { B } } \\cdot \\mathbf { \\bar { x } } _ { \\perp } \\hat { \\mathbf { e } } _ { \\perp + } ^ { \\mathbf { B } } ] \\cdot [ \\hat { \\mathbf { x } } _ { \\perp } \\hat { \\mathbf { e } } _ { \\perp + } ^ { \\mathbf { B } } ] \\cdot [ \\hat { \\mathbf { x } } _ { \\perp - } \\mathbf { \\bar { x } } _ { \\perp } \\hat { \\mathbf { x } } _ { \\perp } \\hat { \\mathbf { x } } _ { \\perp } ] \\hat { \\mathcal { Q } } _ { \\perp - } } \\\\ & \\phantom { = } \\frac \\nabla \\cdot \\mathbf { S } _ { \\perp + } [ \\hat { \\mathbf { x } } _ \\ \\end{array}", + "type": "interline_equation", + "image_path": "7647db2ac043d0446feb6e873c5a35be596580529f0e35ed83643835f0b5eb85.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 162, + 409, + 514, + 492.3333333333333 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 162, + 492.3333333333333, + 514, + 575.6666666666666 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 162, + 575.6666666666666, + 514, + 659.0 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 673, + 506, + 733 + ], + "lines": [ + { + "bbox": [ + 105, + 672, + 507, + 687 + ], + "spans": [ + { + "bbox": [ + 105, + 672, + 417, + 687 + ], + "score": 1.0, + "content": "where the first inequality is by Lemma H.11, the second inequality is by", + "type": "text" + }, + { + "bbox": [ + 417, + 674, + 429, + 685 + ], + "score": 0.88, + "content": "\\mathcal { T } _ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 672, + 507, + 687 + ], + "score": 1.0, + "content": "is non-expansive,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 685, + 506, + 698 + ], + "spans": [ + { + "bbox": [ + 105, + 685, + 506, + 698 + ], + "score": 1.0, + "content": "the third inequality is by Lemma C.8, the next equality is by Markovian property, and the fourth", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 695, + 504, + 708 + ], + "spans": [ + { + "bbox": [ + 105, + 695, + 443, + 708 + ], + "score": 1.0, + "content": "inequality is by Lemma C.7 and Lemma C.10. The fifth inequality uses definition", + "type": "text" + }, + { + "bbox": [ + 443, + 695, + 504, + 708 + ], + "score": 0.89, + "content": "\\sigma _ { h , V } ( s , a ) ^ { 2 } : =", + "type": "inline_equation" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 707, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 209, + 721 + ], + "score": 0.85, + "content": "\\operatorname* { m a x } \\{ 1 , \\operatorname { V a r } _ { P _ { h } } ( V ) ( s , a ) \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 708, + 345, + 722 + ], + "score": 1.0, + "content": "and the last one is by condition", + "type": "text" + }, + { + "bbox": [ + 346, + 707, + 417, + 722 + ], + "score": 0.93, + "content": "K \\ge \\widetilde { O } ( H ^ { 6 } d / \\kappa )", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 708, + 437, + 722 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 438, + 709, + 505, + 722 + ], + "score": 0.89, + "content": "\\sigma _ { h , V ^ { \\star } } ( s , a ) ^ { 2 } : =", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 110, + 733 + ], + "score": 0.0, + "content": "", + "type": "text" + }, + { + "bbox": [ + 110, + 721, + 237, + 733 + ], + "score": 0.88, + "content": "\\operatorname* { n a x } \\{ 1 , \\operatorname { V a r } _ { P _ { h } } ( V ^ { \\star } ) ( s , a ) \\} \\ \\geq \\ 1", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 720, + 506, + 733 + ], + "score": 1.0, + "content": ". Thus, by Bernstein inequality for self-normalized martingale", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21 + } + ], + "page_idx": 20, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 26, + 293, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 310, + 761 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "21", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 80, + 385, + 96 + ], + "lines": [ + { + "bbox": [ + 102, + 78, + 387, + 99 + ], + "spans": [ + { + "bbox": [ + 102, + 78, + 115, + 99 + ], + "score": 1.0, + "content": "If", + "type": "text" + }, + { + "bbox": [ + 115, + 81, + 181, + 96 + ], + "score": 0.92, + "content": "K \\ge 4 C _ { H , d , \\kappa , K } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 78, + 199, + 99 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 200, + 80, + 268, + 95 + ], + "score": 0.92, + "content": "K \\ge \\widetilde { \\cal O } ( H ^ { 6 } d / \\kappa ) .", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 78, + 357, + 99 + ], + "score": 1.0, + "content": ", then with probability", + "type": "text" + }, + { + "bbox": [ + 358, + 83, + 381, + 93 + ], + "score": 0.7, + "content": "1 - \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 78, + 387, + 99 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 102, + 78, + 387, + 99 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 115, + 103, + 493, + 138 + ], + "lines": [ + { + "bbox": [ + 115, + 103, + 493, + 138 + ], + "spans": [ + { + "bbox": [ + 115, + 103, + 493, + 138 + ], + "score": 0.92, + "content": "\\left\\| \\sum _ { \\tau = 1 } ^ { K } x _ { \\tau } \\eta _ { \\tau } \\right\\| _ { \\widehat { \\Lambda } ^ { - 1 } } \\leq 1 6 \\sqrt { d \\log \\left( 1 + \\frac { K } { \\lambda d } \\right) } \\cdot \\log \\left( \\frac { 4 K ^ { 2 } } { \\delta } \\right) + 4 \\xi \\log \\left( \\frac { 4 K ^ { 2 } } { \\delta } \\right) \\leq \\widetilde { O } \\operatorname* { m a x } \\left\\{ \\sqrt { d } , \\xi \\right\\} ,", + "type": "interline_equation", + "image_path": "87a788e78e4a4624516bff7ea43a860cc5631671d1b210be0cd2e6b2fa86694d.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 115, + 103, + 493, + 114.66666666666667 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 115, + 114.66666666666667, + 493, + 126.33333333333334 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 115, + 126.33333333333334, + 493, + 138.0 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 147, + 308, + 160 + ], + "lines": [ + { + "bbox": [ + 105, + 145, + 309, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 145, + 133, + 162 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 146, + 142, + 158 + ], + "score": 0.84, + "content": "\\widetilde O", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 145, + 309, + 162 + ], + "score": 1.0, + "content": "absorbs the constants and Polylog terms.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4, + "bbox_fs": [ + 105, + 145, + 309, + 162 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 177, + 505, + 203 + ], + "lines": [ + { + "bbox": [ + 105, + 177, + 505, + 192 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 305, + 192 + ], + "score": 1.0, + "content": "Proof of Lemma C.4. By construction, we have", + "type": "text" + }, + { + "bbox": [ + 306, + 178, + 394, + 191 + ], + "score": 0.92, + "content": "\\| x _ { \\tau } \\| \\le \\| \\phi / \\widehat { \\sigma } \\| \\le 1", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 177, + 505, + 192 + ], + "score": 1.0, + "content": "and by Lemma C.7, with", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 190, + 190, + 203 + ], + "spans": [ + { + "bbox": [ + 105, + 190, + 153, + 203 + ], + "score": 1.0, + "content": "probability", + "type": "text" + }, + { + "bbox": [ + 153, + 190, + 186, + 202 + ], + "score": 0.92, + "content": "1 - \\delta / 3", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 190, + 190, + 203 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5, + "bbox_fs": [ + 105, + 177, + 505, + 203 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 123, + 208, + 487, + 252 + ], + "lines": [ + { + "bbox": [ + 123, + 208, + 487, + 252 + ], + "spans": [ + { + "bbox": [ + 123, + 208, + 487, + 252 + ], + "score": 0.94, + "content": "\\left\\| \\sigma _ { \\widehat { V } _ { h + 1 } } - \\widehat { \\sigma } _ { h } \\right\\| _ { \\infty } = \\operatorname* { s u p } _ { s , a } \\left| \\frac { \\sigma _ { \\widehat { V } _ { h + 1 } } ^ { 2 } ( s , a ) - \\widehat { \\sigma } _ { h } ^ { 2 } ( s , a ) } { \\sigma _ { \\widehat { V } _ { h + 1 } } ( s , a ) + \\widehat { \\sigma } _ { h } ( s , a ) } \\right| \\leq \\frac { 1 } { 2 } \\left\\| \\sigma _ { \\widehat { V } _ { h + 1 } } ^ { 2 } - \\widehat { \\sigma } _ { h } ^ { 2 } \\right\\| _ { \\infty } \\leq C _ { H , d , \\kappa , K } \\sqrt { \\frac { 1 } { K } }", + "type": "interline_equation", + "image_path": "1a05682e6bf307b8706bc1a9a1b48122d0ebfe3bee7380fbfaf98bbd7a772305.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 123, + 208, + 487, + 222.66666666666666 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 123, + 222.66666666666666, + 487, + 237.33333333333331 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 123, + 237.33333333333331, + 487, + 251.99999999999997 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 259, + 455, + 279 + ], + "lines": [ + { + "bbox": [ + 104, + 258, + 457, + 281 + ], + "spans": [ + { + "bbox": [ + 104, + 258, + 174, + 281 + ], + "score": 1.0, + "content": "Therefore, when", + "type": "text" + }, + { + "bbox": [ + 174, + 258, + 410, + 280 + ], + "score": 0.93, + "content": "K \\geq 4 C _ { H , d , \\kappa , K } ^ { 2 } , C _ { H , d , \\kappa , K } \\sqrt { \\frac { 1 } { K } } \\leq 1 / 2 \\leq \\sigma _ { \\widehat { V } _ { h + 1 } } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 258, + 457, + 281 + ], + "score": 1.0, + "content": "and hence", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10, + "bbox_fs": [ + 104, + 258, + 457, + 281 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 286, + 505, + 357 + ], + "lines": [ + { + "bbox": [ + 111, + 286, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 111, + 286, + 505, + 357 + ], + "score": 0.92, + "content": "\\begin{array} { r l } & { \\eta _ { \\tau } | \\leq \\left| \\frac { r _ { h } ^ { \\tau } + \\widehat { V } _ { h + 1 } \\left( s _ { h + 1 } ^ { \\tau } \\right) - \\left( \\mathcal { T } _ { h } \\widehat { V } _ { h + 1 } \\right) \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) } { \\sigma _ { \\widehat { V } _ { h + 1 } } \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) - \\frac { C _ { H , d , \\kappa , K } } { K ^ { 1 / 2 } } } \\right| \\leq 2 \\left| \\frac { r _ { h } ^ { \\tau } + \\widehat { V } _ { h + 1 } \\left( s _ { h + 1 } ^ { \\tau } \\right) - \\left( \\mathcal { T } _ { h } \\widehat { V } _ { h + 1 } \\right) \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) } { \\sigma _ { \\widehat { V } _ { h + 1 } } \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) } \\right| } \\\\ & { \\quad \\leq 2 \\underset { V \\in [ 0 , H ] , s ^ { \\prime } \\sim P _ { h } ( s , a ) } { \\operatorname* { s u p } } \\left| \\frac { r + V \\left( s ^ { \\prime } \\right) - \\left( \\mathcal { T } _ { h } V \\right) \\left( s , a \\right) } { \\sigma _ { V } \\left( s , a \\right) } \\right| : = \\xi . } \\end{array}", + "type": "interline_equation", + "image_path": "649ba0c516232a49c240d91990639d87ff0b65dea268745a3f6f1f097c46ea3d.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 111, + 286, + 505, + 309.6666666666667 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 111, + 309.6666666666667, + 505, + 333.33333333333337 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 111, + 333.33333333333337, + 505, + 357.00000000000006 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 378, + 506, + 402 + ], + "lines": [ + { + "bbox": [ + 104, + 376, + 507, + 392 + ], + "spans": [ + { + "bbox": [ + 104, + 376, + 206, + 392 + ], + "score": 1.0, + "content": "Next, for a fixed function", + "type": "text" + }, + { + "bbox": [ + 206, + 379, + 215, + 389 + ], + "score": 0.79, + "content": "V", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 376, + 338, + 392 + ], + "score": 1.0, + "content": ", we define the Bellman error as", + "type": "text" + }, + { + "bbox": [ + 339, + 379, + 503, + 391 + ], + "score": 0.92, + "content": "B _ { h } ( V ) ( s , a ) = r _ { h } + V ( s ^ { \\prime } ) - ( T _ { h } V ) ( s , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 376, + 507, + 392 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 388, + 128, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 128, + 402 + ], + "score": 1.0, + "content": "then", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5, + "bbox_fs": [ + 104, + 376, + 507, + 402 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 162, + 409, + 514, + 659 + ], + "lines": [ + { + "bbox": [ + 162, + 409, + 514, + 659 + ], + "spans": [ + { + "bbox": [ + 162, + 409, + 514, + 659 + ], + "score": 0.78, + "content": "\\begin{array} { r l } & { - \\frac { \\nabla \\cdot \\mathbf { S } _ { \\perp + } [ \\hat { \\mathbf { x } } _ { \\perp } + \\hat { \\mathbf { x } } _ { \\perp } - ( \\hat { \\mathbf { x } } _ { \\perp } - [ ( \\hat { \\mathbf { x } } _ { \\perp } \\hat { \\mathbf { e } } _ { \\perp + } ^ { - \\mathbf { A } } ) ] ) \\cdot \\mathbf { \\bar { x } } _ { \\perp } \\hat { \\mathbf { e } } _ { \\perp } ^ { - \\mathbf { A } } ] \\cdot [ \\hat { \\mathbf { x } } _ { \\perp - } ] } { \\hat { \\mathcal { Q } } _ { \\perp + } [ \\hat { \\mathbf { x } } _ { \\perp } + \\hat { \\mathbf { x } } _ { \\perp } ] } } \\\\ & { - \\frac { \\nabla \\cdot \\mathbf { S } _ { \\perp + } [ \\hat { \\mathbf { x } } _ { \\perp } + \\hat { \\mathbf { x } } _ { \\perp } - \\hat { \\mathbf { x } } _ { \\perp } ] } { 2 } \\cdot \\frac { \\partial \\hat { \\mathbf { x } } _ { \\perp + } [ \\hat { \\mathbf { x } } _ { \\perp } + \\hat { \\mathbf { x } } _ { \\perp } ] \\cdot ( \\hat { \\mathbf { x } } _ { \\perp } - \\hat { \\mathbf { x } } _ { \\perp } ) } { \\partial \\hat { \\mathbf { x } } _ { \\perp } [ \\hat { \\mathbf { x } } _ { \\perp } + \\hat { \\mathbf { x } } _ { \\perp } ] } [ \\hat { \\mathbf { x } } _ { \\perp } \\hat { \\mathbf { e } } _ { \\perp + } ^ { \\mathbf { A } } ] \\hat { \\mathcal { Q } } _ { \\perp - } \\Bigg ] } \\\\ & \\phantom { = } \\frac { \\nabla \\cdot \\mathbf { S } _ { \\perp + } [ \\hat { \\mathbf { x } } _ { \\perp + } ^ { \\mathbf { B } } \\cdot \\mathbf { \\bar { x } } _ { \\perp } \\hat { \\mathbf { e } } _ { \\perp + } ^ { \\mathbf { B } } ] \\cdot [ \\hat { \\mathbf { x } } _ { \\perp } \\hat { \\mathbf { e } } _ { \\perp + } ^ { \\mathbf { B } } ] \\cdot [ \\hat { \\mathbf { x } } _ { \\perp - } \\mathbf { \\bar { x } } _ { \\perp } \\hat { \\mathbf { x } } _ { \\perp } \\hat { \\mathbf { x } } _ { \\perp } ] \\hat { \\mathcal { Q } } _ { \\perp - } } \\\\ & \\phantom { = } \\frac \\nabla \\cdot \\mathbf { S } _ { \\perp + } [ \\hat { \\mathbf { x } } _ \\ \\end{array}", + "type": "interline_equation", + "image_path": "7647db2ac043d0446feb6e873c5a35be596580529f0e35ed83643835f0b5eb85.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 162, + 409, + 514, + 492.3333333333333 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 162, + 492.3333333333333, + 514, + 575.6666666666666 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 162, + 575.6666666666666, + 514, + 659.0 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 673, + 506, + 733 + ], + "lines": [ + { + "bbox": [ + 105, + 672, + 507, + 687 + ], + "spans": [ + { + "bbox": [ + 105, + 672, + 417, + 687 + ], + "score": 1.0, + "content": "where the first inequality is by Lemma H.11, the second inequality is by", + "type": "text" + }, + { + "bbox": [ + 417, + 674, + 429, + 685 + ], + "score": 0.88, + "content": "\\mathcal { T } _ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 672, + 507, + 687 + ], + "score": 1.0, + "content": "is non-expansive,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 685, + 506, + 698 + ], + "spans": [ + { + "bbox": [ + 105, + 685, + 506, + 698 + ], + "score": 1.0, + "content": "the third inequality is by Lemma C.8, the next equality is by Markovian property, and the fourth", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 695, + 504, + 708 + ], + "spans": [ + { + "bbox": [ + 105, + 695, + 443, + 708 + ], + "score": 1.0, + "content": "inequality is by Lemma C.7 and Lemma C.10. The fifth inequality uses definition", + "type": "text" + }, + { + "bbox": [ + 443, + 695, + 504, + 708 + ], + "score": 0.89, + "content": "\\sigma _ { h , V } ( s , a ) ^ { 2 } : =", + "type": "inline_equation" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 707, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 209, + 721 + ], + "score": 0.85, + "content": "\\operatorname* { m a x } \\{ 1 , \\operatorname { V a r } _ { P _ { h } } ( V ) ( s , a ) \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 708, + 345, + 722 + ], + "score": 1.0, + "content": "and the last one is by condition", + "type": "text" + }, + { + "bbox": [ + 346, + 707, + 417, + 722 + ], + "score": 0.93, + "content": "K \\ge \\widetilde { O } ( H ^ { 6 } d / \\kappa )", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 708, + 437, + 722 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 438, + 709, + 505, + 722 + ], + "score": 0.89, + "content": "\\sigma _ { h , V ^ { \\star } } ( s , a ) ^ { 2 } : =", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 110, + 733 + ], + "score": 0.0, + "content": "", + "type": "text" + }, + { + "bbox": [ + 110, + 721, + 237, + 733 + ], + "score": 0.88, + "content": "\\operatorname* { n a x } \\{ 1 , \\operatorname { V a r } _ { P _ { h } } ( V ^ { \\star } ) ( s , a ) \\} \\ \\geq \\ 1", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 720, + 506, + 733 + ], + "score": 1.0, + "content": ". Thus, by Bernstein inequality for self-normalized martingale", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 672, + 507, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 263, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 264, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 236, + 96 + ], + "score": 1.0, + "content": "(Lemma H.3),9 with probability", + "type": "text" + }, + { + "bbox": [ + 236, + 83, + 259, + 93 + ], + "score": 0.81, + "content": "1 - \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 80, + 264, + 96 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 100, + 501, + 136 + ], + "lines": [ + { + "bbox": [ + 111, + 100, + 501, + 136 + ], + "spans": [ + { + "bbox": [ + 111, + 100, + 501, + 136 + ], + "score": 0.92, + "content": "\\left\\| \\sum _ { \\tau = 1 } ^ { K } x _ { \\tau } \\eta _ { \\tau } \\right\\| _ { \\widehat { \\Lambda } ^ { - 1 } } \\leq \\widetilde { \\mathcal { O } } \\left( \\sqrt { d \\log \\left( 1 + \\frac { K } { \\lambda d } \\right) } \\cdot \\log \\left( \\frac { 4 K ^ { 2 } } { \\delta } \\right) \\right) + 4 \\xi \\log \\left( \\frac { 4 K ^ { 2 } } { \\delta } \\right) \\leq \\widetilde { \\mathcal { O } } \\operatorname* { m a x } \\left\\{ \\sqrt { d } , \\xi \\right\\}", + "type": "interline_equation", + "image_path": "c035e458bda461595ff07b2ae3671e844ae1bf072f841fda7aa562c98230b031.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 111, + 100, + 501, + 112.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 111, + 112.0, + 501, + 124.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 111, + 124.0, + 501, + 136.0 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 144, + 307, + 157 + ], + "lines": [ + { + "bbox": [ + 105, + 142, + 308, + 159 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 133, + 159 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 143, + 142, + 155 + ], + "score": 0.86, + "content": "\\widetilde O", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 142, + 308, + 159 + ], + "score": 1.0, + "content": "absorbs the constants and Polylog terms.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 106, + 173, + 505, + 186 + ], + "lines": [ + { + "bbox": [ + 105, + 172, + 506, + 187 + ], + "spans": [ + { + "bbox": [ + 105, + 172, + 134, + 187 + ], + "score": 1.0, + "content": "Recall", + "type": "text" + }, + { + "bbox": [ + 135, + 174, + 214, + 185 + ], + "score": 0.92, + "content": "\\mathcal { M } _ { 1 } , \\mathcal { M } _ { 2 } , \\mathcal { M } _ { 3 } , \\mathcal { M } _ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 172, + 506, + 187 + ], + "score": 1.0, + "content": "in List A. Based on the above results, we have the following key lemma:", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 106, + 190, + 506, + 240 + ], + "lines": [ + { + "bbox": [ + 105, + 189, + 507, + 204 + ], + "spans": [ + { + "bbox": [ + 105, + 189, + 200, + 204 + ], + "score": 1.0, + "content": "Lemma C.5. Assume", + "type": "text" + }, + { + "bbox": [ + 201, + 190, + 336, + 203 + ], + "score": 0.9, + "content": "K > \\operatorname* { m a x } \\{ \\mathcal { M } _ { 1 } , \\mathcal { M } _ { 2 } , \\mathcal { M } _ { 3 } , \\mathcal { M } _ { 4 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 189, + 374, + 204 + ], + "score": 1.0, + "content": ", for any", + "type": "text" + }, + { + "bbox": [ + 374, + 191, + 426, + 201 + ], + "score": 0.87, + "content": "0 < \\lambda < \\kappa ,", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 189, + 466, + 204 + ], + "score": 1.0, + "content": ", suppose", + "type": "text" + }, + { + "bbox": [ + 467, + 189, + 503, + 203 + ], + "score": 0.91, + "content": "\\sqrt { d } > \\xi", + "type": "inline_equation" + }, + { + "bbox": [ + 504, + 189, + 507, + 204 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 133, + 203, + 506, + 228 + ], + "spans": [ + { + "bbox": [ + 133, + 203, + 361, + 228 + ], + "score": 0.81, + "content": "\\xi : = \\operatorname* { s u p } _ { V \\in [ 0 , H ] , \\ s ^ { \\prime } \\sim P _ { h } ( s , a ) , \\ h \\in [ H ] } \\left| \\frac { r _ { h } + V \\big ( s ^ { \\prime } \\big ) - ( T _ { h } V ) ( s , a ) } { \\sigma _ { V } ( s , a ) } \\right| .", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 205, + 451, + 223 + ], + "score": 1.0, + "content": "Then with probability", + "type": "text" + }, + { + "bbox": [ + 451, + 209, + 475, + 220 + ], + "score": 0.67, + "content": "1 - \\delta _ { : }", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 205, + 506, + 223 + ], + "score": 1.0, + "content": ", for all", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 226, + 202, + 239 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 199, + 239 + ], + "score": 0.75, + "content": "h , s , a \\in [ H ] \\times \\mathcal { S } \\times \\mathcal { A } ,", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 226, + 202, + 239 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7 + }, + { + "type": "interline_equation", + "bbox": [ + 155, + 246, + 455, + 274 + ], + "lines": [ + { + "bbox": [ + 155, + 246, + 455, + 274 + ], + "spans": [ + { + "bbox": [ + 155, + 246, + 455, + 274 + ], + "score": 0.9, + "content": "\\left| ( { \\cal T } _ { h } \\widehat V _ { h + 1 } - \\widehat T _ { h } \\widehat V _ { h + 1 } ) ( s , a ) \\right| \\leq \\widetilde O \\left( \\sqrt { d } \\sqrt { \\phi ( s , a ) \\Lambda _ { h } ^ { - 1 } \\phi ( s , a ) } \\right) + \\frac { 2 H ^ { 3 } \\sqrt { d } } { K } ,", + "type": "interline_equation", + "image_path": "5ca4ed36e8c44d91c97ff962c4ad7623fd3c949f127a55a8acddea465251d4ab.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 155, + 246, + 455, + 274 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 280, + 504, + 310 + ], + "lines": [ + { + "bbox": [ + 102, + 276, + 509, + 300 + ], + "spans": [ + { + "bbox": [ + 102, + 276, + 132, + 300 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 132, + 281, + 352, + 299 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\Lambda _ { h } = \\sum _ { \\tau = 1 } ^ { K } \\phi ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) ^ { \\top } \\phi ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) / \\sigma _ { \\widehat { V } _ { h + 1 } } ^ { 2 } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) + \\lambda I } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 276, + 370, + 300 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 370, + 281, + 379, + 294 + ], + "score": 0.82, + "content": "\\widetilde O", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 276, + 509, + 300 + ], + "score": 1.0, + "content": "absorbs the universal constants", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 297, + 184, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 184, + 311 + ], + "score": 1.0, + "content": "and Polylog terms.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 107, + 325, + 505, + 349 + ], + "lines": [ + { + "bbox": [ + 106, + 325, + 506, + 339 + ], + "spans": [ + { + "bbox": [ + 106, + 325, + 506, + 339 + ], + "score": 1.0, + "content": "Proof of Lemma C.5. Combing equation 9, Lemma C.2, equation 10, Lemma C.3 and C.4 and a", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 337, + 504, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 337, + 235, + 349 + ], + "score": 1.0, + "content": "union bound to finish the proof.", + "type": "text" + }, + { + "bbox": [ + 496, + 339, + 504, + 347 + ], + "score": 0.998, + "content": "□", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5 + }, + { + "type": "title", + "bbox": [ + 106, + 363, + 322, + 376 + ], + "lines": [ + { + "bbox": [ + 106, + 363, + 322, + 376 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 322, + 376 + ], + "score": 1.0, + "content": "C.3 PROOF OF THE FIRST PART OF THEOREM 3.2", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 106, + 383, + 506, + 443 + ], + "lines": [ + { + "bbox": [ + 105, + 383, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 289, + 398 + ], + "score": 1.0, + "content": "Theorem C.6 (First part of Theorem 3.2). Let", + "type": "text" + }, + { + "bbox": [ + 290, + 385, + 300, + 395 + ], + "score": 0.74, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 383, + 443, + 398 + ], + "score": 1.0, + "content": "be the number of episodes. Suppose", + "type": "text" + }, + { + "bbox": [ + 443, + 383, + 475, + 397 + ], + "score": 0.92, + "content": "\\sqrt { d } > \\xi", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 383, + 505, + 398 + ], + "score": 1.0, + "content": ", where", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 396, + 506, + 422 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 339, + 422 + ], + "score": 0.68, + "content": "\\begin{array} { r } { \\xi : = \\operatorname* { s u p } _ { V \\in [ 0 , H ] , \\ s ^ { \\prime } \\sim P _ { h } ( s , a ) , \\ h \\in [ H ] } \\left| \\frac { r _ { h } + V \\left( s ^ { \\prime } \\right) - ( \\mathcal T _ { h } V ) ( s , a ) } { \\sigma _ { V } ( s , a ) } \\right| } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 400, + 360, + 415 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 360, + 402, + 503, + 415 + ], + "score": 0.88, + "content": "K > \\operatorname* { m a x } \\{ \\mathcal { M } _ { 1 } , \\mathcal { M } _ { 2 } , \\mathcal { M } _ { 3 } , \\mathcal { M } _ { 4 } \\} ^ { 1 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 400, + 506, + 415 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 419, + 506, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 162, + 433 + ], + "score": 1.0, + "content": "Then for any", + "type": "text" + }, + { + "bbox": [ + 162, + 420, + 209, + 431 + ], + "score": 0.9, + "content": "0 < \\lambda < \\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 419, + 281, + 433 + ], + "score": 1.0, + "content": ", with probability", + "type": "text" + }, + { + "bbox": [ + 282, + 421, + 306, + 431 + ], + "score": 0.33, + "content": "1 - 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v ^ { \\widehat { \\pi } } \\leq \\widetilde { O } \\left( \\sqrt { d } \\cdot \\sum _ { h = 1 } ^ { H } \\mathbb { E } _ { \\pi } \\left[ \\left( \\phi ( \\cdot , \\cdot ) ^ { \\top } \\Lambda _ { h } ^ { - 1 } \\phi ( \\cdot , \\cdot ) \\right) ^ { 1 / 2 } \\right] \\right) + \\frac { 2 H ^ { 4 } \\sqrt { d } } { K }", + "type": "interline_equation", + "image_path": "4163ed88fbfb0446b698a10f32143c31548b24a7dda8bc846707cd7f2e1e8390.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 168, + 450, + 443, + 461.3333333333333 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 168, + 461.3333333333333, + 443, + 472.66666666666663 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 168, + 472.66666666666663, + 443, + 483.99999999999994 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 491, + 506, + 523 + ], + "lines": [ + { + "bbox": [ + 101, + 485, + 507, + 515 + ], + "spans": [ + { + "bbox": [ + 101, + 485, + 132, + 514 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 490, + 286, + 515 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\Lambda _ { h } = \\sum _ { \\tau = 1 } ^ { K } \\frac { \\phi ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) \\cdot \\phi ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) ^ { \\top } } { \\sigma _ { \\hat { V } _ { h + 1 } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) } ^ { 2 } } + \\lambda I _ { d } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 490, + 507, + 509 + ], + "score": 1.0, + "content": "1 φ(sτh,aτh)·φ(sτh,aτh)>2 + λId and O absorbs the universal constants and the Polylog", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 511, + 134, + 524 + ], + "spans": [ + { + "bbox": [ + 104, + 511, + 134, + 524 + ], + "score": 1.0, + "content": "terms.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 105, + 539, + 504, + 563 + ], + "lines": [ + { + "bbox": [ + 105, + 537, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 505, + 554 + ], + "score": 1.0, + "content": "Proof of Theorem C.6. Combing Lemma C.1 and Lemma C.5, we directly have with probability", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 550, + 260, + 564 + ], + "spans": [ + { + "bbox": [ + 106, + 552, + 129, + 562 + ], + "score": 0.88, + "content": "1 - \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 550, + 186, + 564 + ], + "score": 1.0, + "content": ", for all policy", + "type": "text" + }, + { + "bbox": [ + 187, + 554, + 194, + 561 + ], + "score": 0.77, + "content": "\\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 550, + 260, + 564 + ], + "score": 1.0, + "content": "simultaneously,", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5 + }, + { + "type": "interline_equation", + "bbox": [ + 128, + 569, + 466, + 604 + ], + "lines": [ + { + "bbox": [ + 128, + 569, + 466, + 604 + ], + "spans": [ + { + "bbox": [ + 128, + 569, + 466, + 604 + ], + "score": 0.94, + "content": "V _ { 1 } ^ { \\pi } ( s ) - V _ { 1 } ^ { \\widehat { \\pi } } ( s ) \\leq \\widetilde { O } \\left( \\sqrt { d } \\cdot \\sum _ { h = 1 } ^ { H } \\mathbb { E } _ { \\pi } \\left[ \\left( \\phi ( \\cdot , \\cdot ) ^ { \\top } \\Lambda _ { h } ^ { - 1 } \\phi ( \\cdot , \\cdot ) \\right) ^ { 1 / 2 } \\middle | s _ { 1 } = s \\right] \\right) + \\frac { 2 H ^ { 4 } \\sqrt { d } } { K } ,", + "type": "interline_equation", + "image_path": "a0a189412cb0543c770e196f77cccb7d9349335e0a73429fc8248c1912cd7bed.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 128, + 569, + 466, + 580.6666666666666 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 128, + 580.6666666666666, + 466, + 592.3333333333333 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 128, + 592.3333333333333, + 466, + 603.9999999999999 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 610, + 393, + 622 + ], + "lines": [ + { + "bbox": [ + 105, + 609, + 392, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 233, + 623 + ], + "score": 1.0, + "content": "now take the initial distribution", + "type": "text" + }, + { + "bbox": [ + 234, + 611, + 244, + 621 + ], + "score": 0.88, + "content": "d _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 609, + 392, + 623 + ], + "score": 1.0, + "content": "on both sides to get the stated result.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "title", + "bbox": [ + 108, + 636, + 262, + 648 + ], + "lines": [ + { + "bbox": [ + 106, + 636, + 262, + 649 + ], + "spans": [ + { + "bbox": [ + 106, + 636, + 262, + 649 + ], + "score": 1.0, + "content": "C.4 TWO INTERMEDIATE RESULTS", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 108, + 657, + 426, + 670 + ], + "lines": [ + { + "bbox": [ + 105, + 656, + 428, + 671 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 428, + 671 + ], + "score": 1.0, + "content": "The next two lemmas provide intermediate results in finishing the whole proofs.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + } + ], + "page_idx": 21, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 26, + 293, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 765 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 765 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 15 + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 494, + 145, + 505, + 156 + ], + "lines": [ + { + "bbox": [ + 496, + 147, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 496, + 147, + 505, + 156 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 109, + 680, + 506, + 721 + ], + "lines": [ + { + "bbox": [ + 118, + 678, + 506, + 693 + ], + "spans": [ + { + "bbox": [ + 118, + 678, + 506, + 693 + ], + "score": 1.0, + "content": "9To be rigorous, Lemma H.3 needs to be modified since the absolute value bound and the variance bound", + "type": "text" + } + ] + }, + { + "bbox": [ + 107, + 691, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 107, + 691, + 505, + 701 + ], + "score": 1.0, + "content": "here are in the high probability sense. However, this will not affect the validity of the result as the weaker version", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 700, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 506, + 712 + ], + "score": 1.0, + "content": "can also be obtained (see Chung and Lu (2006) and a related discussion in Yin et al. (2021) Remark E.7.) To", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 709, + 417, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 417, + 722 + ], + "score": 1.0, + "content": "make the proof more readable, we do not include them here to avoid over-technicality.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 494, + 610, + 505, + 621 + ], + "lines": [ + { + "bbox": [ + 495, + 612, + 505, + 621 + ], + "spans": [ + { + "bbox": [ + 495, + 612, + 505, + 621 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ] + } + ] + }, + { + "type": "text", + "bbox": [ + 118, + 721, + 245, + 731 + ], + "lines": [] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 263, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 264, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 236, + 96 + ], + "score": 1.0, + "content": "(Lemma H.3),9 with probability", + "type": "text" + }, + { + "bbox": [ + 236, + 83, + 259, + 93 + ], + "score": 0.81, + "content": "1 - \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 80, + 264, + 96 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 105, + 80, + 264, + 96 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 100, + 501, + 136 + ], + "lines": [ + { + "bbox": [ + 111, + 100, + 501, + 136 + ], + "spans": [ + { + "bbox": [ + 111, + 100, + 501, + 136 + ], + "score": 0.92, + "content": "\\left\\| \\sum _ { \\tau = 1 } ^ { K } x _ { \\tau } \\eta _ { \\tau } \\right\\| _ { \\widehat { \\Lambda } ^ { - 1 } } \\leq \\widetilde { \\mathcal { O } } \\left( \\sqrt { d \\log \\left( 1 + \\frac { K } { \\lambda d } \\right) } \\cdot \\log \\left( \\frac { 4 K ^ { 2 } } { \\delta } \\right) \\right) + 4 \\xi \\log \\left( \\frac { 4 K ^ { 2 } } { \\delta } \\right) \\leq \\widetilde { \\mathcal { O } } \\operatorname* { m a x } \\left\\{ \\sqrt { d } , \\xi \\right\\}", + "type": "interline_equation", + "image_path": "c035e458bda461595ff07b2ae3671e844ae1bf072f841fda7aa562c98230b031.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 111, + 100, + 501, + 112.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 111, + 112.0, + 501, + 124.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 111, + 124.0, + 501, + 136.0 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 144, + 307, + 157 + ], + "lines": [ + { + "bbox": [ + 105, + 142, + 308, + 159 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 133, + 159 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 143, + 142, + 155 + ], + "score": 0.86, + "content": "\\widetilde O", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 142, + 308, + 159 + ], + "score": 1.0, + "content": "absorbs the constants and Polylog terms.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4, + "bbox_fs": [ + 105, + 142, + 308, + 159 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 173, + 505, + 186 + ], + "lines": [ + { + "bbox": [ + 105, + 172, + 506, + 187 + ], + "spans": [ + { + "bbox": [ + 105, + 172, + 134, + 187 + ], + "score": 1.0, + "content": "Recall", + "type": "text" + }, + { + "bbox": [ + 135, + 174, + 214, + 185 + ], + "score": 0.92, + "content": "\\mathcal { M } _ { 1 } , \\mathcal { M } _ { 2 } , \\mathcal { M } _ { 3 } , \\mathcal { M } _ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 172, + 506, + 187 + ], + "score": 1.0, + "content": "in List A. Based on the above results, we have the following key lemma:", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5, + "bbox_fs": [ + 105, + 172, + 506, + 187 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 190, + 506, + 240 + ], + "lines": [ + { + "bbox": [ + 105, + 189, + 507, + 204 + ], + "spans": [ + { + "bbox": [ + 105, + 189, + 200, + 204 + ], + "score": 1.0, + "content": "Lemma C.5. Assume", + "type": "text" + }, + { + "bbox": [ + 201, + 190, + 336, + 203 + ], + "score": 0.9, + "content": "K > \\operatorname* { m a x } \\{ \\mathcal { M } _ { 1 } , \\mathcal { M } _ { 2 } , \\mathcal { M } _ { 3 } , \\mathcal { M } _ { 4 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 189, + 374, + 204 + ], + "score": 1.0, + "content": ", for any", + "type": "text" + }, + { + "bbox": [ + 374, + 191, + 426, + 201 + ], + "score": 0.87, + "content": "0 < \\lambda < \\kappa ,", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 189, + 466, + 204 + ], + "score": 1.0, + "content": ", suppose", + "type": "text" + }, + { + "bbox": [ + 467, + 189, + 503, + 203 + ], + "score": 0.91, + "content": "\\sqrt { d } > \\xi", + "type": "inline_equation" + }, + { + "bbox": [ + 504, + 189, + 507, + 204 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 133, + 203, + 506, + 228 + ], + "spans": [ + { + "bbox": [ + 133, + 203, + 361, + 228 + ], + "score": 0.81, + "content": "\\xi : = \\operatorname* { s u p } _ { V \\in [ 0 , H ] , \\ s ^ { \\prime } \\sim P _ { h } ( s , a ) , \\ h \\in [ H ] } \\left| \\frac { r _ { h } + V \\big ( s ^ { \\prime } \\big ) - ( T _ { h } V ) ( s , a ) } { \\sigma _ { V } ( s , a ) } \\right| .", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 205, + 451, + 223 + ], + "score": 1.0, + "content": "Then with probability", + "type": "text" + }, + { + "bbox": [ + 451, + 209, + 475, + 220 + ], + "score": 0.67, + "content": "1 - \\delta _ { : }", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 205, + 506, + 223 + ], + "score": 1.0, + "content": ", for all", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 226, + 202, + 239 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 199, + 239 + ], + "score": 0.75, + "content": "h , s , a \\in [ H ] \\times \\mathcal { S } \\times \\mathcal { A } ,", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 226, + 202, + 239 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 189, + 507, + 239 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 155, + 246, + 455, + 274 + ], + "lines": [ + { + "bbox": [ + 155, + 246, + 455, + 274 + ], + "spans": [ + { + "bbox": [ + 155, + 246, + 455, + 274 + ], + "score": 0.9, + "content": "\\left| ( { \\cal T } _ { h } \\widehat V _ { h + 1 } - \\widehat T _ { h } \\widehat V _ { h + 1 } ) ( s , a ) \\right| \\leq \\widetilde O \\left( \\sqrt { d } \\sqrt { \\phi ( s , a ) \\Lambda _ { h } ^ { - 1 } \\phi ( s , a ) } \\right) + \\frac { 2 H ^ { 3 } \\sqrt { d } } { K } ,", + "type": "interline_equation", + "image_path": "5ca4ed36e8c44d91c97ff962c4ad7623fd3c949f127a55a8acddea465251d4ab.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 155, + 246, + 455, + 274 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 280, + 504, + 310 + ], + "lines": [ + { + "bbox": [ + 102, + 276, + 509, + 300 + ], + "spans": [ + { + "bbox": [ + 102, + 276, + 132, + 300 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 132, + 281, + 352, + 299 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\Lambda _ { h } = \\sum _ { \\tau = 1 } ^ { K } \\phi ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) ^ { \\top } \\phi ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) / \\sigma _ { \\widehat { V } _ { h + 1 } } ^ { 2 } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) + \\lambda I } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 276, + 370, + 300 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 370, + 281, + 379, + 294 + ], + "score": 0.82, + "content": "\\widetilde O", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 276, + 509, + 300 + ], + "score": 1.0, + "content": "absorbs the universal constants", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 297, + 184, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 184, + 311 + ], + "score": 1.0, + "content": "and Polylog terms.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5, + "bbox_fs": [ + 102, + 276, + 509, + 311 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 325, + 505, + 349 + ], + "lines": [ + { + "bbox": [ + 106, + 325, + 506, + 339 + ], + "spans": [ + { + "bbox": [ + 106, + 325, + 506, + 339 + ], + "score": 1.0, + "content": "Proof of Lemma C.5. Combing equation 9, Lemma C.2, equation 10, Lemma C.3 and C.4 and a", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 337, + 504, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 337, + 235, + 349 + ], + "score": 1.0, + "content": "union bound to finish the proof.", + "type": "text" + }, + { + "bbox": [ + 496, + 339, + 504, + 347 + ], + "score": 0.998, + "content": "□", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 325, + 506, + 349 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 363, + 322, + 376 + ], + "lines": [ + { + "bbox": [ + 106, + 363, + 322, + 376 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 322, + 376 + ], + "score": 1.0, + "content": "C.3 PROOF OF THE FIRST PART OF THEOREM 3.2", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 106, + 383, + 506, + 443 + ], + "lines": [ + { + "bbox": [ + 105, + 383, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 289, + 398 + ], + "score": 1.0, + "content": "Theorem C.6 (First part of Theorem 3.2). Let", + "type": "text" + }, + { + "bbox": [ + 290, + 385, + 300, + 395 + ], + "score": 0.74, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 383, + 443, + 398 + ], + "score": 1.0, + "content": "be the number of episodes. Suppose", + "type": "text" + }, + { + "bbox": [ + 443, + 383, + 475, + 397 + ], + "score": 0.92, + "content": "\\sqrt { d } > \\xi", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 383, + 505, + 398 + ], + "score": 1.0, + "content": ", where", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 396, + 506, + 422 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 339, + 422 + ], + "score": 0.68, + "content": "\\begin{array} { r } { \\xi : = \\operatorname* { s u p } _ { V \\in [ 0 , H ] , \\ s ^ { \\prime } \\sim P _ { h } ( s , a ) , \\ h \\in [ H ] } \\left| \\frac { r _ { h } + V \\left( s ^ { \\prime } \\right) - ( \\mathcal T _ { h } V ) ( s , a ) } { \\sigma _ { V } ( s , a ) } \\right| } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 400, + 360, + 415 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 360, + 402, + 503, + 415 + ], + "score": 0.88, + "content": "K > \\operatorname* { m a x } \\{ \\mathcal { M } _ { 1 } , \\mathcal { M } _ { 2 } , \\mathcal { M } _ { 3 } , \\mathcal { M } _ { 4 } \\} ^ { 1 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 400, + 506, + 415 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 419, + 506, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 162, + 433 + ], + "score": 1.0, + "content": "Then for any", + "type": "text" + }, + { + "bbox": [ + 162, + 420, + 209, + 431 + ], + "score": 0.9, + "content": "0 < \\lambda < \\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 419, + 281, + 433 + ], + "score": 1.0, + "content": ", with probability", + "type": "text" + }, + { + "bbox": [ + 282, + 421, + 306, + 431 + ], + "score": 0.33, + "content": "1 - 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v ^ { \\widehat { \\pi } } \\leq \\widetilde { O } \\left( \\sqrt { d } \\cdot \\sum _ { h = 1 } ^ { H } \\mathbb { E } _ { \\pi } \\left[ \\left( \\phi ( \\cdot , \\cdot ) ^ { \\top } \\Lambda _ { h } ^ { - 1 } \\phi ( \\cdot , \\cdot ) \\right) ^ { 1 / 2 } \\right] \\right) + \\frac { 2 H ^ { 4 } \\sqrt { d } } { K }", + "type": "interline_equation", + "image_path": "4163ed88fbfb0446b698a10f32143c31548b24a7dda8bc846707cd7f2e1e8390.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 168, + 450, + 443, + 461.3333333333333 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 168, + 461.3333333333333, + 443, + 472.66666666666663 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 168, + 472.66666666666663, + 443, + 483.99999999999994 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 491, + 506, + 523 + ], + "lines": [ + { + "bbox": [ + 101, + 485, + 507, + 515 + ], + "spans": [ + { + "bbox": [ + 101, + 485, + 132, + 514 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 490, + 286, + 515 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\Lambda _ { h } = \\sum _ { \\tau = 1 } ^ { K } \\frac { \\phi ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) \\cdot \\phi ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) ^ { \\top } } { \\sigma _ { \\hat { V } _ { h + 1 } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) } ^ { 2 } } + \\lambda I _ { d } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 490, + 507, + 509 + ], + "score": 1.0, + "content": "1 φ(sτh,aτh)·φ(sτh,aτh)>2 + λId and O absorbs the universal constants and the Polylog", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 511, + 134, + 524 + ], + "spans": [ + { + "bbox": [ + 104, + 511, + 134, + 524 + ], + "score": 1.0, + "content": "terms.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5, + "bbox_fs": [ + 101, + 485, + 507, + 524 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 539, + 504, + 563 + ], + "lines": [ + { + "bbox": [ + 105, + 537, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 505, + 554 + ], + "score": 1.0, + "content": "Proof of Theorem C.6. Combing Lemma C.1 and Lemma C.5, we directly have with probability", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 550, + 260, + 564 + ], + "spans": [ + { + "bbox": [ + 106, + 552, + 129, + 562 + ], + "score": 0.88, + "content": "1 - \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 550, + 186, + 564 + ], + "score": 1.0, + "content": ", for all policy", + "type": "text" + }, + { + "bbox": [ + 187, + 554, + 194, + 561 + ], + "score": 0.77, + "content": "\\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 550, + 260, + 564 + ], + "score": 1.0, + "content": "simultaneously,", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 537, + 505, + 564 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 128, + 569, + 466, + 604 + ], + "lines": [ + { + "bbox": [ + 128, + 569, + 466, + 604 + ], + "spans": [ + { + "bbox": [ + 128, + 569, + 466, + 604 + ], + "score": 0.94, + "content": "V _ { 1 } ^ { \\pi } ( s ) - V _ { 1 } ^ { \\widehat { \\pi } } ( s ) \\leq \\widetilde { O } \\left( \\sqrt { d } \\cdot \\sum _ { h = 1 } ^ { H } \\mathbb { E } _ { \\pi } \\left[ \\left( \\phi ( \\cdot , \\cdot ) ^ { \\top } \\Lambda _ { h } ^ { - 1 } \\phi ( \\cdot , \\cdot ) \\right) ^ { 1 / 2 } \\middle | s _ { 1 } = s \\right] \\right) + \\frac { 2 H ^ { 4 } \\sqrt { d } } { K } ,", + "type": "interline_equation", + "image_path": "a0a189412cb0543c770e196f77cccb7d9349335e0a73429fc8248c1912cd7bed.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 128, + 569, + 466, + 580.6666666666666 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 128, + 580.6666666666666, + 466, + 592.3333333333333 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 128, + 592.3333333333333, + 466, + 603.9999999999999 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 610, + 393, + 622 + ], + "lines": [ + { + "bbox": [ + 105, + 609, + 392, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 233, + 623 + ], + "score": 1.0, + "content": "now take the initial distribution", + "type": "text" + }, + { + "bbox": [ + 234, + 611, + 244, + 621 + ], + "score": 0.88, + "content": "d _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 609, + 392, + 623 + ], + "score": 1.0, + "content": "on both sides to get the stated result.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 609, + 392, + 623 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 636, + 262, + 648 + ], + "lines": [ + { + "bbox": [ + 106, + 636, + 262, + 649 + ], + "spans": [ + { + "bbox": [ + 106, + 636, + 262, + 649 + ], + "score": 1.0, + "content": "C.4 TWO INTERMEDIATE RESULTS", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 108, + 657, + 426, + 670 + ], + "lines": [ + { + "bbox": [ + 105, + 656, + 428, + 671 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 428, + 671 + ], + "score": 1.0, + "content": "The next two lemmas provide intermediate results in finishing the whole proofs.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 656, + 428, + 671 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 82, + 257, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 257, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 257, + 95 + ], + "score": 1.0, + "content": "C.4.1 BOUNDING THE VARIANCE", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 105, + 99, + 506, + 163 + ], + "lines": [ + { + "bbox": [ + 105, + 97, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 97, + 252, + 117 + ], + "score": 1.0, + "content": "Lemma C.7. Recall the definition", + "type": "text" + }, + { + "bbox": [ + 252, + 99, + 421, + 114 + ], + "score": 0.91, + "content": "\\widehat { \\sigma } _ { h } ( \\cdot , \\cdot ) ^ { 2 } = \\operatorname* { m a x } \\{ 1 , \\widehat { \\mathrm { V a r } } _ { P _ { h } } \\widehat { V } _ { h + 1 } ( \\cdot , \\cdot ) \\} + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 97, + 442, + 117 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 442, + 100, + 505, + 117 + ], + "score": 0.88, + "content": "\\sigma _ { \\widehat { V } _ { h + 1 } } ( \\cdot , \\cdot ) ^ { 2 } : =", + "type": "inline_equation" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 112, + 503, + 136 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 236, + 130 + ], + "score": 0.88, + "content": "\\operatorname * { m a x } \\{ 1 , \\mathrm { V a r } _ { P _ { h } } \\widehat { V } _ { h + 1 } ( \\cdot , \\cdot ) \\} \\ + \\ 1 .", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 112, + 305, + 136 + ], + "score": 1.0, + "content": "Moreover,", + "type": "text" + }, + { + "bbox": [ + 306, + 117, + 503, + 134 + ], + "score": 0.61, + "content": "\\begin{array} { r l r } { \\left[ \\widehat { \\mathrm { V a r } } _ { h } \\widehat { V } _ { h + 1 } \\right] ( \\cdot , \\cdot ) } & { { } = } & { \\big \\langle \\phi ( \\cdot , \\cdot ) , \\bar { \\beta } _ { h } \\big \\rangle _ { \\left[ 0 , ( H - h + 1 ) ^ { 2 } \\right] } - } \\end{array}", + "type": "inline_equation" + } + ], + "index": 2 + }, + { + "bbox": [ + 108, + 131, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 108, + 133, + 200, + 149 + ], + "score": 0.9, + "content": "\\left[ \\left. \\phi ( \\cdot , \\cdot ) , \\bar { \\theta } _ { h } \\right. _ { [ 0 , H - h + 1 ] } \\right] ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 131, + 241, + 150 + ], + "score": 1.0, + "content": "(where", + "type": "text" + }, + { + "bbox": [ + 241, + 133, + 254, + 146 + ], + "score": 0.81, + "content": "\\bar { \\beta } _ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 131, + 282, + 150 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 282, + 133, + 294, + 146 + ], + "score": 0.87, + "content": "{ \\bar { \\theta } } _ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 131, + 473, + 150 + ], + "score": 1.0, + "content": "are defined in Algorithm 1). Let", + "type": "text" + }, + { + "bbox": [ + 473, + 134, + 505, + 146 + ], + "score": 0.74, + "content": "K \\_ { \\mathbf { \\alpha } } \\geq", + "type": "inline_equation" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 147, + 371, + 163 + ], + "spans": [ + { + "bbox": [ + 106, + 147, + 127, + 163 + ], + "score": 1.0, + "content": "max", + "type": "text" + }, + { + "bbox": [ + 128, + 148, + 253, + 163 + ], + "score": 0.89, + "content": "\\textstyle \\left\\{ 5 1 2 ( 1 / \\kappa ) ^ { 2 } \\log \\left( { \\frac { 4 H d } { \\delta } } \\right) , 4 \\lambda / \\kappa \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 147, + 342, + 163 + ], + "score": 1.0, + "content": ", then with probability", + "type": "text" + }, + { + "bbox": [ + 342, + 149, + 365, + 160 + ], + "score": 0.78, + "content": "1 - \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 147, + 371, + 163 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "interline_equation", + "bbox": [ + 153, + 165, + 457, + 199 + ], + "lines": [ + { + "bbox": [ + 153, + 165, + 457, + 199 + ], + "spans": [ + { + "bbox": [ + 153, + 165, + 457, + 199 + ], + "score": 0.93, + "content": "\\operatorname* { s u p } _ { h } \\lvert | \\widehat { \\sigma } _ { h } ^ { 2 } - \\sigma _ { \\widehat { V } _ { h + 1 } } ^ { 2 } | | _ { \\infty } \\leq 3 6 \\sqrt { \\frac { H ^ { 4 } d ^ { 3 } } { \\kappa K } \\log \\left( \\frac { ( \\lambda + K ) 2 K d H ^ { 2 } } { \\lambda \\delta } \\right) } + 1 2 \\lambda \\frac { H ^ { 2 } \\sqrt { d } } { \\kappa K } .", + "type": "interline_equation", + "image_path": "4959c8b6eeb0f2e8c592f6e868a69cc424c54159b8eb9bf92754632b30b8f55f.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 153, + 165, + 457, + 176.33333333333334 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 153, + 176.33333333333334, + 457, + 187.66666666666669 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 153, + 187.66666666666669, + 457, + 199.00000000000003 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 208, + 435, + 222 + ], + "lines": [ + { + "bbox": [ + 105, + 208, + 434, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 248, + 223 + ], + "score": 1.0, + "content": "Proof. 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Note", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 281, + 514, + 444 + ], + "lines": [ + { + "bbox": [ + 111, + 281, + 514, + 444 + ], + "spans": [ + { + "bbox": [ + 111, + 281, + 514, + 444 + ], + "score": 0.93, + "content": "\\begin{array} { r l } & { | \\phi ( s , a ) , \\widetilde { \\beta } _ { h } [ 0 , ( J - h + 1 ) ^ { 2 } ] - \\widetilde { \\Gamma } _ { h } ( \\widehat { V } _ { h + 1 } ) ^ { 2 } ( s , a ) | \\leq | \\phi ( s , a ) , \\widetilde { \\beta } _ { h } - \\widetilde { \\Gamma } _ { h } ( \\widehat { V } _ { h + 1 } ) ^ { 2 } ( s , a ) | } \\\\ & { = | \\phi ( s , a ) ^ { \\top } \\widetilde { \\Sigma } _ { h } ^ { - 1 } \\displaystyle \\sum _ { \\tau = 1 } ^ { K } \\phi ( \\widehat { s } _ { h } ^ { \\top } , \\widetilde { a } _ { h } ^ { \\top } ) \\cdot \\widehat { V } _ { h + 1 } ( \\widehat { s } _ { h + 1 } ^ { \\top } ) ^ { 2 } - \\widetilde { \\Gamma } _ { h } ( \\widehat { V } _ { h + 1 } ) ^ { 2 } ( s , a ) | } \\\\ & { = | \\phi ( s , a ) ^ { \\top } \\widetilde { \\Sigma } _ { h } ^ { - 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1 } } \\left\\| \\sum _ { \\tau = 1 } ^ { K } \\phi ( \\bar { s } _ { h } ^ { \\tau } , \\bar { a } _ { h } ^ { \\tau } ) \\cdot \\Big ( \\widehat { V } _ { h + 1 } ( \\bar { s } _ { h + 1 } ^ { \\tau } ) ^ { 2 } - \\mathbb { P } _ { h } ( \\widehat { V } _ { h + 1 } ) ^ { 2 } ( \\bar { s } _ { h } ^ { \\tau } , \\bar { a } _ { h } ^ { \\tau } ) \\Big ) \\right\\| _ { \\bar { \\Sigma } _ { h } ^ { - 1 } }", + "type": "interline_equation", + "image_path": "9b8bbc961017fefa852e03bb274a44859133d2cfb6014ec838728bfaac9f8d25.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 144, + 574, + 468, + 585.6666666666666 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 144, + 585.6666666666666, + 468, + 597.3333333333333 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 144, + 597.3333333333333, + 468, + 608.9999999999999 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 613, + 504, + 649 + ], + "lines": [ + { + "bbox": [ + 105, + 612, + 506, + 627 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 288, + 627 + ], + "score": 1.0, + "content": "Bounding using covering. 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C _ { 1 } \\sqrt { d \\cdot \\phi ( \\cdot , \\cdot ) ^ { \\top } \\widehat \\Lambda _ { h } ^ { - 1 } \\phi ( \\cdot , \\cdot ) } - C _ { 2 } , H - h + 1 \\} ^ { + } \\} \\} . } \\end{array}", + "type": "inline_equation", + "image_path": "76e7fb9ab2e27ddf960aec8959c5901309e4e1bc414d0c13aad7001e9f46f3e9.jpg" + }, + { + "bbox": [ + 432, + 709, + 507, + 736 + ], + "score": 1.0, + "content": ". 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Recall the definition", + "type": "text" + }, + { + "bbox": [ + 252, + 99, + 421, + 114 + ], + "score": 0.91, + "content": "\\widehat { \\sigma } _ { h } ( \\cdot , \\cdot ) ^ { 2 } = \\operatorname* { m a x } \\{ 1 , \\widehat { \\mathrm { V a r } } _ { P _ { h } } \\widehat { V } _ { h + 1 } ( \\cdot , \\cdot ) \\} + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 97, + 442, + 117 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 442, + 100, + 505, + 117 + ], + "score": 0.88, + "content": "\\sigma _ { \\widehat { V } _ { h + 1 } } ( \\cdot , \\cdot ) ^ { 2 } : =", + "type": "inline_equation" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 112, + 503, + 136 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 236, + 130 + ], + "score": 0.88, + "content": "\\operatorname * { m a x } \\{ 1 , \\mathrm { V a r } _ { P _ { h } } \\widehat { V } _ { h + 1 } ( \\cdot , \\cdot ) \\} \\ + \\ 1 .", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 112, + 305, + 136 + ], + "score": 1.0, + "content": "Moreover,", + "type": "text" + }, + { + "bbox": [ + 306, + 117, + 503, + 134 + ], + "score": 0.61, + "content": "\\begin{array} { r l r } { \\left[ \\widehat { \\mathrm { V a r } } _ { h } \\widehat { V } _ { h + 1 } \\right] ( \\cdot , \\cdot ) } & { { } = } & { \\big \\langle \\phi ( \\cdot , \\cdot ) , \\bar { \\beta } _ { h } \\big \\rangle _ { \\left[ 0 , ( H - h + 1 ) ^ { 2 } \\right] } - } \\end{array}", + "type": "inline_equation" + } + ], + "index": 2 + }, + { + "bbox": [ + 108, + 131, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 108, + 133, + 200, + 149 + ], + "score": 0.9, + "content": "\\left[ \\left. \\phi ( \\cdot , \\cdot ) , \\bar { \\theta } _ { h } \\right. _ { [ 0 , H - h + 1 ] } \\right] ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 131, + 241, + 150 + ], + "score": 1.0, + "content": "(where", + "type": "text" + }, + { + "bbox": [ + 241, + 133, + 254, + 146 + ], + "score": 0.81, + "content": "\\bar { \\beta } _ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 131, + 282, + 150 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 282, + 133, + 294, + 146 + ], + "score": 0.87, + "content": "{ \\bar { \\theta } } _ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 131, + 473, + 150 + ], + "score": 1.0, + "content": "are defined in Algorithm 1). Let", + "type": "text" + }, + { + "bbox": [ + 473, + 134, + 505, + 146 + ], + "score": 0.74, + "content": "K \\_ { \\mathbf { \\alpha } } \\geq", + "type": "inline_equation" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 147, + 371, + 163 + ], + "spans": [ + { + "bbox": [ + 106, + 147, + 127, + 163 + ], + "score": 1.0, + "content": "max", + "type": "text" + }, + { + "bbox": [ + 128, + 148, + 253, + 163 + ], + "score": 0.89, + "content": "\\textstyle \\left\\{ 5 1 2 ( 1 / \\kappa ) ^ { 2 } \\log \\left( { \\frac { 4 H d } { \\delta } } \\right) , 4 \\lambda / \\kappa \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 147, + 342, + 163 + ], + "score": 1.0, + "content": ", then with probability", + "type": "text" + }, + { + "bbox": [ + 342, + 149, + 365, + 160 + ], + "score": 0.78, + "content": "1 - \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 147, + 371, + 163 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2.5, + "bbox_fs": [ + 105, + 97, + 505, + 163 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 153, + 165, + 457, + 199 + ], + "lines": [ + { + "bbox": [ + 153, + 165, + 457, + 199 + ], + "spans": [ + { + "bbox": [ + 153, + 165, + 457, + 199 + ], + "score": 0.93, + "content": "\\operatorname* { s u p } _ { h } \\lvert | \\widehat { \\sigma } _ { h } ^ { 2 } - \\sigma _ { \\widehat { V } _ { h + 1 } } ^ { 2 } | | _ { \\infty } \\leq 3 6 \\sqrt { \\frac { H ^ { 4 } d ^ { 3 } } { \\kappa K } \\log \\left( \\frac { ( \\lambda + K ) 2 K d H ^ { 2 } } { \\lambda \\delta } \\right) } + 1 2 \\lambda \\frac { H ^ { 2 } \\sqrt { d } } { \\kappa K } .", + "type": "interline_equation", + "image_path": "4959c8b6eeb0f2e8c592f6e868a69cc424c54159b8eb9bf92754632b30b8f55f.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 153, + 165, + 457, + 176.33333333333334 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 153, + 176.33333333333334, + 457, + 187.66666666666669 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 153, + 187.66666666666669, + 457, + 199.00000000000003 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 208, + 435, + 222 + ], + "lines": [ + { + "bbox": [ + 105, + 208, + 434, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 248, + 223 + ], + "score": 1.0, + "content": "Proof. 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( \\Sigma _ { h } ^ { p } ) ^ { - 1 } \\right. ^ { 1 / 2 } } { \\sqrt { K } } \\leq \\frac { 2 } { \\sqrt { \\kappa K } } .", + "type": "interline_equation", + "image_path": "85fbd2fd64ba5f536e7c4cb62dcd3c083b8df47cc3300324d11c0a6bf19910af.jpg" + } + ] + } + ], + "index": 19.5, + "virtual_lines": [ + { + "bbox": [ + 219, + 387, + 394, + 403.0 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 219, + 403.0, + 394, + 419.0 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 422, + 504, + 446 + ], + "lines": [ + { + "bbox": [ + 106, + 421, + 506, + 435 + ], + "spans": [ + { + "bbox": [ + 106, + 421, + 467, + 435 + ], + "score": 1.0, + "content": "Combing equation 13, equation 14, equation 15 and equation 16 we obtain with probability", + "type": "text" + }, + { + "bbox": [ + 467, + 423, + 490, + 433 + ], + "score": 0.85, + "content": "1 - \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 490, + 421, + 506, + 435 + ], + "score": 1.0, + "content": "for", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 431, + 215, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 431, + 119, + 446 + ], + "score": 1.0, + "content": "all", + "type": "text" + }, + { + "bbox": [ + 119, + 433, + 210, + 445 + ], + "score": 0.92, + "content": "h , s , a \\in [ H ] \\times \\mathcal { S } \\times \\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 431, + 215, + 446 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 421, + 506, + 446 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 449, + 506, + 483 + ], + "lines": [ + { + "bbox": [ + 111, + 449, + 506, + 483 + ], + "spans": [ + { + "bbox": [ + 111, + 449, + 506, + 483 + ], + "score": 0.92, + "content": "\\langle \\phi ( s , a ) , \\bar { \\beta } _ { h } \\rangle _ { [ 0 , ( H - 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By equation 17,", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 604, + 239, + 620 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 621, + 514, + 701 + ], + "lines": [ + { + "bbox": [ + 111, + 621, + 514, + 701 + ], + "spans": [ + { + "bbox": [ + 111, + 621, + 514, + 701 + ], + "score": 0.92, + "content": "\\begin{array} { r l } & { \\quad \\left| \\left[ \\big < \\phi ( \\cdot , \\cdot ) , \\bar { \\theta } _ { h } \\big > _ { [ 0 , H - h + 1 ] } \\right] ^ { 2 } - \\left[ \\mathbb { P } _ { h } ( \\widehat { V } _ { h + 1 } ) ( s , a ) \\right] ^ { 2 } \\right| } \\\\ & { = \\left| \\big < \\phi ( s , a ) , \\bar { \\theta } _ { h } \\big > _ { [ 0 , H - h + 1 ] } + \\mathbb { P } _ { h } ( \\widehat { V } _ { h + 1 } ) ( s , a ) \\right| \\cdot \\left| \\big < \\phi ( s , a ) , \\bar { \\theta } _ { h } \\big > _ { [ 0 , H - h + 1 ] } - \\mathbb { P } _ { h } ( \\widehat { V } _ { h + 1 } ) ( s , a ) \\right| } \\\\ & { \\leq 2 H \\cdot \\left| \\big < \\phi ( s , a ) , \\bar { \\theta } _ { h } \\big > _ { [ 0 , H - h + 1 ] } - \\mathbb { P } _ { h } ( \\widehat { V } _ { h + 1 } ) ( s , a ) \\right| \\leq 2 4 \\sqrt { \\frac { H ^ { 4 } d ^ { 3 } } { \\kappa K } \\log \\left( \\frac { \\big ( \\lambda + K \\big ) 2 K d H ^ { 2 } } { \\lambda \\delta } \\right) } + 8 \\lambda \\frac { H ^ { 2 } \\sqrt { \\lambda + 1 } } { \\kappa K } } \\end{array}", + "type": "interline_equation", + "image_path": "0e384c5d91264a0a26a3ddf234eeb18e3d63e6b2aec171b7b4df4d87f7954227.jpg" + } + ] + } + ], + "index": 34, + "virtual_lines": [ + { + "bbox": [ + 111, + 621, + 514, + 647.6666666666666 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 111, + 647.6666666666666, + 514, + 674.3333333333333 + ], + "spans": [], + "index": 34 + }, + { + "bbox": [ + 111, + 674.3333333333333, + 514, + 700.9999999999999 + ], + "spans": [], + "index": 35 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 453, + 95 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 452, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 261, + 96 + ], + "score": 1.0, + "content": "Combining this with Step1 we receive", + "type": "text" + }, + { + "bbox": [ + 261, + 82, + 358, + 95 + ], + "score": 0.92, + "content": "\\forall h , s , a \\in [ H ] \\times S \\times A", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 81, + 428, + 96 + ], + "score": 1.0, + "content": ", with probability", + "type": "text" + }, + { + "bbox": [ + 429, + 83, + 452, + 93 + ], + "score": 0.72, + "content": "1 - 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\\widehat { V } _ { h } \\rvert | _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 178, + 312, + 198 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 106, + 203, + 506, + 241 + ], + "lines": [ + { + "bbox": [ + 102, + 201, + 504, + 229 + ], + "spans": [ + { + "bbox": [ + 102, + 201, + 215, + 229 + ], + "score": 1.0, + "content": "Lemma C.8. Define", + "type": "text" + }, + { + "bbox": [ + 216, + 203, + 504, + 229 + ], + "score": 0.66, + "content": "\\begin{array} { r l r } { \\widehat { \\sigma } _ { h } ( s , a ) } & { { } = } & { \\sqrt { \\operatorname* { m a x } \\left\\{ 1 , \\widehat { \\mathrm { V a r } } _ { P _ { h } } \\widehat { V } _ { h + 1 } ( s , a ) \\right\\} + 1 } , \\quad i f \\quad K \\quad \\geq } \\end{array}", + "type": "inline_equation" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 225, + 446, + 243 + ], + "spans": [ + { + "bbox": [ + 106, + 228, + 215, + 240 + ], + "score": 0.89, + "content": "\\operatorname* { m a x } \\{ \\mathcal { M } _ { 1 } , \\mathcal { M } _ { 2 } , \\mathcal { M } _ { 3 } , \\mathcal { M } _ { 4 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 225, + 234, + 243 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 234, + 228, + 296, + 239 + ], + "score": 0.91, + "content": "K > C \\cdot H ^ { 4 } \\kappa ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 225, + 417, + 243 + ], + "score": 1.0, + "content": ", then with probability at least", + "type": "text" + }, + { + "bbox": [ + 417, + 228, + 440, + 239 + ], + "score": 0.77, + "content": "1 - \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 225, + 446, + 243 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5 + }, + { + "type": "interline_equation", + "bbox": [ + 232, + 246, + 379, + 279 + ], + "lines": [ + { + "bbox": [ + 232, + 246, + 379, + 279 + ], + "spans": [ + { + "bbox": [ + 232, + 246, + 379, + 279 + ], + "score": 0.93, + "content": "\\operatorname* { s u p } _ { h } \\left\\| V _ { h } ^ { \\star } - \\widehat { V } _ { h } \\right\\| _ { \\infty } \\leq \\widetilde { O } \\left( \\frac { H ^ { 2 } \\sqrt { d } } { \\sqrt { \\kappa K } } \\right) .", + "type": "interline_equation", + "image_path": "3c7cadfe349706768342616197eb287aa483058a78bdf4afd5994dc2d3682e27.jpg" + } + ] + } + ], + "index": 8.5, + "virtual_lines": [ + { + "bbox": [ + 232, + 246, + 379, + 262.5 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 232, + 262.5, + 379, + 279.0 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 103, + 293, + 471, + 311 + ], + "lines": [ + { + "bbox": [ + 100, + 287, + 471, + 319 + ], + "spans": [ + { + "bbox": [ + 100, + 287, + 302, + 319 + ], + "score": 1.0, + "content": "Proof. 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\\widehat { V } _ { h } \\rvert | _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 178, + 312, + 198 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5, + "bbox_fs": [ + 105, + 178, + 312, + 198 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 203, + 506, + 241 + ], + "lines": [ + { + "bbox": [ + 102, + 201, + 504, + 229 + ], + "spans": [ + { + "bbox": [ + 102, + 201, + 215, + 229 + ], + "score": 1.0, + "content": "Lemma C.8. Define", + "type": "text" + }, + { + "bbox": [ + 216, + 203, + 504, + 229 + ], + "score": 0.66, + "content": "\\begin{array} { r l r } { \\widehat { \\sigma } _ { h } ( s , a ) } & { { } = } & { \\sqrt { \\operatorname* { m a x } \\left\\{ 1 , \\widehat { \\mathrm { V a r } } _ { P _ { h } } \\widehat { V } _ { h + 1 } ( s , a ) \\right\\} + 1 } , \\quad i f \\quad K \\quad \\geq } \\end{array}", + "type": "inline_equation" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 225, + 446, + 243 + ], + "spans": [ + { + "bbox": [ + 106, + 228, + 215, + 240 + ], + "score": 0.89, + "content": "\\operatorname* { m a x } \\{ \\mathcal { M } _ { 1 } , \\mathcal { M } _ { 2 } , \\mathcal { M } _ { 3 } , \\mathcal { M } _ { 4 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 225, + 234, + 243 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 234, + 228, + 296, + 239 + ], + "score": 0.91, + "content": "K > C \\cdot H ^ { 4 } \\kappa ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 225, + 417, + 243 + ], + "score": 1.0, + "content": ", then with probability at least", + "type": "text" + }, + { + "bbox": [ + 417, + 228, + 440, + 239 + ], + "score": 0.77, + "content": "1 - \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 225, + 446, + 243 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5, + "bbox_fs": [ + 102, + 201, + 504, + 243 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 232, + 246, + 379, + 279 + ], + "lines": [ + { + "bbox": [ + 232, + 246, + 379, + 279 + ], + "spans": [ + { + "bbox": [ + 232, + 246, + 379, + 279 + ], + "score": 0.93, + "content": "\\operatorname* { s u p } _ { h } \\left\\| V _ { h } ^ { \\star } - \\widehat { V } _ { h } \\right\\| _ { \\infty } \\leq \\widetilde { O } \\left( \\frac { H ^ { 2 } \\sqrt { d } } { \\sqrt { \\kappa K } } \\right) .", + "type": "interline_equation", + "image_path": "3c7cadfe349706768342616197eb287aa483058a78bdf4afd5994dc2d3682e27.jpg" + } + ] + } + ], + "index": 8.5, + "virtual_lines": [ + { + "bbox": [ + 232, + 246, + 379, + 262.5 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 232, + 262.5, + 379, + 279.0 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 103, + 293, + 471, + 311 + ], + "lines": [ + { + "bbox": [ + 100, + 287, + 471, + 319 + ], + "spans": [ + { + "bbox": [ + 100, + 287, + 302, + 319 + ], + "score": 1.0, + "content": "Proof. 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V _ { h } ^ { \\widehat { \\pi } } ( s ) \\leq \\widetilde O \\left( \\sqrt { d } \\cdot \\displaystyle \\sum _ { t = h } ^ { H } \\mathbb { E } _ { \\pi ^ { * } } \\left[ \\left( \\phi ( \\cdot , \\cdot ) ^ { \\top } \\Lambda _ { t } ^ { - 1 } \\phi ( \\cdot , \\cdot ) \\right) ^ { 1 / 2 } \\Big | s _ { h } = s \\right] \\right) + \\frac { 2 H ^ { 4 } \\sqrt { d } } { K } } } \\\\ { \\displaystyle \\leq \\widetilde O \\left( \\frac { H ^ { 2 } \\sqrt { d } } { \\sqrt { \\kappa K } } \\right) + \\frac { 2 H ^ { 4 } \\sqrt { d } } { K } . } \\end{array}", + "type": "interline_equation", + "image_path": "ab74c525c6a278049b9876c8be6f74a9eb95027342b1013941c6ecd34eadc214.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 117, + 467, + 478, + 490.0 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 117, + 490.0, + 478, + 513.0 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 117, + 513.0, + 478, + 536.0 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 540, + 430, + 554 + ], + "lines": [ + { + "bbox": [ + 104, + 539, + 432, + 556 + ], + "spans": [ + { + "bbox": [ + 104, + 539, + 249, + 556 + ], + "score": 1.0, + "content": "This implies by using the condition", + "type": "text" + }, + { + "bbox": [ + 250, + 541, + 312, + 552 + ], + "score": 0.92, + "content": "K > C \\cdot H ^ { 4 } \\kappa ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 539, + 432, + 556 + ], + "score": 1.0, + "content": ", we finish the proof of Step1.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23, + "bbox_fs": [ + 104, + 539, + 432, + 556 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 558, + 411, + 579 + ], + "lines": [ + { + "bbox": [ + 100, + 552, + 409, + 585 + ], + "spans": [ + { + "bbox": [ + 100, + 552, + 241, + 585 + ], + "score": 1.0, + "content": "Step2: We show with probability", + "type": "text" + }, + { + "bbox": [ + 241, + 558, + 409, + 579 + ], + "score": 0.76, + "content": "\\begin{array} { r } { 1 - \\delta , \\operatorname* { s u p } _ { h } { \\left\\| { \\widehat { V } } _ { h } - V _ { h } ^ { \\widehat { \\pi } } \\right\\| _ { \\infty } } \\leq \\widetilde { O } \\left( \\frac { H ^ { 2 } \\sqrt { d } } { \\sqrt { \\kappa K } } \\right) . } \\end{array}", + "type": "inline_equation" + } + ], + "index": 24 + } + ], + "index": 24, + "bbox_fs": [ + 100, + 552, + 409, + 585 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 583, + 507, + 606 + ], + "lines": [ + { + "bbox": [ + 105, + 582, + 507, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 344, + 597 + ], + "score": 1.0, + "content": "Indeed, applying Extended Value Difference Lemma H.7 for", + "type": "text" + }, + { + "bbox": [ + 344, + 584, + 393, + 594 + ], + "score": 0.9, + "content": "\\pi = \\pi ^ { \\prime } = \\widehat { \\pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 582, + 481, + 597 + ], + "score": 1.0, + "content": ", then with probability", + "type": "text" + }, + { + "bbox": [ + 481, + 584, + 502, + 595 + ], + "score": 0.85, + "content": "1 - 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( \\mathcal { T } _ { h } \\widehat { V } _ { h + 1 } ) ( s _ { h } , a _ { h } ) | s _ { h } = s ] | } \\\\ & { \\leq \\displaystyle \\sum _ { t = h } ^ { H } \\| ( \\widehat { \\mathcal { T } } _ { h } \\widehat { V } _ { h + 1 } - \\mathcal { T } _ { h } \\widehat { V } _ { h + 1 } ) ( s , a ) \\| + \\| \\Gamma _ { h } ( s , a ) \\| } \\\\ & { { \\leq } \\widetilde { O } ( H \\sqrt { d } \\| \\sqrt { \\phi ( s , a ) \\Lambda _ { h } ^ { - 1 } \\phi ( s , a ) } \\| ) + \\frac { 4 H ^ { 4 } \\sqrt { d } } { K } \\leq \\widetilde { O } ( \\frac { H ^ { 2 } \\sqrt { d } } { \\sqrt { \\kappa K } } ) } \\end{array}", + "type": "interline_equation", + "image_path": "424f3fc11ee126d5be85b3f4c29be9f2eaf8103d3bae63b44fca45864c1dc6d7.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 151, + 612, + 459, + 646.6666666666666 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 151, + 646.6666666666666, + 459, + 681.3333333333333 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 151, + 681.3333333333333, + 459, + 715.9999999999999 + ], + "spans": [], + "index": 29 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 505, + 106 + ], + "lines": [ + { + "bbox": [ + 104, + 79, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 104, + 79, + 270, + 96 + ], + "score": 1.0, + "content": "where the second inequality uses Lemma", + "type": "text" + }, + { + "bbox": [ + 270, + 82, + 293, + 93 + ], + "score": 0.82, + "content": "{ \\mathrm { C } } . 5 ^ { 1 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 79, + 506, + 96 + ], + "score": 1.0, + "content": "and the last inequality follows the same procedure as", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 104, + 92, + 136, + 108 + ], + "spans": [ + { + "bbox": [ + 104, + 92, + 136, + 108 + ], + "score": 1.0, + "content": "Step1.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 105, + 110, + 505, + 133 + ], + "lines": [ + { + "bbox": [ + 105, + 109, + 506, + 124 + ], + "spans": [ + { + "bbox": [ + 105, + 109, + 506, + 124 + ], + "score": 1.0, + "content": "Step3: Combine Step1 and Step2, by triangular inequality and a union bound we finish the proof of", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 120, + 154, + 134 + ], + "spans": [ + { + "bbox": [ + 105, + 120, + 154, + 134 + ], + "score": 1.0, + "content": "the lemma.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 106, + 157, + 506, + 259 + ], + "lines": [ + { + "bbox": [ + 104, + 156, + 506, + 171 + ], + "spans": [ + { + "bbox": [ + 104, + 156, + 347, + 171 + ], + "score": 1.0, + "content": "Remark C.9. Note as an intermediate calculation, equation", + "type": "text" + }, + { + "bbox": [ + 348, + 159, + 358, + 168 + ], + "score": 0.51, + "content": "^ { 1 9 }", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 156, + 506, + 171 + ], + "score": 1.0, + "content": "ensures a learning bound with order", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 107, + 167, + 507, + 189 + ], + "spans": [ + { + "bbox": [ + 107, + 169, + 147, + 187 + ], + "score": 0.92, + "content": "\\widetilde { \\cal O } \\big ( \\frac { H ^ { 2 } \\sqrt { d } } { \\sqrt { \\kappa K } } \\big )", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 167, + 387, + 189 + ], + "score": 1.0, + "content": ". Here, the convergence rate is the standard statistical rate", + "type": "text" + }, + { + "bbox": [ + 388, + 171, + 404, + 187 + ], + "score": 0.92, + "content": "\\textstyle { \\frac { 1 } { \\sqrt { K } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 167, + 438, + 189 + ], + "score": 1.0, + "content": "and the", + "type": "text" + }, + { + "bbox": [ + 439, + 171, + 453, + 182 + ], + "score": 0.86, + "content": "H ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 167, + 507, + 189 + ], + "score": 1.0, + "content": "dependence", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 186, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 280, + 200 + ], + "score": 1.0, + "content": "is loose. However, the feature dependence", + "type": "text" + }, + { + "bbox": [ + 281, + 186, + 307, + 200 + ], + "score": 0.92, + "content": "\\sqrt { d / \\kappa }", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 187, + 505, + 200 + ], + "score": 1.0, + "content": "is roughly tight, since, in the well-explored case", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 199, + 506, + 215 + ], + "spans": [ + { + "bbox": [ + 104, + 200, + 262, + 215 + ], + "score": 1.0, + "content": "(Assumption 2 of Wang et al. (2021a)), √", + "type": "text" + }, + { + "bbox": [ + 263, + 201, + 298, + 213 + ], + "score": 0.91, + "content": "\\kappa = 1 / d", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 200, + 332, + 215 + ], + "score": 1.0, + "content": "and the", + "type": "text" + }, + { + "bbox": [ + 332, + 199, + 389, + 214 + ], + "score": 0.93, + "content": "\\sqrt { d / \\kappa } = \\sqrt { d ^ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 200, + 506, + 215 + ], + "score": 1.0, + "content": "recovers the optimal feature", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 212, + 506, + 228 + ], + "spans": [ + { + "bbox": [ + 104, + 212, + 155, + 228 + ], + "score": 1.0, + "content": "dependence", + "type": "text" + }, + { + "bbox": [ + 155, + 213, + 186, + 225 + ], + "score": 0.91, + "content": "d H \\sqrt { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 212, + 358, + 228 + ], + "score": 1.0, + "content": "in the online setting (Zhou et al., 2021a). If", + "type": "text" + }, + { + "bbox": [ + 358, + 214, + 396, + 226 + ], + "score": 0.88, + "content": "\\kappa \\ll 1 / d ,", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 212, + 506, + 228 + ], + "score": 1.0, + "content": ", then doing offline learning", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 226, + 505, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 256, + 237 + ], + "score": 1.0, + "content": "requires sample size proportional to", + "type": "text" + }, + { + "bbox": [ + 256, + 226, + 273, + 237 + ], + "score": 0.88, + "content": "d / \\kappa ,", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 226, + 362, + 237 + ], + "score": 1.0, + "content": ", which reveals offline", + "type": "text" + }, + { + "bbox": [ + 363, + 226, + 377, + 236 + ], + "score": 0.56, + "content": "R L", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 226, + 505, + 237 + ], + "score": 1.0, + "content": "is harder when the exploration", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 236, + 506, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 271, + 249 + ], + "score": 1.0, + "content": "of behavior policy is insufficient. When", + "type": "text" + }, + { + "bbox": [ + 271, + 237, + 298, + 246 + ], + "score": 0.88, + "content": "\\kappa = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 236, + 506, + 249 + ], + "score": 1.0, + "content": ", learning the optimal policy accurately cannot be", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 247, + 318, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 279, + 261 + ], + "score": 1.0, + "content": "guaranteed even if the sample/episode size", + "type": "text" + }, + { + "bbox": [ + 279, + 248, + 315, + 258 + ], + "score": 0.89, + "content": "K \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 247, + 318, + 261 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 107, + 272, + 333, + 285 + ], + "lines": [ + { + "bbox": [ + 106, + 272, + 334, + 285 + ], + "spans": [ + { + "bbox": [ + 106, + 272, + 334, + 285 + ], + "score": 1.0, + "content": "C.5 PROOF OF THE SECOND PART OF THEOREM 3.2", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 106, + 291, + 506, + 340 + ], + "lines": [ + { + "bbox": [ + 104, + 291, + 507, + 317 + ], + "spans": [ + { + "bbox": [ + 104, + 294, + 199, + 315 + ], + "score": 1.0, + "content": "Lemma C.10. Recall", + "type": "text" + }, + { + "bbox": [ + 199, + 291, + 342, + 317 + ], + "score": 0.94, + "content": "\\widehat { \\sigma } _ { h } = \\sqrt { \\operatorname* { m a x } \\left\\{ 1 , \\widehat { \\mathrm { V a r } } _ { P _ { h } } \\widehat { V } _ { h + 1 } \\right\\} + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 294, + 361, + 315 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 361, + 294, + 504, + 315 + ], + "score": 0.84, + "content": "\\sigma _ { h } ^ { \\star } = \\sqrt { \\operatorname* { m a x } \\left\\{ 1 , \\mathrm { V a r } _ { P _ { h } } V _ { h + 1 } ^ { \\star } \\right\\} + 1 } .", + "type": "inline_equation" + }, + { + "bbox": [ + 504, + 294, + 507, + 315 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 315, + 507, + 333 + ], + "spans": [ + { + "bbox": [ + 104, + 315, + 121, + 333 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 121, + 317, + 290, + 331 + ], + "score": 0.89, + "content": "\\begin{array} { r } { K \\ge \\operatorname* { m a x } \\left\\{ 5 1 2 ( 1 / \\kappa ) ^ { 2 } \\log \\left( \\frac { 4 H d } { \\delta } \\right) , 4 \\lambda / \\kappa \\right\\} } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 315, + 308, + 333 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 309, + 318, + 440, + 330 + ], + "score": 0.91, + "content": "K \\ge \\operatorname* { m a x } \\{ \\mathcal { M } _ { 1 } , \\mathcal { M } _ { 2 } , \\mathcal { M } _ { 3 } , \\mathcal { M } _ { 4 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 315, + 507, + 333 + ], + "score": 1.0, + "content": ", then with prob-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 328, + 163, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 328, + 134, + 342 + ], + "score": 1.0, + "content": "ability", + "type": "text" + }, + { + "bbox": [ + 135, + 330, + 157, + 340 + ], + "score": 0.83, + "content": "1 - \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 328, + 163, + 342 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14 + }, + { + "type": "interline_equation", + "bbox": [ + 232, + 339, + 379, + 373 + ], + "lines": [ + { + "bbox": [ + 232, + 339, + 379, + 373 + ], + "spans": [ + { + "bbox": [ + 232, + 339, + 379, + 373 + ], + "score": 0.94, + "content": "\\operatorname* { s u p } _ { h } \\vert \\vert \\widehat { \\sigma } _ { h } ^ { 2 } - \\sigma _ { h } ^ { \\star 2 } \\vert \\vert _ { \\infty } \\leq \\widetilde { O } \\left( \\frac { H ^ { 3 } \\sqrt { d } } { \\sqrt { \\kappa K } } \\right) .", + "type": "interline_equation", + "image_path": "ec14809aea10c1a5dbf9381ecb8244a76d3a73b1d50835ed0d928ab437f01e6f.jpg" + } + ] + } + ], + "index": 16.5, + "virtual_lines": [ + { + "bbox": [ + 232, + 339, + 379, + 356.0 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 232, + 356.0, + 379, + 373.0 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 385, + 409, + 398 + ], + "lines": [ + { + "bbox": [ + 106, + 385, + 408, + 399 + ], + "spans": [ + { + "bbox": [ + 106, + 385, + 311, + 399 + ], + "score": 1.0, + "content": "Proof. By definition and the non-expansiveness of", + "type": "text" + }, + { + "bbox": [ + 311, + 386, + 369, + 398 + ], + "score": 0.89, + "content": "\\operatorname* { m a x } \\{ 1 , \\cdot \\} + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 385, + 408, + 399 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "interline_equation", + "bbox": [ + 156, + 403, + 455, + 505 + ], + "lines": [ + { + "bbox": [ + 156, + 403, + 455, + 505 + ], + "spans": [ + { + "bbox": [ + 156, + 403, + 455, + 505 + ], + "score": 0.95, + "content": "\\begin{array} { r l } & { \\quad \\left\\| \\sigma _ { \\widehat { V } _ { h + 1 } } ^ { 2 } - \\sigma _ { h } ^ { \\star 2 } \\right\\| _ { \\infty } \\leq \\left\\| \\operatorname { V a r } \\widehat { V } _ { h + 1 } - \\operatorname { V a r } V _ { h + 1 } ^ { \\star } \\right\\| _ { \\infty } } \\\\ & { \\leq \\left\\| \\mathbb { P } _ { h } \\left( \\widehat { V } _ { h + 1 } ^ { 2 } - V _ { h + 1 } ^ { \\star 2 } \\right) \\right\\| _ { \\infty } + \\left\\| ( \\mathbb { P } _ { h } \\widehat { V } _ { h + 1 } ) ^ { 2 } - ( \\mathbb { P } _ { h } V _ { h + 1 } ^ { \\star } ) ^ { 2 } \\right\\| _ { \\infty } } \\\\ & { \\leq \\left\\| \\widehat { V } _ { h + 1 } ^ { 2 } - V _ { h + 1 } ^ { \\star 2 } \\right\\| _ { \\infty } + \\left\\| ( \\mathbb { P } _ { h } \\widehat { V } _ { h + 1 } + \\mathbb { P } _ { h } V _ { h + 1 } ^ { \\star } ) ( \\mathbb { P } _ { h } \\widehat { V } _ { h + 1 } - \\mathbb { P } _ { h } V _ { h + 1 } ^ { \\star } ) \\right\\| _ { \\infty } } \\\\ & { \\leq 2 H \\left\\| \\widehat { V } _ { h + 1 } - V _ { h + 1 } ^ { \\star } \\right\\| _ { \\infty } + 2 H \\left\\| \\mathbb { P } _ { h } \\widehat { V } _ { h + 1 } - \\mathbb { P } _ { h } V _ { h + 1 } ^ { \\star } \\right\\| _ { \\infty } \\leq \\widetilde { \\mathcal { O } } \\left( \\frac { H ^ { 3 } \\sqrt { d } } { \\sqrt { \\kappa K } } \\right) . } \\end{array}", + "type": "interline_equation", + "image_path": "22eb4cd699d7db194269f8a2277efcf636aaccf41f32ef2b4e65064d1f788669.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 156, + 403, + 455, + 437.0 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 156, + 437.0, + 455, + 471.0 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 156, + 471.0, + 455, + 505.0 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 510, + 505, + 533 + ], + "lines": [ + { + "bbox": [ + 105, + 509, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 171, + 524 + ], + "score": 1.0, + "content": "with probability", + "type": "text" + }, + { + "bbox": [ + 172, + 511, + 195, + 521 + ], + "score": 0.87, + "content": "1 - \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 509, + 222, + 524 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 222, + 511, + 255, + 522 + ], + "score": 0.92, + "content": "h \\in [ H ]", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 509, + 506, + 524 + ], + "score": 1.0, + "content": ", where the last inequality comes from Lemma C.8. Combining", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 521, + 298, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 521, + 298, + 533 + ], + "score": 1.0, + "content": "this with Lemma C.7, we have the stated result.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 106, + 540, + 505, + 581 + ], + "lines": [ + { + "bbox": [ + 105, + 540, + 506, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 264, + 555 + ], + "score": 1.0, + "content": "Lemma C.11. Denote the quantities", + "type": "text" + }, + { + "bbox": [ + 264, + 541, + 485, + 554 + ], + "score": 0.9, + "content": "C _ { 1 } = \\operatorname* { m a x } \\{ 2 \\lambda , 1 2 8 \\log ( 2 d / \\delta ) , 1 2 8 H ^ { 4 } \\log ( 2 d / \\delta ) / \\kappa ^ { 2 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 540, + 506, + 555 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 550, + 507, + 571 + ], + "spans": [ + { + "bbox": [ + 106, + 552, + 368, + 569 + ], + "score": 0.73, + "content": "\\begin{array} { r } { C _ { 2 } = \\operatorname* { m a x } \\{ \\frac { \\lambda ^ { 2 } } { \\kappa \\log ( ( \\lambda + K ) H / \\lambda \\delta ) } , 9 6 ^ { 2 } H ^ { 1 2 } d \\log ( ( \\lambda + K ) H / \\lambda \\delta ) / \\kappa ^ { 5 } \\} } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 550, + 507, + 571 + ], + "score": 1.0, + "content": ". Suppose the number of episode", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 107, + 567, + 348, + 582 + ], + "spans": [ + { + "bbox": [ + 107, + 569, + 117, + 578 + ], + "score": 0.69, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 567, + 152, + 582 + ], + "score": 1.0, + "content": "satisfies", + "type": "text" + }, + { + "bbox": [ + 152, + 569, + 231, + 581 + ], + "score": 0.85, + "content": "K > \\operatorname* { m a x } \\{ C _ { 1 } , C _ { 2 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 567, + 321, + 582 + ], + "score": 1.0, + "content": ", then with probability", + "type": "text" + }, + { + "bbox": [ + 322, + 569, + 344, + 579 + ], + "score": 0.8, + "content": "1 - \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 567, + 348, + 582 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25 + }, + { + "type": "interline_equation", + "bbox": [ + 171, + 586, + 439, + 608 + ], + "lines": [ + { + "bbox": [ + 171, + 586, + 439, + 608 + ], + "spans": [ + { + "bbox": [ + 171, + 586, + 439, + 608 + ], + "score": 0.89, + "content": "\\sqrt { \\phi ( s , a ) \\Lambda _ { h } ^ { - 1 } \\phi ( s , a ) } \\le 2 \\sqrt { \\phi ( s , a ) \\Lambda _ { h } ^ { \\star - 1 } \\phi ( s , a ) } , \\quad \\forall s , a \\in \\mathcal { S } \\times \\mathcal { A } ,", + "type": "interline_equation", + "image_path": "0afe9a095521196b73798b4aadea8daeb7346d645eaaa22cd1a14bbb516186f7.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 171, + 586, + 439, + 608 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 622, + 461, + 641 + ], + "lines": [ + { + "bbox": [ + 104, + 622, + 464, + 643 + ], + "spans": [ + { + "bbox": [ + 104, + 622, + 255, + 643 + ], + "score": 1.0, + "content": "Proof of Lemma C.11. By definition", + "type": "text" + }, + { + "bbox": [ + 255, + 622, + 407, + 642 + ], + "score": 0.93, + "content": "\\sqrt { \\phi ( s , a ) \\Lambda _ { h } ^ { - 1 } \\phi ( s , a ) } = \\| \\phi ( s , a ) \\| _ { \\Lambda _ { h } ^ { - 1 } } .", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 622, + 464, + 643 + ], + "score": 1.0, + "content": ". Then denote", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "interline_equation", + "bbox": [ + 245, + 647, + 365, + 671 + ], + "lines": [ + { + "bbox": [ + 245, + 647, + 365, + 671 + ], + "spans": [ + { + "bbox": [ + 245, + 647, + 365, + 671 + ], + "score": 0.93, + "content": "\\Lambda _ { h } ^ { \\prime } = \\frac { 1 } { K } \\Lambda _ { h } , \\quad { \\Lambda _ { h } ^ { \\star } } ^ { \\prime } = \\frac { 1 } { K } \\Lambda _ { h } ^ { \\star } ,", + "type": "interline_equation", + "image_path": "4cf2251ae3e307003067d53ebd4cbe5b6a64e97ebcbe36ca70502c914b7f81ed.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 245, + 647, + 365, + 671 + ], + "spans": [], + "index": 29 + } + ] + } + ], + "page_idx": 25, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 679, + 506, + 732 + ], + "lines": [ + { + "bbox": [ + 115, + 676, + 506, + 694 + ], + "spans": [ + { + "bbox": [ + 115, + 676, + 506, + 694 + ], + "score": 1.0, + "content": "12To be absolutely rigorous, we cannot directly apply Lemma C.5 here since the crude bound has already", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 688, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 506, + 702 + ], + "score": 1.0, + "content": "been used in Lemma C.4. However, this can be resolved completely by first deriving an even cruder bound for", + "type": "text" + } + ] + }, + { + "bbox": [ + 107, + 699, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 107, + 699, + 177, + 712 + ], + "score": 0.9, + "content": "\\operatorname* { s u p } _ { h } \\lvert | V _ { h } ^ { \\star } - \\widehat { V } _ { h } \\rvert | _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 700, + 208, + 713 + ], + "score": 1.0, + "content": "that has", + "type": "text" + }, + { + "bbox": [ + 209, + 700, + 235, + 712 + ], + "score": 0.93, + "content": "1 / \\sqrt { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 700, + 438, + 713 + ], + "score": 1.0, + "content": "rate without using Lemma C.5 (which we call it Lemma", + "type": "text" + }, + { + "bbox": [ + 438, + 701, + 456, + 711 + ], + "score": 0.83, + "content": "C . 8 ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 700, + 506, + 713 + ], + "score": 1.0, + "content": "), and we can", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 711, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 149, + 723 + ], + "score": 1.0, + "content": "use Lemma", + "type": "text" + }, + { + "bbox": [ + 150, + 712, + 167, + 721 + ], + "score": 0.77, + "content": "C . 8 ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 711, + 282, + 723 + ], + "score": 1.0, + "content": "to show a similar result Lemma", + "type": "text" + }, + { + "bbox": [ + 282, + 712, + 300, + 721 + ], + "score": 0.79, + "content": "{ \\mathbf { C . 5 ^ { * } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 711, + 403, + 723 + ], + "score": 1.0, + "content": ". Finally, we can use Lemma", + "type": "text" + }, + { + "bbox": [ + 403, + 712, + 421, + 721 + ], + "score": 0.86, + "content": "C . 5 ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 711, + 506, + 723 + ], + "score": 1.0, + "content": "here to finish the proof", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 720, + 425, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 425, + 734 + ], + "score": 1.0, + "content": "of this Lemma C.8. However, we avoid explicitly doing this to prevent over-technicality.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 107, + 26, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "score": 1.0, + "content": "26", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 494, + 138, + 505, + 149 + ], + "lines": [ + { + "bbox": [ + 495, + 140, + 505, + 149 + ], + "spans": [ + { + "bbox": [ + 495, + 140, + 505, + 149 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 505, + 106 + ], + "lines": [ + { + "bbox": [ + 104, + 79, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 104, + 79, + 270, + 96 + ], + "score": 1.0, + "content": "where the second inequality uses Lemma", + "type": "text" + }, + { + "bbox": [ + 270, + 82, + 293, + 93 + ], + "score": 0.82, + "content": "{ \\mathrm { C } } . 5 ^ { 1 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 79, + 506, + 96 + ], + "score": 1.0, + "content": "and the last inequality follows the same procedure as", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 104, + 92, + 136, + 108 + ], + "spans": [ + { + "bbox": [ + 104, + 92, + 136, + 108 + ], + "score": 1.0, + "content": "Step1.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 104, + 79, + 506, + 108 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 110, + 505, + 133 + ], + "lines": [ + { + "bbox": [ + 105, + 109, + 506, + 124 + ], + "spans": [ + { + "bbox": [ + 105, + 109, + 506, + 124 + ], + "score": 1.0, + "content": "Step3: Combine Step1 and Step2, by triangular inequality and a union bound we finish the proof of", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 120, + 154, + 134 + ], + "spans": [ + { + "bbox": [ + 105, + 120, + 154, + 134 + ], + "score": 1.0, + "content": "the lemma.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5, + "bbox_fs": [ + 105, + 109, + 506, + 134 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 157, + 506, + 259 + ], + "lines": [ + { + "bbox": [ + 104, + 156, + 506, + 171 + ], + "spans": [ + { + "bbox": [ + 104, + 156, + 347, + 171 + ], + "score": 1.0, + "content": "Remark C.9. Note as an intermediate calculation, equation", + "type": "text" + }, + { + "bbox": [ + 348, + 159, + 358, + 168 + ], + "score": 0.51, + "content": "^ { 1 9 }", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 156, + 506, + 171 + ], + "score": 1.0, + "content": "ensures a learning bound with order", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 107, + 167, + 507, + 189 + ], + "spans": [ + { + "bbox": [ + 107, + 169, + 147, + 187 + ], + "score": 0.92, + "content": "\\widetilde { \\cal O } \\big ( \\frac { H ^ { 2 } \\sqrt { d } } { \\sqrt { \\kappa K } } \\big )", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 167, + 387, + 189 + ], + "score": 1.0, + "content": ". Here, the convergence rate is the standard statistical rate", + "type": "text" + }, + { + "bbox": [ + 388, + 171, + 404, + 187 + ], + "score": 0.92, + "content": "\\textstyle { \\frac { 1 } { \\sqrt { K } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 167, + 438, + 189 + ], + "score": 1.0, + "content": "and the", + "type": "text" + }, + { + "bbox": [ + 439, + 171, + 453, + 182 + ], + "score": 0.86, + "content": "H ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 167, + 507, + 189 + ], + "score": 1.0, + "content": "dependence", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 186, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 280, + 200 + ], + "score": 1.0, + "content": "is loose. However, the feature dependence", + "type": "text" + }, + { + "bbox": [ + 281, + 186, + 307, + 200 + ], + "score": 0.92, + "content": "\\sqrt { d / \\kappa }", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 187, + 505, + 200 + ], + "score": 1.0, + "content": "is roughly tight, since, in the well-explored case", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 199, + 506, + 215 + ], + "spans": [ + { + "bbox": [ + 104, + 200, + 262, + 215 + ], + "score": 1.0, + "content": "(Assumption 2 of Wang et al. (2021a)), √", + "type": "text" + }, + { + "bbox": [ + 263, + 201, + 298, + 213 + ], + "score": 0.91, + "content": "\\kappa = 1 / d", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 200, + 332, + 215 + ], + "score": 1.0, + "content": "and the", + "type": "text" + }, + { + "bbox": [ + 332, + 199, + 389, + 214 + ], + "score": 0.93, + "content": "\\sqrt { d / \\kappa } = \\sqrt { d ^ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 200, + 506, + 215 + ], + "score": 1.0, + "content": "recovers the optimal feature", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 212, + 506, + 228 + ], + "spans": [ + { + "bbox": [ + 104, + 212, + 155, + 228 + ], + "score": 1.0, + "content": "dependence", + "type": "text" + }, + { + "bbox": [ + 155, + 213, + 186, + 225 + ], + "score": 0.91, + "content": "d H \\sqrt { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 212, + 358, + 228 + ], + "score": 1.0, + "content": "in the online setting (Zhou et al., 2021a). If", + "type": "text" + }, + { + "bbox": [ + 358, + 214, + 396, + 226 + ], + "score": 0.88, + "content": "\\kappa \\ll 1 / d ,", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 212, + 506, + 228 + ], + "score": 1.0, + "content": ", then doing offline learning", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 226, + 505, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 256, + 237 + ], + "score": 1.0, + "content": "requires sample size proportional to", + "type": "text" + }, + { + "bbox": [ + 256, + 226, + 273, + 237 + ], + "score": 0.88, + "content": "d / \\kappa ,", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 226, + 362, + 237 + ], + "score": 1.0, + "content": ", which reveals offline", + "type": "text" + }, + { + "bbox": [ + 363, + 226, + 377, + 236 + ], + "score": 0.56, + "content": "R L", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 226, + 505, + 237 + ], + "score": 1.0, + "content": "is harder when the exploration", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 236, + 506, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 271, + 249 + ], + "score": 1.0, + "content": "of behavior policy is insufficient. When", + "type": "text" + }, + { + "bbox": [ + 271, + 237, + 298, + 246 + ], + "score": 0.88, + "content": "\\kappa = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 236, + 506, + 249 + ], + "score": 1.0, + "content": ", learning the optimal policy accurately cannot be", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 247, + 318, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 279, + 261 + ], + "score": 1.0, + "content": "guaranteed even if the sample/episode size", + "type": "text" + }, + { + "bbox": [ + 279, + 248, + 315, + 258 + ], + "score": 0.89, + "content": "K \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 247, + 318, + 261 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 7.5, + "bbox_fs": [ + 104, + 156, + 507, + 261 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 272, + 333, + 285 + ], + "lines": [ + { + "bbox": [ + 106, + 272, + 334, + 285 + ], + "spans": [ + { + "bbox": [ + 106, + 272, + 334, + 285 + ], + "score": 1.0, + "content": "C.5 PROOF OF THE SECOND PART OF THEOREM 3.2", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12, + "bbox_fs": [ + 106, + 272, + 334, + 285 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 291, + 506, + 340 + ], + "lines": [ + { + "bbox": [ + 104, + 291, + 507, + 317 + ], + "spans": [ + { + "bbox": [ + 104, + 294, + 199, + 315 + ], + "score": 1.0, + "content": "Lemma C.10. Recall", + "type": "text" + }, + { + "bbox": [ + 199, + 291, + 342, + 317 + ], + "score": 0.94, + "content": "\\widehat { \\sigma } _ { h } = \\sqrt { \\operatorname* { m a x } \\left\\{ 1 , \\widehat { \\mathrm { V a r } } _ { P _ { h } } \\widehat { V } _ { h + 1 } \\right\\} + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 294, + 361, + 315 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 361, + 294, + 504, + 315 + ], + "score": 0.84, + "content": "\\sigma _ { h } ^ { \\star } = \\sqrt { \\operatorname* { m a x } \\left\\{ 1 , \\mathrm { V a r } _ { P _ { h } } V _ { h + 1 } ^ { \\star } \\right\\} + 1 } .", + "type": "inline_equation" + }, + { + "bbox": [ + 504, + 294, + 507, + 315 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 315, + 507, + 333 + ], + "spans": [ + { + "bbox": [ + 104, + 315, + 121, + 333 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 121, + 317, + 290, + 331 + ], + "score": 0.89, + "content": "\\begin{array} { r } { K \\ge \\operatorname* { m a x } \\left\\{ 5 1 2 ( 1 / \\kappa ) ^ { 2 } \\log \\left( \\frac { 4 H d } { \\delta } \\right) , 4 \\lambda / \\kappa \\right\\} } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 315, + 308, + 333 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 309, + 318, + 440, + 330 + ], + "score": 0.91, + "content": "K \\ge \\operatorname* { m a x } \\{ \\mathcal { M } _ { 1 } , \\mathcal { M } _ { 2 } , \\mathcal { M } _ { 3 } , \\mathcal { M } _ { 4 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 315, + 507, + 333 + ], + "score": 1.0, + "content": ", then with prob-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 328, + 163, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 328, + 134, + 342 + ], + "score": 1.0, + "content": "ability", + "type": "text" + }, + { + "bbox": [ + 135, + 330, + 157, + 340 + ], + "score": 0.83, + "content": "1 - \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 328, + 163, + 342 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14, + "bbox_fs": [ + 104, + 291, + 507, + 342 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 232, + 339, + 379, + 373 + ], + "lines": [ + { + "bbox": [ + 232, + 339, + 379, + 373 + ], + "spans": [ + { + "bbox": [ + 232, + 339, + 379, + 373 + ], + "score": 0.94, + "content": "\\operatorname* { s u p } _ { h } \\vert \\vert \\widehat { \\sigma } _ { h } ^ { 2 } - \\sigma _ { h } ^ { \\star 2 } \\vert \\vert _ { \\infty } \\leq \\widetilde { O } \\left( \\frac { H ^ { 3 } \\sqrt { d } } { \\sqrt { \\kappa K } } \\right) .", + "type": "interline_equation", + "image_path": "ec14809aea10c1a5dbf9381ecb8244a76d3a73b1d50835ed0d928ab437f01e6f.jpg" + } + ] + } + ], + "index": 16.5, + "virtual_lines": [ + { + "bbox": [ + 232, + 339, + 379, + 356.0 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 232, + 356.0, + 379, + 373.0 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 385, + 409, + 398 + ], + "lines": [ + { + "bbox": [ + 106, + 385, + 408, + 399 + ], + "spans": [ + { + "bbox": [ + 106, + 385, + 311, + 399 + ], + "score": 1.0, + "content": "Proof. By definition and the non-expansiveness of", + "type": "text" + }, + { + "bbox": [ + 311, + 386, + 369, + 398 + ], + "score": 0.89, + "content": "\\operatorname* { m a x } \\{ 1 , \\cdot \\} + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 385, + 408, + 399 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18, + "bbox_fs": [ + 106, + 385, + 408, + 399 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 156, + 403, + 455, + 505 + ], + "lines": [ + { + "bbox": [ + 156, + 403, + 455, + 505 + ], + "spans": [ + { + "bbox": [ + 156, + 403, + 455, + 505 + ], + "score": 0.95, + "content": "\\begin{array} { r l } & { \\quad \\left\\| \\sigma _ { \\widehat { V } _ { h + 1 } } ^ { 2 } - \\sigma _ { h } ^ { \\star 2 } \\right\\| _ { \\infty } \\leq \\left\\| \\operatorname { V a r } \\widehat { V } _ { h + 1 } - \\operatorname { V a r } V _ { h + 1 } ^ { \\star } \\right\\| _ { \\infty } } \\\\ & { \\leq \\left\\| \\mathbb { P } _ { h } \\left( \\widehat { V } _ { h + 1 } ^ { 2 } - V _ { h + 1 } ^ { \\star 2 } \\right) \\right\\| _ { \\infty } + \\left\\| ( \\mathbb { P } _ { h } \\widehat { V } _ { h + 1 } ) ^ { 2 } - ( \\mathbb { P } _ { h } V _ { h + 1 } ^ { \\star } ) ^ { 2 } \\right\\| _ { \\infty } } \\\\ & { \\leq \\left\\| \\widehat { V } _ { h + 1 } ^ { 2 } - V _ { h + 1 } ^ { \\star 2 } \\right\\| _ { \\infty } + \\left\\| ( \\mathbb { P } _ { h } \\widehat { V } _ { h + 1 } + \\mathbb { P } _ { h } V _ { h + 1 } ^ { \\star } ) ( \\mathbb { P } _ { h } \\widehat { V } _ { h + 1 } - \\mathbb { P } _ { h } V _ { h + 1 } ^ { \\star } ) \\right\\| _ { \\infty } } \\\\ & { \\leq 2 H \\left\\| \\widehat { V } _ { h + 1 } - V _ { h + 1 } ^ { \\star } \\right\\| _ { \\infty } + 2 H \\left\\| \\mathbb { P } _ { h } \\widehat { V } _ { h + 1 } - \\mathbb { P } _ { h } V _ { h + 1 } ^ { \\star } \\right\\| _ { \\infty } \\leq \\widetilde { \\mathcal { O } } \\left( \\frac { H ^ { 3 } \\sqrt { d } } { \\sqrt { \\kappa K } } \\right) . } \\end{array}", + "type": "interline_equation", + "image_path": "22eb4cd699d7db194269f8a2277efcf636aaccf41f32ef2b4e65064d1f788669.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 156, + 403, + 455, + 437.0 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 156, + 437.0, + 455, + 471.0 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 156, + 471.0, + 455, + 505.0 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 510, + 505, + 533 + ], + "lines": [ + { + "bbox": [ + 105, + 509, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 171, + 524 + ], + "score": 1.0, + "content": "with probability", + "type": "text" + }, + { + "bbox": [ + 172, + 511, + 195, + 521 + ], + "score": 0.87, + "content": "1 - \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 509, + 222, + 524 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 222, + 511, + 255, + 522 + ], + "score": 0.92, + "content": "h \\in [ H ]", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 509, + 506, + 524 + ], + "score": 1.0, + "content": ", where the last inequality comes from Lemma C.8. Combining", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 521, + 298, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 521, + 298, + 533 + ], + "score": 1.0, + "content": "this with Lemma C.7, we have the stated result.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 509, + 506, + 533 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 540, + 505, + 581 + ], + "lines": [ + { + "bbox": [ + 105, + 540, + 506, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 264, + 555 + ], + "score": 1.0, + "content": "Lemma C.11. Denote the quantities", + "type": "text" + }, + { + "bbox": [ + 264, + 541, + 485, + 554 + ], + "score": 0.9, + "content": "C _ { 1 } = \\operatorname* { m a x } \\{ 2 \\lambda , 1 2 8 \\log ( 2 d / \\delta ) , 1 2 8 H ^ { 4 } \\log ( 2 d / \\delta ) / \\kappa ^ { 2 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 540, + 506, + 555 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 550, + 507, + 571 + ], + "spans": [ + { + "bbox": [ + 106, + 552, + 368, + 569 + ], + "score": 0.73, + "content": "\\begin{array} { r } { C _ { 2 } = \\operatorname* { m a x } \\{ \\frac { \\lambda ^ { 2 } } { \\kappa \\log ( ( \\lambda + K ) H / \\lambda \\delta ) } , 9 6 ^ { 2 } H ^ { 1 2 } d \\log ( ( \\lambda + K ) H / \\lambda \\delta ) / \\kappa ^ { 5 } \\} } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 550, + 507, + 571 + ], + "score": 1.0, + "content": ". Suppose the number of episode", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 107, + 567, + 348, + 582 + ], + "spans": [ + { + "bbox": [ + 107, + 569, + 117, + 578 + ], + "score": 0.69, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 567, + 152, + 582 + ], + "score": 1.0, + "content": "satisfies", + "type": "text" + }, + { + "bbox": [ + 152, + 569, + 231, + 581 + ], + "score": 0.85, + "content": "K > \\operatorname* { m a x } \\{ C _ { 1 } , C _ { 2 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 567, + 321, + 582 + ], + "score": 1.0, + "content": ", then with probability", + "type": "text" + }, + { + "bbox": [ + 322, + 569, + 344, + 579 + ], + "score": 0.8, + "content": "1 - \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 567, + 348, + 582 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 540, + 507, + 582 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 171, + 586, + 439, + 608 + ], + "lines": [ + { + "bbox": [ + 171, + 586, + 439, + 608 + ], + "spans": [ + { + "bbox": [ + 171, + 586, + 439, + 608 + ], + "score": 0.89, + "content": "\\sqrt { \\phi ( s , a ) \\Lambda _ { h } ^ { - 1 } \\phi ( s , a ) } \\le 2 \\sqrt { \\phi ( s , a ) \\Lambda _ { h } ^ { \\star - 1 } \\phi ( s , a ) } , \\quad \\forall s , a \\in \\mathcal { S } \\times \\mathcal { A } ,", + "type": "interline_equation", + "image_path": "0afe9a095521196b73798b4aadea8daeb7346d645eaaa22cd1a14bbb516186f7.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 171, + 586, + 439, + 608 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 622, + 461, + 641 + ], + "lines": [ + { + "bbox": [ + 104, + 622, + 464, + 643 + ], + "spans": [ + { + "bbox": [ + 104, + 622, + 255, + 643 + ], + "score": 1.0, + "content": "Proof of Lemma C.11. By definition", + "type": "text" + }, + { + "bbox": [ + 255, + 622, + 407, + 642 + ], + "score": 0.93, + "content": "\\sqrt { \\phi ( s , a ) \\Lambda _ { h } ^ { - 1 } \\phi ( s , a ) } = \\| \\phi ( s , a ) \\| _ { \\Lambda _ { h } ^ { - 1 } } .", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 622, + 464, + 643 + ], + "score": 1.0, + "content": ". Then denote", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28, + "bbox_fs": [ + 104, + 622, + 464, + 643 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 245, + 647, + 365, + 671 + ], + "lines": [ + { + "bbox": [ + 245, + 647, + 365, + 671 + ], + "spans": [ + { + "bbox": [ + 245, + 647, + 365, + 671 + ], + "score": 0.93, + "content": "\\Lambda _ { h } ^ { \\prime } = \\frac { 1 } { K } \\Lambda _ { h } , \\quad { \\Lambda _ { h } ^ { \\star } } ^ { \\prime } = \\frac { 1 } { K } \\Lambda _ { h } ^ { \\star } ,", + "type": "interline_equation", + "image_path": "4cf2251ae3e307003067d53ebd4cbe5b6a64e97ebcbe36ca70502c914b7f81ed.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 245, + 647, + 365, + 671 + ], + "spans": [], + "index": 29 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 79, + 505, + 108 + ], + "lines": [ + { + "bbox": [ + 102, + 75, + 508, + 99 + ], + "spans": [ + { + "bbox": [ + 102, + 75, + 135, + 99 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 135, + 81, + 366, + 98 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\Lambda _ { h } \\ = \\ \\sum _ { \\tau = 1 } ^ { K } \\phi ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) ^ { \\top } \\phi ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) / \\sigma _ { V _ { h + 1 } ^ { \\star } } ^ { 2 } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) + \\lambda I . } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 75, + 476, + 99 + ], + "score": 1.0, + "content": ". Under the condition of", + "type": "text" + }, + { + "bbox": [ + 477, + 83, + 487, + 92 + ], + "score": 0.8, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 75, + 508, + 99 + ], + "score": 1.0, + "content": ", by", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 95, + 253, + 109 + ], + "spans": [ + { + "bbox": [ + 105, + 95, + 230, + 109 + ], + "score": 1.0, + "content": "Lemma C.10, with probability", + "type": "text" + }, + { + "bbox": [ + 230, + 97, + 253, + 107 + ], + "score": 0.74, + "content": "1 - \\delta", + "type": "inline_equation" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "interline_equation", + "bbox": [ + 132, + 111, + 462, + 214 + ], + "lines": [ + { + "bbox": [ + 132, + 111, + 462, + 214 + ], + "spans": [ + { + "bbox": [ + 132, + 111, + 462, + 214 + ], + "score": 0.93, + "content": "\\begin{array} { r l } & { \\left. \\Lambda _ { h } ^ { \\star ^ { \\prime } } - \\Lambda _ { h } ^ { \\prime } \\right. \\leq \\underset { s , a } { \\operatorname* { s u p } } \\left. \\frac { \\phi ( s , a ) \\phi ( s , a ) ^ { \\top } } { \\sigma _ { h } ^ { \\star 2 } ( s , a ) } - \\frac { \\phi ( s , a ) \\phi ( s , a ) ^ { \\top } } { \\sigma _ { \\hat { V } _ { h + 1 } } ^ { 2 } ( s , a ) } \\right. } \\\\ & { \\leq \\underset { s , a } { \\operatorname* { s u p } } \\left. \\frac { \\sigma _ { h } ^ { \\star 2 } ( s , a ) - \\sigma _ { \\hat { V } _ { h + 1 } } ^ { 2 } ( s , a ) } { \\sigma _ { h } ^ { \\star 2 } ( s , a ) \\sigma _ { \\hat { V } _ { h + 1 } } ^ { 2 } ( s , a ) } \\right. \\cdot \\left. \\phi ( s , a ) \\right. ^ { 2 } \\leq \\underset { s , a } { \\operatorname* { s u p } } \\left. \\frac { \\sigma _ { h } ^ { \\star 2 } ( s , a ) - \\sigma _ { \\hat { V } _ { h + 1 } } ^ { 2 } ( s , a ) } { 1 } \\right. \\cdot 1 } \\\\ & { \\leq \\tilde { O } \\left( \\frac { H ^ { 3 } \\sqrt { d } } { \\sqrt { \\kappa K } } \\right) . } \\end{array}", + "type": "interline_equation", + "image_path": "526db28629d9a68511d8dba74ba4673be24b340046629ae03743fb39351a09f9.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 132, + 111, + 462, + 145.33333333333334 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 132, + 145.33333333333334, + 462, + 179.66666666666669 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 132, + 179.66666666666669, + 462, + 214.00000000000003 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 218, + 460, + 233 + ], + "lines": [ + { + "bbox": [ + 104, + 217, + 460, + 235 + ], + "spans": [ + { + "bbox": [ + 104, + 217, + 214, + 235 + ], + "score": 1.0, + "content": "Next by Lemma H.5 (with", + "type": "text" + }, + { + "bbox": [ + 214, + 220, + 222, + 231 + ], + "score": 0.85, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 217, + 245, + 235 + ], + "score": 1.0, + "content": "to be", + "type": "text" + }, + { + "bbox": [ + 245, + 220, + 281, + 234 + ], + "score": 0.93, + "content": "\\phi / \\sigma _ { V _ { h + 1 } ^ { \\star } }", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 217, + 299, + 235 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 299, + 220, + 327, + 230 + ], + "score": 0.87, + "content": "C = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 217, + 432, + 235 + ], + "score": 1.0, + "content": "), it holds with probability", + "type": "text" + }, + { + "bbox": [ + 432, + 220, + 455, + 230 + ], + "score": 0.83, + "content": "1 - 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\\frac { 4 \\sqrt { 2 } } { \\sqrt { K } } \\left( \\log \\frac { 2 d } { \\delta } \\right) ^ { 1 / 2 } } & \\\\ { \\le \\lambda { \\operatorname* { m i n } } \\left( { \\mathbb { E } _ { \\rho , h } } [ \\phi ( s , a ) \\phi ( s , a ) ] ^ { \\top } \\rho _ { \\mathcal { N } _ { k + 1 } ^ { \\prime } ( s , a ) } ^ { \\sigma } | \\right) - \\frac { 4 \\sqrt { 2 } } { \\sqrt { K } } \\left( \\log \\frac { 2 d } { \\delta } \\right) ^ { 1 / 2 } } & \\\\ \\ge \\frac { \\kappa } { L } - \\frac { 4 \\sqrt { 2 } } \\end{array}", + "type": "interline_equation", + "image_path": "dcf54322bd8ce9562fd57dbefd2e43c8878238c74f9ab8763019dab9c5972d88.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 112, + 298, + 505, + 359.0 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 112, + 359.0, + 505, + 420.0 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 112, + 420.0, + 505, + 481.0 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 483, + 506, + 522 + ], + "lines": [ + { + "bbox": [ + 104, + 482, + 507, + 504 + ], + "spans": [ + { + "bbox": [ + 104, + 482, + 205, + 504 + ], + "score": 1.0, + "content": "Hence with probability", + "type": "text" + }, + { + "bbox": [ + 206, + 486, + 230, + 498 + ], + "score": 0.51, + "content": "1 - 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The claimed result follows straightfor-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 702, + 337, + 716 + ], + "spans": [ + { + "bbox": [ + 105, + 702, + 197, + 716 + ], + "score": 1.0, + "content": "wardly by multiplying", + "type": "text" + }, + { + "bbox": [ + 198, + 702, + 226, + 716 + ], + "score": 0.91, + "content": "1 / \\sqrt { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 702, + 337, + 716 + ], + "score": 1.0, + "content": "on both sides of the above.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20 + } + ], + "page_idx": 26, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 26, + 293, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 312, + 765 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 312, + 765 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 14 + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 494, + 721, + 505, + 731 + ], + "lines": [ + { + "bbox": [ + 496, + 722, + 505, + 731 + ], + "spans": [ + { + "bbox": [ + 496, + 722, + 505, + 731 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 79, + 505, + 108 + ], + "lines": [ + { + "bbox": [ + 102, + 75, + 508, + 99 + ], + "spans": [ + { + "bbox": [ + 102, + 75, + 135, + 99 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 135, + 81, + 366, + 98 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\Lambda _ { h } \\ = \\ \\sum _ { \\tau = 1 } ^ { K } \\phi ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) ^ { \\top } \\phi ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) / \\sigma _ { V _ { h + 1 } ^ { \\star } } ^ { 2 } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) + \\lambda I . } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 75, + 476, + 99 + ], + "score": 1.0, + "content": ". Under the condition of", + "type": "text" + }, + { + "bbox": [ + 477, + 83, + 487, + 92 + ], + "score": 0.8, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 75, + 508, + 99 + ], + "score": 1.0, + "content": ", by", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 95, + 253, + 109 + ], + "spans": [ + { + "bbox": [ + 105, + 95, + 230, + 109 + ], + "score": 1.0, + "content": "Lemma C.10, with probability", + "type": "text" + }, + { + "bbox": [ + 230, + 97, + 253, + 107 + ], + "score": 0.74, + "content": "1 - \\delta", + "type": "inline_equation" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 102, + 75, + 508, + 109 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 132, + 111, + 462, + 214 + ], + "lines": [ + { + "bbox": [ + 132, + 111, + 462, + 214 + ], + "spans": [ + { + "bbox": [ + 132, + 111, + 462, + 214 + ], + "score": 0.93, + "content": "\\begin{array} { r l } & { \\left. \\Lambda _ { h } ^ { \\star ^ { \\prime } } - \\Lambda _ { h } ^ { \\prime } \\right. \\leq \\underset { s , a } { \\operatorname* { s u p } } \\left. \\frac { \\phi ( s , a ) \\phi ( s , a ) ^ { \\top } } { \\sigma _ { h } ^ { \\star 2 } ( s , a ) } - \\frac { \\phi ( s , a ) \\phi ( s , a ) ^ { \\top } } { \\sigma _ { \\hat { V } _ { h + 1 } } ^ { 2 } ( s , a ) } \\right. } \\\\ & { \\leq \\underset { s , a } { \\operatorname* { s u p } } \\left. \\frac { \\sigma _ { h } ^ { \\star 2 } ( s , a ) - \\sigma _ { \\hat { V } _ { h + 1 } } ^ { 2 } ( s , a ) } { \\sigma _ { h } ^ { \\star 2 } ( s , a ) \\sigma _ { \\hat { V } _ { h + 1 } } ^ { 2 } ( s , a ) } \\right. \\cdot \\left. \\phi ( s , a ) \\right. ^ { 2 } \\leq \\underset { s , a } { \\operatorname* { s u p } } \\left. \\frac { \\sigma _ { h } ^ { \\star 2 } ( s , a ) - \\sigma _ { \\hat { V } _ { h + 1 } } ^ { 2 } ( s , a ) } { 1 } \\right. \\cdot 1 } \\\\ & { \\leq \\tilde { O } \\left( \\frac { H ^ { 3 } \\sqrt { d } } { \\sqrt { \\kappa K } } \\right) . } \\end{array}", + "type": "interline_equation", + "image_path": "526db28629d9a68511d8dba74ba4673be24b340046629ae03743fb39351a09f9.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 132, + 111, + 462, + 145.33333333333334 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 132, + 145.33333333333334, + 462, + 179.66666666666669 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 132, + 179.66666666666669, + 462, + 214.00000000000003 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 218, + 460, + 233 + ], + "lines": [ + { + "bbox": [ + 104, + 217, + 460, + 235 + ], + "spans": [ + { + "bbox": [ + 104, + 217, + 214, + 235 + ], + "score": 1.0, + "content": "Next by Lemma H.5 (with", + "type": "text" + }, + { + "bbox": [ + 214, + 220, + 222, + 231 + ], + "score": 0.85, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 217, + 245, + 235 + ], + "score": 1.0, + "content": "to be", + "type": "text" + }, + { + "bbox": [ + 245, + 220, + 281, + 234 + ], + "score": 0.93, + "content": "\\phi / \\sigma _ { V _ { h + 1 } ^ { \\star } }", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 217, + 299, + 235 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 299, + 220, + 327, + 230 + ], + "score": 0.87, + "content": "C = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 217, + 432, + 235 + ], + "score": 1.0, + "content": "), it holds with probability", + "type": "text" + }, + { + "bbox": [ + 432, + 220, + 455, + 230 + ], + "score": 0.83, + "content": "1 - \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 217, + 460, + 235 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5, + "bbox_fs": [ + 104, + 217, + 460, + 235 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 144, + 237, + 465, + 268 + ], + "lines": [ + { + "bbox": [ + 144, + 237, + 465, + 268 + ], + "spans": [ + { + "bbox": [ + 144, + 237, + 465, + 268 + ], + "score": 0.9, + "content": "\\left\\| \\Lambda _ { h } ^ { \\star ^ { \\prime } } - \\left( \\mathbb { E } _ { \\mu , h } [ \\phi ( s , a ) \\phi ( s , a ) ^ { \\top } / \\sigma _ { V _ { h + 1 } ^ { \\star } } ^ { 2 } ( s , a ) ] + \\frac { \\lambda } { K } I _ { d } \\right) \\right\\| \\leq \\frac { 4 \\sqrt { 2 } } { \\sqrt { K } } \\left( \\log \\frac { 2 d } { \\delta } \\right) ^ { 1 / 2 } .", + "type": "interline_equation", + "image_path": "652fa363c9f42c0535cfb6248feb0df6df83e64beb364647e1bf65a4883a66ef.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 144, + 237, + 465, + 268 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 270, + 503, + 294 + ], + "lines": [ + { + "bbox": [ + 106, + 268, + 505, + 283 + ], + "spans": [ + { + "bbox": [ + 106, + 269, + 275, + 283 + ], + "score": 1.0, + "content": "Therefore by Weyl’s spectrum", + "type": "text" + }, + { + "bbox": [ + 286, + 270, + 325, + 282 + ], + "score": 1.0, + "content": "theorem", + "type": "text" + }, + { + "bbox": [ + 337, + 270, + 387, + 282 + ], + "score": 1.0, + "content": "and the", + "type": "text" + }, + { + "bbox": [ + 398, + 268, + 455, + 283 + ], + "score": 1.0, + "content": "condition", + "type": "text" + }, + { + "bbox": [ + 456, + 270, + 466, + 280 + ], + "score": 0.41, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 272, + 505, + 281 + ], + "score": 1.0, + "content": ">", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 280, + 376, + 294 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 299, + 294 + ], + "score": 0.88, + "content": "\\operatorname* { m a x } \\{ 2 \\lambda , 1 2 8 \\log ( 2 d / \\delta ) , 1 2 8 H ^ { 4 } \\log ( 2 d / \\delta ) / \\kappa ^ { 2 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 280, + 376, + 294 + ], + "score": 1.0, + "content": ", the above implies", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5, + "bbox_fs": [ + 106, + 268, + 505, + 294 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 112, + 298, + 505, + 481 + ], + "lines": [ + { + "bbox": [ + 112, + 298, + 505, + 481 + ], + "spans": [ + { + "bbox": [ + 112, + 298, + 505, + 481 + ], + "score": 0.92, + "content": "\\begin{array} { r l } { \\left\\| { { { \\Lambda } _ { k } ^ { \\prime } } } \\right\\| = \\lambda { \\operatorname* { m s u p } _ { i } } ( \\Lambda _ { k } ^ { \\prime } ) \\le \\lambda { \\operatorname* { m s u p } _ { i } } \\left( { \\operatorname* { \\sum } _ { n , h } } [ \\phi ( s , a ) \\phi ( s , a ) ] ^ { \\top } \\rho _ { \\mathcal { N } _ { k + 1 } ^ { \\prime } } ^ { \\sigma } ( s , a ) \\right) + \\frac { \\lambda } { K } + \\frac { 4 \\sqrt { 2 } } { \\sqrt { K } } \\left( \\log \\frac { 2 d } { \\delta } \\right) ^ { 1 / 2 } } & \\\\ { \\le \\left\\| { { \\mathbb { E } _ { \\rho , h } } [ \\phi ( s , a ) \\phi ( s , a ) ] ^ { \\top } } \\rho _ { \\mathcal { N } _ { k + 1 } ^ { \\prime } } ^ { \\sigma } ( s , a ) \\right\\| + \\frac { \\lambda } { K } + \\frac { 4 \\sqrt { 2 } } { \\sqrt { K } } \\left( \\log \\frac { 2 d } { \\delta } \\right) ^ { 1 / 2 } } & \\\\ { \\le \\| \\phi ( s , a ) \\| ^ { 2 } + \\frac { \\lambda } { K } + \\frac { 4 \\sqrt { 2 } } { \\sqrt { K } } \\left( \\log \\frac { 2 d } { \\delta } \\right) ^ { 1 / 2 } \\le 1 + \\frac { \\lambda } { K } + \\frac { 4 \\sqrt { 2 } } { \\sqrt { K } } \\left( \\log \\frac { 2 d } { \\delta } \\right) ^ { 1 / 2 } \\le 2 , } & \\\\ { \\operatorname* { m i n } ( \\Lambda _ { k } ^ { \\prime } ) \\le \\lambda { \\operatorname* { m i n } } \\left( { \\mathbb { E } _ { \\rho , h } } [ \\phi ( s , a ) \\phi ( s , a ) ] ^ { \\top } \\rho _ { \\mathcal { N } _ { k + 1 } ^ { \\prime } ( s , a ) } ^ { \\sigma } | \\right) + \\frac { \\lambda } { K } - \\frac { 4 \\sqrt { 2 } } { \\sqrt { K } } \\left( \\log \\frac { 2 d } { \\delta } \\right) ^ { 1 / 2 } } & \\\\ { \\le \\lambda { \\operatorname* { m i n } } \\left( { \\mathbb { E } _ { \\rho , h } } [ \\phi ( s , a ) \\phi ( s , a ) ] ^ { \\top } \\rho _ { \\mathcal { N } _ { k + 1 } ^ { \\prime } ( s , a ) } ^ { \\sigma } | \\right) - \\frac { 4 \\sqrt { 2 } } { \\sqrt { K } } \\left( \\log \\frac { 2 d } { \\delta } \\right) ^ { 1 / 2 } } & \\\\ \\ge \\frac { \\kappa } { L } - \\frac { 4 \\sqrt { 2 } } \\end{array}", + "type": "interline_equation", + "image_path": "dcf54322bd8ce9562fd57dbefd2e43c8878238c74f9ab8763019dab9c5972d88.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 112, + 298, + 505, + 359.0 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 112, + 359.0, + 505, + 420.0 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 112, + 420.0, + 505, + 481.0 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 483, + 506, + 522 + ], + "lines": [ + { + "bbox": [ + 104, + 482, + 507, + 504 + ], + "spans": [ + { + "bbox": [ + 104, + 482, + 205, + 504 + ], + "score": 1.0, + "content": "Hence with probability", + "type": "text" + }, + { + "bbox": [ + 206, + 486, + 230, + 498 + ], + "score": 0.51, + "content": "1 - \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 482, + 235, + 504 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 236, + 483, + 285, + 504 + ], + "score": 0.86, + "content": "\\left\\| \\Lambda _ { h } ^ { \\star ^ { \\prime } } \\right\\| \\leq 2", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 482, + 305, + 504 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 305, + 483, + 456, + 504 + ], + "score": 0.94, + "content": "\\left\\| { \\boldsymbol { \\Lambda } } _ { h } ^ { \\star ^ { \\prime } - 1 } \\right\\| = 1 / { \\lambda _ { \\operatorname* { m i n } } ( \\boldsymbol { \\Lambda } _ { h } ^ { \\star ^ { \\prime } } ) } \\le 2 H ^ { 2 } / \\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 482, + 507, + 504 + ], + "score": 1.0, + "content": ". Similarly,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 500, + 275, + 525 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 182, + 523 + ], + "score": 0.93, + "content": "\\left\\| \\Lambda _ { h } ^ { ' - 1 } \\right\\| \\le 2 H ^ { 2 } / \\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 500, + 275, + 525 + ], + "score": 1.0, + "content": "with high probability.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5, + "bbox_fs": [ + 104, + 482, + 507, + 525 + ] + }, + { + "type": "text", + "bbox": [ + 101, + 528, + 505, + 553 + ], + "lines": [ + { + "bbox": [ + 105, + 527, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 214, + 542 + ], + "score": 1.0, + "content": "Now apply Lemma H.4 to", + "type": "text" + }, + { + "bbox": [ + 214, + 527, + 229, + 542 + ], + "score": 0.9, + "content": "\\Lambda _ { h } ^ { \\star ^ { \\prime } }", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 528, + 248, + 542 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 248, + 529, + 261, + 542 + ], + "score": 0.9, + "content": "\\Lambda _ { h } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 528, + 451, + 542 + ], + "score": 1.0, + "content": "and a union bound, we obtain with probability", + "type": "text" + }, + { + "bbox": [ + 451, + 529, + 474, + 540 + ], + "score": 0.84, + "content": "1 - \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 528, + 505, + 542 + ], + "score": 1.0, + "content": ", for all", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 107, + 543, + 122, + 551 + ], + "spans": [ + { + "bbox": [ + 107, + 543, + 122, + 551 + ], + "score": 0.8, + "content": "s , a", + "type": "inline_equation" + } + ], + "index": 15 + } + ], + "index": 14.5, + "bbox_fs": [ + 105, + 527, + 505, + 551 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 133, + 554, + 477, + 659 + ], + "lines": [ + { + "bbox": [ + 133, + 554, + 477, + 659 + ], + "spans": [ + { + "bbox": [ + 133, + 554, + 477, + 659 + ], + "score": 0.95, + "content": "\\begin{array} { r l } { \\| \\phi ( s , a ) \\| _ { \\Lambda _ { h } ^ { \\prime - 1 } } \\leq \\left[ 1 + \\sqrt { \\left\\| \\Lambda _ { h } ^ { \\star ^ { \\prime } - 1 } \\right\\| \\left\\| \\Lambda _ { h } ^ { \\star ^ { \\prime } } \\right\\| \\cdot \\left\\| \\Lambda _ { h } ^ { \\prime - 1 } \\right\\| \\cdot \\left\\| \\Lambda _ { h } ^ { \\star ^ { \\prime } } - \\Lambda _ { h } ^ { \\prime } \\right\\| } \\right] \\cdot \\| \\phi ( s , a ) \\| _ { \\Lambda _ { h } ^ { \\star ^ { \\prime } - 1 } } } & { } \\\\ { \\leq \\left[ 1 + \\sqrt { \\frac { 2 H ^ { 2 } } { \\kappa } \\cdot 1 \\cdot \\frac { 2 H ^ { 2 } } { \\kappa } \\cdot \\left\\| \\Lambda _ { h } ^ { \\star ^ { \\prime } } - \\Lambda _ { h } ^ { \\prime } \\right\\| } \\right] \\cdot \\| \\phi ( s , a ) \\| _ { \\Lambda _ { h } ^ { \\star ^ { \\prime } - 1 } } } & { } \\\\ { \\leq \\left[ 1 + \\sqrt { \\frac { H ^ { 4 } } { \\kappa ^ { 2 } } \\left[ \\tilde { \\mathcal { O } } \\left( \\frac { H ^ { 3 } \\sqrt { d } } { \\sqrt { \\kappa K } } \\right) \\right] } \\right] \\cdot \\| \\phi ( s , a ) \\| _ { \\Lambda _ { h } ^ { \\star ^ { \\prime } - 1 } } \\leq 2 \\left\\| \\phi ( s , a ) \\right\\| _ { \\Lambda _ { h } ^ { \\star ^ { \\prime } - 1 } } } & { } \\end{array}", + "type": "interline_equation", + "image_path": "27fb18c65eff53dc82250ad63a633e3cb2aaf15e407851db0caeaac205d94b02.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 133, + 554, + 477, + 589.0 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 133, + 589.0, + 477, + 624.0 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 133, + 624.0, + 477, + 659.0 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 674, + 506, + 716 + ], + "lines": [ + { + "bbox": [ + 104, + 673, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 104, + 673, + 471, + 689 + ], + "score": 1.0, + "content": "where the third inequality uses equation 20 and the last inequality uses", + "type": "text" + }, + { + "bbox": [ + 471, + 675, + 505, + 687 + ], + "score": 0.8, + "content": "K \\_ { > }", + "type": "inline_equation" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 684, + 508, + 704 + ], + "spans": [ + { + "bbox": [ + 106, + 685, + 343, + 703 + ], + "score": 0.82, + "content": "\\begin{array} { r } { \\operatorname* { m a x } \\{ \\frac { \\lambda ^ { 2 } } { \\kappa \\log ( ( \\lambda + K ) H / \\lambda \\delta ) } , \\overset { } { 9 } 6 ^ { 2 } \\dot { H } ^ { 1 2 } d \\log ( ( \\lambda + K ) H / \\lambda \\delta ) / \\kappa ^ { 5 } \\} } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 684, + 508, + 704 + ], + "score": 1.0, + "content": ". The claimed result follows straightfor-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 702, + 337, + 716 + ], + "spans": [ + { + "bbox": [ + 105, + 702, + 197, + 716 + ], + "score": 1.0, + "content": "wardly by multiplying", + "type": "text" + }, + { + "bbox": [ + 198, + 702, + 226, + 716 + ], + "score": 0.91, + "content": "1 / \\sqrt { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 702, + 337, + 716 + ], + "score": 1.0, + "content": "on both sides of the above.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20, + "bbox_fs": [ + 104, + 673, + 508, + 716 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 104, + 82, + 505, + 105 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 94 + ], + "score": 1.0, + "content": "Proof of Theorem 3.2. The first part of the theorem has been shown in Theorem C.6. For the second", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 104, + 92, + 376, + 106 + ], + "spans": [ + { + "bbox": [ + 104, + 92, + 227, + 106 + ], + "score": 1.0, + "content": "part, apply Theorem C.6 with", + "type": "text" + }, + { + "bbox": [ + 227, + 94, + 258, + 104 + ], + "score": 0.93, + "content": "\\pi = \\pi ^ { \\star }", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 92, + 348, + 106 + ], + "score": 1.0, + "content": ", then with probability", + "type": "text" + }, + { + "bbox": [ + 348, + 94, + 371, + 104 + ], + "score": 0.86, + "content": "1 - \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 92, + 376, + 106 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "interline_equation", + "bbox": [ + 162, + 107, + 448, + 142 + ], + "lines": [ + { + "bbox": [ + 162, + 107, + 448, + 142 + ], + "spans": [ + { + "bbox": [ + 162, + 107, + 448, + 142 + ], + "score": 0.93, + "content": "v ^ { \\pi ^ { \\star } } - v ^ { \\widehat { \\pi } } \\leq \\widetilde O \\left( \\sqrt { d } \\cdot \\sum _ { h = 1 } ^ { H } \\mathbb E _ { \\pi ^ { \\star } } \\left[ \\left( \\phi ( \\cdot , \\cdot ) ^ { \\top } \\Lambda _ { h } ^ { - 1 } \\phi ( \\cdot , \\cdot ) \\right) ^ { 1 / 2 } \\right] \\right) + \\frac { 2 H ^ { 4 } \\sqrt { d } } { K } ,", + "type": "interline_equation", + "image_path": "6fa7eacbb016f34f419b8c971fc787a07f9badeea298ee2ab92255c68893d28f.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 162, + 107, + 448, + 118.66666666666667 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 162, + 118.66666666666667, + 448, + 130.33333333333334 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 162, + 130.33333333333334, + 448, + 142.0 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 143, + 379, + 155 + ], + "lines": [ + { + "bbox": [ + 105, + 142, + 380, + 157 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 353, + 157 + ], + "score": 1.0, + "content": "Now apply Lemma C.11 and a union bound, with probability", + "type": "text" + }, + { + "bbox": [ + 353, + 144, + 376, + 154 + ], + "score": 0.8, + "content": "1 - \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 142, + 380, + 157 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "interline_equation", + "bbox": [ + 154, + 157, + 457, + 192 + ], + "lines": [ + { + "bbox": [ + 154, + 157, + 457, + 192 + ], + "spans": [ + { + "bbox": [ + 154, + 157, + 457, + 192 + ], + "score": 0.93, + "content": "0 \\leq v ^ { \\star } - v ^ { \\widehat { \\pi } } \\leq \\widetilde { O } \\left( \\sqrt { d } \\cdot \\sum _ { h = 1 } ^ { H } \\mathbb { E } _ { \\pi ^ { \\star } } \\left[ \\left( \\phi ( \\cdot , \\cdot ) ^ { \\top } \\Lambda _ { h } ^ { \\star - 1 } \\phi ( \\cdot , \\cdot ) \\right) ^ { 1 / 2 } \\right] \\right) + \\frac { 2 H ^ { 4 } \\sqrt { d } } { K } .", + "type": "interline_equation", + "image_path": "f267a949e8c56d1b709e3869b520b7e1a83149d97bc5bed955e0d89e65dd1ed0.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 154, + 157, + 457, + 168.66666666666666 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 154, + 168.66666666666666, + 457, + 180.33333333333331 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 154, + 180.33333333333331, + 457, + 191.99999999999997 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "title", + "bbox": [ + 107, + 226, + 255, + 239 + ], + "lines": [ + { + "bbox": [ + 106, + 226, + 256, + 240 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 256, + 240 + ], + "score": 1.0, + "content": "D PROOF OF THEOREM 3.3", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 107, + 250, + 277, + 262 + ], + "lines": [ + { + "bbox": [ + 105, + 249, + 278, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 278, + 264 + ], + "score": 1.0, + "content": "First of all, we show the following lemma.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 106, + 264, + 356, + 277 + ], + "lines": [ + { + "bbox": [ + 105, + 262, + 357, + 280 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 199, + 280 + ], + "score": 1.0, + "content": "Lemma D.1. Suppose", + "type": "text" + }, + { + "bbox": [ + 199, + 264, + 330, + 277 + ], + "score": 0.91, + "content": "K > \\operatorname* { m a x } \\{ \\mathcal { M } _ { 1 } , \\mathcal { M } _ { 2 } , \\mathcal { M } _ { 3 } , \\mathcal { M } _ { 4 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 262, + 357, + 280 + ], + "score": 1.0, + "content": ". Plug", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 279, + 514, + 319 + ], + "lines": [ + { + "bbox": [ + 111, + 279, + 514, + 319 + ], + "spans": [ + { + "bbox": [ + 111, + 279, + 514, + 319 + ], + "score": 0.89, + "content": "\\mathbf { \\Sigma } _ { h } ^ { \\scriptscriptstyle { T } } ( s , a ) \\gets \\phi ( s , a ) ^ { \\scriptscriptstyle { T } } \\left| \\widehat { \\Lambda } _ { h } ^ { - 1 } \\sum _ { \\tau = 1 } ^ { K } \\frac { \\phi \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\cdot \\left( r _ { h } ^ { \\tau } + \\widehat { V } _ { h + 1 } \\left( s _ { h + 1 } ^ { \\tau } \\right) - \\left( \\widehat { T } _ { h } \\widehat { V } _ { h + 1 } \\right) \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\right) } { \\widehat { \\sigma } _ { h } ^ { 2 } \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) } \\right| + \\widetilde { O } ( \\frac { H ^ { 3 } d / \\kappa } { K } )", + "type": "interline_equation", + "image_path": "146979c426a5db4602cc39be48033f2f69fbfe2197072e8f93e047713e381914.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 111, + 279, + 514, + 292.3333333333333 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 111, + 292.3333333333333, + 514, + 305.66666666666663 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 111, + 305.66666666666663, + 514, + 318.99999999999994 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 322, + 506, + 346 + ], + "lines": [ + { + "bbox": [ + 105, + 322, + 507, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 161, + 338 + ], + "score": 1.0, + "content": "in Algorithm", + "type": "text" + }, + { + "bbox": [ + 161, + 325, + 167, + 334 + ], + "score": 0.29, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 322, + 199, + 338 + ], + "score": 1.0, + "content": "and let", + "type": "text" + }, + { + "bbox": [ + 199, + 324, + 211, + 335 + ], + "score": 0.87, + "content": "\\mathcal { T } _ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 322, + 332, + 338 + ], + "score": 1.0, + "content": "be the Bellman operator and", + "type": "text" + }, + { + "bbox": [ + 332, + 322, + 344, + 335 + ], + "score": 0.89, + "content": "\\widehat { \\mathcal { T } } _ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 322, + 507, + 338 + ], + "score": 1.0, + "content": "be the approximated Bellman operator.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 334, + 257, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 334, + 229, + 347 + ], + "score": 1.0, + "content": "Then we have with probability", + "type": "text" + }, + { + "bbox": [ + 229, + 335, + 252, + 345 + ], + "score": 0.87, + "content": "1 - \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 334, + 257, + 347 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15.5 + }, + { + "type": "interline_equation", + "bbox": [ + 187, + 348, + 424, + 365 + ], + "lines": [ + { + "bbox": [ + 187, + 348, + 424, + 365 + ], + "spans": [ + { + "bbox": [ + 187, + 348, + 424, + 365 + ], + "score": 0.87, + "content": "\\vert ( \\mathcal T _ { h } \\widehat V _ { h + 1 } - \\widehat T _ { h } \\widehat V _ { h + 1 } ) ( s , a ) \\vert \\le \\Gamma _ { h } ^ { I } ( s , a ) , \\quad \\forall s , a \\in \\mathcal S \\times \\mathcal A .", + "type": "interline_equation", + "image_path": "9f7ff5e2c1095e1bfd32eea1092ce4ba940c47d796bad2abfbf8de4105f980ff.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 187, + 348, + 424, + 365 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 376, + 505, + 403 + ], + "lines": [ + { + "bbox": [ + 106, + 375, + 506, + 391 + ], + "spans": [ + { + "bbox": [ + 106, + 376, + 173, + 391 + ], + "score": 1.0, + "content": "Proof of Lemma", + "type": "text" + }, + { + "bbox": [ + 173, + 378, + 189, + 388 + ], + "score": 0.28, + "content": "D . I", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 376, + 231, + 391 + ], + "score": 1.0, + "content": ". Suppose", + "type": "text" + }, + { + "bbox": [ + 231, + 380, + 244, + 389 + ], + "score": 0.83, + "content": "w _ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 376, + 396, + 391 + ], + "score": 1.0, + "content": "is the coefficient corresponding to the", + "type": "text" + }, + { + "bbox": [ + 396, + 375, + 429, + 389 + ], + "score": 0.93, + "content": "\\mathcal { T } _ { h } \\widehat { V } _ { h + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 376, + 453, + 391 + ], + "score": 1.0, + "content": "(such", + "type": "text" + }, + { + "bbox": [ + 454, + 379, + 467, + 389 + ], + "score": 0.84, + "content": "w _ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 376, + 506, + 391 + ], + "score": 1.0, + "content": "exists by", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 389, + 444, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 177, + 404 + ], + "score": 1.0, + "content": "Lemma H.9), i.e.", + "type": "text" + }, + { + "bbox": [ + 178, + 389, + 248, + 403 + ], + "score": 0.93, + "content": "\\mathcal { T } _ { h } \\widehat { V } _ { h + 1 } = \\phi ^ { \\top } w _ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 389, + 294, + 404 + ], + "score": 1.0, + "content": ", and recall", + "type": "text" + }, + { + "bbox": [ + 294, + 389, + 416, + 403 + ], + "score": 0.91, + "content": "( \\widehat { T } _ { h } \\widehat { V } _ { h + 1 } ) ( s , a ) = \\phi ( s , a ) ^ { \\top } \\widehat { w } _ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 389, + 444, + 404 + ], + "score": 1.0, + "content": ", then:", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5 + }, + { + "type": "interline_equation", + "bbox": [ + 104, + 405, + 445, + 498 + ], + "lines": [ + { + "bbox": [ + 104, + 405, + 445, + 498 + ], + "spans": [ + { + "bbox": [ + 104, + 405, + 445, + 498 + ], + "score": 0.92, + "content": "\\begin{array} { r l } & { \\quad \\left( \\mathcal { T } _ { h } \\widehat { V } _ { h + 1 } \\right) ( s , a ) - 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1 } ( \\sum _ { \\tau = 1 } ^ { K } \\phi ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) \\cdot ( r _ { h } ^ { \\tau } + \\widehat { V } _ { h + 1 } ( s _ { h + 1 } ^ { \\tau } ) - ( \\widehat { T } _ { h } \\widehat { V } _ { h + 1 } ) ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) ) / \\widehat { \\sigma } _ { h } ^ { 2 } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) ) )", + "type": "interline_equation", + "image_path": "491af15c9227ba63aa3ba4f1726761e8542c6afb392743b728f21bad30e276bf.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 134, + 513, + 507, + 524.3333333333334 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 134, + 524.3333333333334, + 507, + 535.6666666666667 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 134, + 535.6666666666667, + 507, + 547.0000000000001 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "interline_equation", + "bbox": [ + 135, + 563, + 510, + 597 + ], + "lines": [ + { + "bbox": [ + 135, + 563, + 510, + 597 + ], + "spans": [ + { + "bbox": [ + 135, + 563, + 510, + 597 + ], + "score": 0.79, + "content": "+ \\phi ( s , a ) ^ { \\top } \\widehat { \\Lambda } _ { h } ^ { - 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\\delta / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 623, + 443, + 647 + ], + "score": 1.0, + "content": ". 13", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 106, + 648, + 226, + 660 + ], + "lines": [ + { + "bbox": [ + 105, + 645, + 227, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 227, + 662 + ], + "score": 1.0, + "content": "For term (ii), it is bounded by", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "interline_equation", + "bbox": [ + 149, + 662, + 461, + 702 + ], + "lines": [ + { + "bbox": [ + 149, + 662, + 461, + 702 + ], + "spans": [ + { + "bbox": [ + 149, + 662, + 461, + 702 + ], + "score": 0.94, + "content": "\\phi ( s , a ) ^ { \\top } \\left| \\widehat { \\Lambda } _ { h } ^ { - 1 } \\sum _ { \\tau = 1 } ^ { K } \\frac { \\phi \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\cdot \\left( r _ { h } ^ { \\tau } + \\widehat { V } _ { h + 1 } \\left( s _ { h + 1 } ^ { \\tau } \\right) - 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The first part of the theorem has been shown in Theorem C.6. For the second", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 104, + 92, + 376, + 106 + ], + "spans": [ + { + "bbox": [ + 104, + 92, + 227, + 106 + ], + "score": 1.0, + "content": "part, apply Theorem C.6 with", + "type": "text" + }, + { + "bbox": [ + 227, + 94, + 258, + 104 + ], + "score": 0.93, + "content": "\\pi = \\pi ^ { \\star }", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 92, + 348, + 106 + ], + "score": 1.0, + "content": ", then with probability", + "type": "text" + }, + { + "bbox": [ + 348, + 94, + 371, + 104 + ], + "score": 0.86, + "content": "1 - \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 92, + 376, + 106 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 104, + 82, + 505, + 106 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 162, + 107, + 448, + 142 + ], + "lines": [ + { + "bbox": [ + 162, + 107, + 448, + 142 + ], + "spans": [ + { + "bbox": [ + 162, + 107, + 448, + 142 + ], + "score": 0.93, + "content": "v ^ { \\pi ^ { \\star } } - v ^ { \\widehat { \\pi } } \\leq \\widetilde O \\left( \\sqrt { d } \\cdot \\sum _ { h = 1 } ^ { H } \\mathbb E _ { \\pi ^ { \\star } } \\left[ \\left( \\phi ( \\cdot , \\cdot ) ^ { \\top } \\Lambda _ { h } ^ { - 1 } \\phi ( \\cdot , \\cdot ) \\right) ^ { 1 / 2 } \\right] \\right) + \\frac { 2 H ^ { 4 } \\sqrt { d } } { K } ,", + "type": "interline_equation", + "image_path": "6fa7eacbb016f34f419b8c971fc787a07f9badeea298ee2ab92255c68893d28f.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 162, + 107, + 448, + 118.66666666666667 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 162, + 118.66666666666667, + 448, + 130.33333333333334 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 162, + 130.33333333333334, + 448, + 142.0 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 143, + 379, + 155 + ], + "lines": [ + { + "bbox": [ + 105, + 142, + 380, + 157 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 353, + 157 + ], + "score": 1.0, + "content": "Now apply Lemma C.11 and a union bound, with probability", + "type": "text" + }, + { + "bbox": [ + 353, + 144, + 376, + 154 + ], + "score": 0.8, + "content": "1 - \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 142, + 380, + 157 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5, + "bbox_fs": [ + 105, + 142, + 380, + 157 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 154, + 157, + 457, + 192 + ], + "lines": [ + { + "bbox": [ + 154, + 157, + 457, + 192 + ], + "spans": [ + { + "bbox": [ + 154, + 157, + 457, + 192 + ], + "score": 0.93, + "content": "0 \\leq v ^ { \\star } - v ^ { \\widehat { \\pi } } \\leq \\widetilde { O } \\left( \\sqrt { d } \\cdot \\sum _ { h = 1 } ^ { H } \\mathbb { E } _ { \\pi ^ { \\star } } \\left[ \\left( \\phi ( \\cdot , \\cdot ) ^ { \\top } \\Lambda _ { h } ^ { \\star - 1 } \\phi ( \\cdot , \\cdot ) \\right) ^ { 1 / 2 } \\right] \\right) + \\frac { 2 H ^ { 4 } \\sqrt { d } } { K } .", + "type": "interline_equation", + "image_path": "f267a949e8c56d1b709e3869b520b7e1a83149d97bc5bed955e0d89e65dd1ed0.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 154, + 157, + 457, + 168.66666666666666 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 154, + 168.66666666666666, + 457, + 180.33333333333331 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 154, + 180.33333333333331, + 457, + 191.99999999999997 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "title", + "bbox": [ + 107, + 226, + 255, + 239 + ], + "lines": [ + { + "bbox": [ + 106, + 226, + 256, + 240 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 256, + 240 + ], + "score": 1.0, + "content": "D PROOF OF THEOREM 3.3", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 107, + 250, + 277, + 262 + ], + "lines": [ + { + "bbox": [ + 105, + 249, + 278, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 278, + 264 + ], + "score": 1.0, + "content": "First of all, we show the following lemma.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10, + "bbox_fs": [ + 105, + 249, + 278, + 264 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 264, + 356, + 277 + ], + "lines": [ + { + "bbox": [ + 105, + 262, + 357, + 280 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 199, + 280 + ], + "score": 1.0, + "content": "Lemma D.1. Suppose", + "type": "text" + }, + { + "bbox": [ + 199, + 264, + 330, + 277 + ], + "score": 0.91, + "content": "K > \\operatorname* { m a x } \\{ \\mathcal { M } _ { 1 } , \\mathcal { M } _ { 2 } , \\mathcal { M } _ { 3 } , \\mathcal { M } _ { 4 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 262, + 357, + 280 + ], + "score": 1.0, + "content": ". 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1 } } } \\\\ { \\displaystyle \\leq \\displaystyle \\frac { 2 H } { \\sqrt { \\kappa K } } \\cdot \\displaystyle \\left\\| \\sum _ { \\tau = 1 } ^ { K } \\phi \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\cdot \\left( \\left( \\widehat { \\gamma } _ { h } \\widehat V _ { h + 1 } \\right) \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) - \\left( \\widehat { \\gamma } _ { h } \\widehat V _ { h + 1 } \\right) \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\right) / \\widehat \\sigma _ { h } ^ { 2 } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) \\right\\| _ { \\widehat \\Lambda _ { h } ^ { - 1 } } } \\\\ \\displaystyle \\leq \\frac { 2 H } { \\sqrt { \\kappa K } } \\cdot \\displaystyle \\tilde { O } ( \\frac { H ^ { 2 } \\sqrt { d / \\kappa } } { \\sqrt { K } } ) \\cdot \\sqrt { d } = \\tilde { O } ( \\frac H ^ { 3 } d / \\kappa \\end{array}", + "type": "interline_equation", + "image_path": "33929071e051bb8174417bc7c0901551d0f698899c850c868627005f4cc0e3b6.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 109, + 97, + 499, + 144.66666666666666 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 109, + 144.66666666666666, + 499, + 192.33333333333331 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 109, + 192.33333333333331, + 499, + 239.99999999999997 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 239, + 506, + 284 + ], + "lines": [ + { + "bbox": [ + 105, + 237, + 505, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 302, + 252 + ], + "score": 1.0, + "content": "where the first inequality is by Lemma H.6 (with", + "type": "text" + }, + { + "bbox": [ + 302, + 239, + 347, + 252 + ], + "score": 0.91, + "content": "\\phi ^ { \\prime } = \\phi / \\widehat { \\sigma } _ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 237, + 365, + 252 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 366, + 239, + 471, + 252 + ], + "score": 0.92, + "content": "\\| \\phi / { \\widehat { \\sigma } } _ { h } \\| \\leq \\| \\phi \\| \\leq 1 : = C )", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 237, + 505, + 252 + ], + "score": 1.0, + "content": "and the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 251, + 508, + 267 + ], + "spans": [ + { + "bbox": [ + 106, + 251, + 190, + 267 + ], + "score": 1.0, + "content": "third inequality uses", + "type": "text" + }, + { + "bbox": [ + 190, + 251, + 394, + 266 + ], + "score": 0.89, + "content": "{ \\sqrt { a ^ { \\top } \\cdot A \\cdot a } } \\leq { \\sqrt { \\left\\| a \\right\\| _ { 2 } \\left\\| A \\right\\| _ { 2 } \\left\\| a \\right\\| _ { 2 } } } = \\left\\| a \\right\\| _ { 2 } { \\sqrt { \\left\\| A \\right\\| _ { 2 } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 251, + 415, + 267 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 415, + 255, + 421, + 263 + ], + "score": 0.78, + "content": "a", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 251, + 470, + 267 + ], + "score": 1.0, + "content": "to be either", + "type": "text" + }, + { + "bbox": [ + 470, + 253, + 478, + 264 + ], + "score": 0.85, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 251, + 489, + 267 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 489, + 255, + 502, + 264 + ], + "score": 0.85, + "content": "w _ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 251, + 508, + 267 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 264, + 449, + 285 + ], + "spans": [ + { + "bbox": [ + 104, + 264, + 150, + 285 + ], + "score": 1.0, + "content": "Moreover,", + "type": "text" + }, + { + "bbox": [ + 150, + 267, + 329, + 281 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\lambda _ { \\operatorname* { m i n } } ( \\widetilde { \\Lambda } _ { h } ^ { p } ) \\ge \\kappa / \\operatorname* { m a x } _ { h , s , a } \\widehat { \\sigma } _ { h } ( s , a ) ^ { 2 } \\ge \\kappa / H ^ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 264, + 362, + 285 + ], + "score": 1.0, + "content": "implies", + "type": "text" + }, + { + "bbox": [ + 363, + 265, + 443, + 285 + ], + "score": 0.9, + "content": "\\left\\| ( \\tilde { \\Lambda } _ { h } ^ { p } ) ^ { - 1 } \\right\\| \\leq H ^ { 2 } / \\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 264, + 449, + 285 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 107, + 289, + 399, + 302 + ], + "lines": [ + { + "bbox": [ + 106, + 288, + 399, + 304 + ], + "spans": [ + { + "bbox": [ + 106, + 288, + 273, + 304 + ], + "score": 1.0, + "content": "The second inequality is true by denoting", + "type": "text" + }, + { + "bbox": [ + 274, + 289, + 380, + 302 + ], + "score": 0.93, + "content": "x _ { \\tau } = \\phi ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) / \\widehat { \\sigma } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } )", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 288, + 399, + 304 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "interline_equation", + "bbox": [ + 177, + 304, + 433, + 326 + ], + "lines": [ + { + "bbox": [ + 177, + 304, + 433, + 326 + ], + "spans": [ + { + "bbox": [ + 177, + 304, + 433, + 326 + ], + "score": 0.9, + "content": "\\eta _ { \\tau } = \\left( \\left( \\widehat { T } _ { h } \\widehat { V } _ { h + 1 } \\right) ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) - \\left( T _ { h } \\widehat { V } _ { h + 1 } \\right) ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) \\right) / \\widehat { \\sigma } _ { h } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } )", + "type": "interline_equation", + "image_path": "f2e7f2135ccad198c5784031582a838b4ac23cd04f0bb387f4b207e3db1409a3.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 177, + 304, + 433, + 326 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 328, + 504, + 351 + ], + "lines": [ + { + "bbox": [ + 105, + 326, + 506, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 506, + 342 + ], + "score": 1.0, + "content": "and use Lemma H.10 as the condition for applying Lemma H.2. By collecting those three terms", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 339, + 219, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 219, + 351 + ], + "score": 1.0, + "content": "together we have the result.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5 + }, + { + "type": "title", + "bbox": [ + 107, + 363, + 239, + 375 + ], + "lines": [ + { + "bbox": [ + 105, + 362, + 240, + 377 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 240, + 377 + ], + "score": 1.0, + "content": "D.1 PROOF OF THEOREM 3.3", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 106, + 384, + 505, + 407 + ], + "lines": [ + { + "bbox": [ + 106, + 384, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 106, + 384, + 476, + 397 + ], + "score": 1.0, + "content": "Proof. Use Lemma D.1 as the condition for Lemma C.1 and average over initial distribution", + "type": "text" + }, + { + "bbox": [ + 477, + 385, + 487, + 396 + ], + "score": 0.88, + "content": "d _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 384, + 505, + 397 + ], + "score": 1.0, + "content": ", we", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 395, + 228, + 408 + ], + "spans": [ + { + "bbox": [ + 106, + 395, + 200, + 408 + ], + "score": 1.0, + "content": "obtain with probability", + "type": "text" + }, + { + "bbox": [ + 201, + 396, + 223, + 406 + ], + "score": 0.88, + "content": "1 - \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 395, + 228, + 408 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5 + }, + { + "type": "interline_equation", + "bbox": [ + 106, + 410, + 154, + 425 + ], + "lines": [], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 106, + 410, + 154, + 425 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "interline_equation", + "bbox": [ + 105, + 414, + 500, + 467 + ], + "lines": [ + { + "bbox": [ + 105, + 414, + 500, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 500, + 467 + ], + "score": 0.83, + "content": "\\begin{array} { l } { v ^ { \\pi } - v ^ { \\pi } \\leq } \\\\ { \\displaystyle \\sum _ { h = 1 } ^ { H } \\mathbb { E } _ { \\pi _ { h } } \\left[ \\phi ( s , a ) \\right] ^ { \\top } \\left| \\widehat { \\Lambda } _ { h } ^ { - 1 } \\displaystyle \\sum _ { \\tau = 1 } ^ { K } \\frac { \\phi \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\cdot \\left( r _ { h } ^ { \\tau } + \\widehat { V } _ { h + 1 } \\left( s _ { h + 1 } ^ { \\tau } \\right) - \\left( \\widehat { \\mathcal { T } } _ { h } \\widehat { V } _ { h + 1 } \\right) \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\right) } { \\widehat { \\sigma } _ { h } ^ { 2 } \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) } \\right| + \\widetilde { O } ( \\frac { H ^ { 4 } d / } { K } ) } \\end{array}", + "type": "interline_equation", + "image_path": "3a0d45f810b808550b45aaab8a91ed8a37d01e24874f8dd71bd4d1f5e5db04cc.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 105, + 414, + 500, + 431.6666666666667 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 105, + 431.6666666666667, + 500, + 449.33333333333337 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 105, + 449.33333333333337, + 500, + 467.00000000000006 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 475, + 386, + 497 + ], + "lines": [ + { + "bbox": [ + 100, + 474, + 361, + 500 + ], + "spans": [ + { + "bbox": [ + 100, + 474, + 138, + 500 + ], + "score": 1.0, + "content": "Denote", + "type": "text" + }, + { + "bbox": [ + 138, + 475, + 361, + 497 + ], + "score": 0.94, + "content": "\\begin{array} { r } { A _ { h } : = \\sum _ { \\tau = 1 } ^ { K } \\frac { \\phi ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) \\cdot \\left( r _ { h } ^ { \\tau } + \\widehat { V } _ { h + 1 } \\left( s _ { h + 1 } ^ { \\tau } \\right) - \\left( \\tau _ { h } \\widehat { V } _ { h + 1 } \\right) \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\right) } { \\widehat { \\sigma } _ { h } ^ { 2 } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) } } \\end{array}", + "type": "inline_equation" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "interline_equation", + "bbox": [ + 112, + 500, + 500, + 583 + ], + "lines": [ + { + "bbox": [ + 112, + 500, + 500, + 583 + ], + "spans": [ + { + "bbox": [ + 112, + 500, + 500, + 583 + ], + "score": 0.91, + "content": "\\begin{array} { r l } & { \\mathbb { E } _ { \\pi _ { h } } \\left[ \\phi ( s , a ) \\right] ^ { \\top } \\left| \\widehat { \\Lambda } _ { h } ^ { - 1 } \\displaystyle \\sum _ { \\tau = 1 } ^ { K } \\frac { \\phi \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\cdot \\left( r _ { h } ^ { \\tau } + \\widehat { V } _ { h + 1 } \\left( s _ { h + 1 } ^ { \\tau } \\right) - \\left( \\widehat { T } _ { h } \\widehat { V } _ { h + 1 } \\right) \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\right) } { \\widehat { \\sigma } _ { h } ^ { 2 } \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) } \\right| } \\\\ & { \\leq \\mathbb { E } _ { \\pi _ { h } } \\left[ \\phi \\right] ^ { \\top } \\cdot \\left| \\widehat { \\Lambda } _ { h } ^ { - 1 } A _ { h } \\right| + \\mathbb { E } _ { \\pi _ { h } } \\left[ \\phi \\right] ^ { \\top } \\left| \\widehat { \\Lambda } _ { h } ^ { - 1 } \\displaystyle \\sum _ { \\tau = 1 } ^ { K } \\frac { \\phi \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\cdot \\left( 7 _ { h } \\widehat { V } _ { h + 1 } \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) - \\widehat { T } _ { h } \\widehat { V } _ { h + 1 } \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\right) } { \\widehat { \\sigma } _ { h } ^ { 2 } \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) } \\right| } \\end{array}", + "type": "interline_equation", + "image_path": "1c1b5aa50ce8df94b8efcae8486f1d57b0c588a509c4625690cc765bc6fbc3b9.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 112, + 500, + 500, + 527.6666666666666 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 112, + 527.6666666666666, + 500, + 555.3333333333333 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 112, + 555.3333333333333, + 500, + 582.9999999999999 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 598, + 504, + 622 + ], + "lines": [ + { + "bbox": [ + 105, + 597, + 506, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 506, + 611 + ], + "score": 1.0, + "content": "For the second term, it can be bounded similar to term (iii) in Lemma D.1 and for the first term we", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 608, + 186, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 186, + 623 + ], + "score": 1.0, + "content": "have the following:", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5 + }, + { + "type": "interline_equation", + "bbox": [ + 123, + 623, + 490, + 669 + ], + "lines": [ + { + "bbox": [ + 123, + 623, + 490, + 669 + ], + "spans": [ + { + "bbox": [ + 123, + 623, + 490, + 669 + ], + "score": 0.87, + "content": "\\begin{array} { r l } & { \\quad \\mathbb E _ { \\pi _ { h } } \\left[ \\phi \\right] ^ { \\top } \\cdot \\left| \\widehat \\Lambda _ { h } ^ { - 1 } A _ { h } \\right| = \\mathbb E _ { \\pi _ { h } } \\left[ \\phi \\right] ^ { \\top } \\cdot \\widehat \\Lambda _ { h } ^ { - 1 } \\cdot \\widehat \\Lambda _ { h } \\left| \\widehat \\Lambda _ { h } ^ { - 1 } A _ { h } \\right| \\leq \\| \\mathbb E _ { \\pi _ { h } } \\left[ \\phi \\right] \\| _ { \\widehat \\Lambda _ { h } ^ { - 1 } } \\cdot \\left\\| \\widehat \\Lambda _ { h } | \\widehat \\Lambda _ { h } ^ { - 1 } A _ { h } \\right| \\Big \\| _ { \\widehat \\Lambda _ { h } ^ { - 1 } } } \\\\ & { \\leq \\| \\mathbb E _ { \\pi _ { h } } [ \\phi ] \\| _ { \\widehat \\Lambda _ { h } ^ { - 1 } } \\cdot \\| | A _ { h } | \\| _ { \\widehat \\Lambda _ { h } ^ { - 1 } } \\leq \\widetilde O ( \\sqrt d ) \\| \\mathbb E _ { \\pi _ { h } } [ \\phi ] \\| _ { \\widehat \\Lambda _ { h } ^ { - 1 } } \\leq \\widetilde O ( \\sqrt d \\| \\mathbb E _ { \\pi _ { h } } [ \\phi ] \\| _ { \\Lambda _ { h } ^ { - 1 } } ) , } \\end{array}", + "type": "interline_equation", + "image_path": "51c2f29c3c26597a16833d6cb7a516babc4504603a4b0f58ce219e96b5ecd7c1.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 123, + 623, + 490, + 638.3333333333334 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 123, + 638.3333333333334, + 490, + 653.6666666666667 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 123, + 653.6666666666667, + 490, + 669.0000000000001 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 673, + 506, + 733 + ], + "lines": [ + { + "bbox": [ + 105, + 671, + 506, + 686 + ], + "spans": [ + { + "bbox": [ + 105, + 673, + 415, + 686 + ], + "score": 1.0, + "content": "where the first inequality uses Cauchy’s inequality, the second inequality uses", + "type": "text" + }, + { + "bbox": [ + 416, + 671, + 429, + 685 + ], + "score": 0.9, + "content": "\\widehat { \\Lambda } _ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 673, + 506, + 686 + ], + "score": 1.0, + "content": "is coordinate-wise", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 684, + 506, + 697 + ], + "spans": [ + { + "bbox": [ + 104, + 684, + 228, + 697 + ], + "score": 1.0, + "content": "positive (since we assume here", + "type": "text" + }, + { + "bbox": [ + 229, + 685, + 254, + 696 + ], + "score": 0.89, + "content": "\\phi \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 684, + 506, + 697 + ], + "score": 1.0, + "content": "), the third inequality is identical to the analysis in Section C.2.2", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 694, + 506, + 709 + ], + "spans": [ + { + "bbox": [ + 104, + 694, + 402, + 709 + ], + "score": 1.0, + "content": "and the fourth inequality is identical to the analysis in Section C.2.1 with", + "type": "text" + }, + { + "bbox": [ + 403, + 696, + 410, + 707 + ], + "score": 0.85, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 694, + 460, + 709 + ], + "score": 1.0, + "content": "replaced by", + "type": "text" + }, + { + "bbox": [ + 461, + 696, + 480, + 707 + ], + "score": 0.89, + "content": "\\mathbb { E } [ \\phi ]", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 694, + 506, + 709 + ], + "score": 1.0, + "content": ". Plug", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 103, + 705, + 504, + 723 + ], + "spans": [ + { + "bbox": [ + 103, + 705, + 475, + 723 + ], + "score": 1.0, + "content": "this back to equation 22 we finish the proof for the first part. For the second part, converting", + "type": "text" + }, + { + "bbox": [ + 475, + 707, + 493, + 720 + ], + "score": 0.92, + "content": "\\Lambda _ { h } ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 705, + 504, + 723 + ], + "score": 1.0, + "content": "to", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 718, + 504, + 735 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 128, + 733 + ], + "score": 0.92, + "content": "\\Lambda _ { h } ^ { \\star - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 718, + 334, + 735 + ], + "score": 1.0, + "content": "is identical to Section C.5. This finishes the proof.", + "type": "text" + }, + { + "bbox": [ + 496, + 722, + 504, + 729 + ], + "score": 0.972, + "content": "□", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29 + } + ], + "page_idx": 28, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 293, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 765 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 765 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 15 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 251, + 95 + ], + "lines": [ + { + "bbox": [ + 105, + 79, + 250, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 250, + 97 + ], + "score": 1.0, + "content": "For term (iii), by Cauchy inequality", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 105, + 79, + 250, + 97 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 109, + 97, + 499, + 240 + ], + "lines": [ + { + "bbox": [ + 109, + 97, + 499, + 240 + ], + "spans": [ + { + "bbox": [ + 109, + 97, + 499, + 240 + ], + "score": 0.89, + "content": "\\begin{array} { l } { \\displaystyle \\phi ( s , a ) ^ { \\top } \\widehat \\Lambda _ { h } ^ { - 1 } \\left( \\displaystyle \\sum _ { \\tau = 1 } ^ { K } \\phi \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\cdot \\left( \\left( \\widehat { \\gamma } _ { h } \\widehat V _ { h + 1 } \\right) \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) - \\left( \\widehat { \\gamma } _ { h } \\widehat V _ { h + 1 } \\right) \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\right) / \\widehat \\sigma _ { h } ^ { 2 } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) \\right) } \\\\ { \\displaystyle \\leq \\left\\| \\phi ( s , a ) \\right\\| _ { \\widehat \\Lambda _ { h } ^ { - 1 } } \\cdot \\displaystyle \\left\\| \\sum _ { \\tau = 1 } ^ { K } \\phi \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\cdot \\left( \\left( \\widehat { \\gamma } _ { h } \\widehat V _ { h + 1 } \\right) \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) - \\left( \\widehat { \\gamma } _ { h } \\widehat V _ { h + 1 } \\right) \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\right) / \\widehat \\sigma _ { h } ^ { 2 } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) \\right\\| _ { \\widehat \\Lambda _ { h } ^ { - 1 } } } \\\\ { \\displaystyle \\leq \\displaystyle \\frac { 2 H } { \\sqrt { \\kappa K } } \\cdot \\displaystyle \\left\\| \\sum _ { \\tau = 1 } ^ { K } \\phi \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\cdot \\left( \\left( \\widehat { \\gamma } _ { h } \\widehat V _ { h + 1 } \\right) \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) - \\left( \\widehat { \\gamma } _ { h } \\widehat V _ { h + 1 } \\right) \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\right) / \\widehat \\sigma _ { h } ^ { 2 } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) \\right\\| _ { \\widehat \\Lambda _ { h } ^ { - 1 } } } \\\\ \\displaystyle \\leq \\frac { 2 H } { \\sqrt { \\kappa K } } \\cdot \\displaystyle \\tilde { O } ( \\frac { H ^ { 2 } \\sqrt { d / \\kappa } } { \\sqrt { K } } ) \\cdot \\sqrt { d } = \\tilde { O } ( \\frac H ^ { 3 } d / \\kappa \\end{array}", + "type": "interline_equation", + "image_path": "33929071e051bb8174417bc7c0901551d0f698899c850c868627005f4cc0e3b6.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 109, + 97, + 499, + 144.66666666666666 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 109, + 144.66666666666666, + 499, + 192.33333333333331 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 109, + 192.33333333333331, + 499, + 239.99999999999997 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 239, + 506, + 284 + ], + "lines": [ + { + "bbox": [ + 105, + 237, + 505, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 302, + 252 + ], + "score": 1.0, + "content": "where the first inequality is by Lemma H.6 (with", + "type": "text" + }, + { + "bbox": [ + 302, + 239, + 347, + 252 + ], + "score": 0.91, + "content": "\\phi ^ { \\prime } = \\phi / \\widehat { \\sigma } _ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 237, + 365, + 252 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 366, + 239, + 471, + 252 + ], + "score": 0.92, + "content": "\\| \\phi / { \\widehat { \\sigma } } _ { h } \\| \\leq \\| \\phi \\| \\leq 1 : = C )", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 237, + 505, + 252 + ], + "score": 1.0, + "content": "and the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 251, + 508, + 267 + ], + "spans": [ + { + "bbox": [ + 106, + 251, + 190, + 267 + ], + "score": 1.0, + "content": "third inequality uses", + "type": "text" + }, + { + "bbox": [ + 190, + 251, + 394, + 266 + ], + "score": 0.89, + "content": "{ \\sqrt { a ^ { \\top } \\cdot A \\cdot a } } \\leq { \\sqrt { \\left\\| a \\right\\| _ { 2 } \\left\\| A \\right\\| _ { 2 } \\left\\| a \\right\\| _ { 2 } } } = \\left\\| a \\right\\| _ { 2 } { \\sqrt { \\left\\| A \\right\\| _ { 2 } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 251, + 415, + 267 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 415, + 255, + 421, + 263 + ], + "score": 0.78, + "content": "a", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 251, + 470, + 267 + ], + "score": 1.0, + "content": "to be either", + "type": "text" + }, + { + "bbox": [ + 470, + 253, + 478, + 264 + ], + "score": 0.85, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 251, + 489, + 267 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 489, + 255, + 502, + 264 + ], + "score": 0.85, + "content": "w _ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 251, + 508, + 267 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 264, + 449, + 285 + ], + "spans": [ + { + "bbox": [ + 104, + 264, + 150, + 285 + ], + "score": 1.0, + "content": "Moreover,", + "type": "text" + }, + { + "bbox": [ + 150, + 267, + 329, + 281 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\lambda _ { \\operatorname* { m i n } } ( \\widetilde { \\Lambda } _ { h } ^ { p } ) \\ge \\kappa / \\operatorname* { m a x } _ { h , s , a } \\widehat { \\sigma } _ { h } ( s , a ) ^ { 2 } \\ge \\kappa / H ^ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 264, + 362, + 285 + ], + "score": 1.0, + "content": "implies", + "type": "text" + }, + { + "bbox": [ + 363, + 265, + 443, + 285 + ], + "score": 0.9, + "content": "\\left\\| ( \\tilde { \\Lambda } _ { h } ^ { p } ) ^ { - 1 } \\right\\| \\leq H ^ { 2 } / \\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 264, + 449, + 285 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5, + "bbox_fs": [ + 104, + 237, + 508, + 285 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 289, + 399, + 302 + ], + "lines": [ + { + "bbox": [ + 106, + 288, + 399, + 304 + ], + "spans": [ + { + "bbox": [ + 106, + 288, + 273, + 304 + ], + "score": 1.0, + "content": "The second inequality is true by denoting", + "type": "text" + }, + { + "bbox": [ + 274, + 289, + 380, + 302 + ], + "score": 0.93, + "content": "x _ { \\tau } = \\phi ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) / \\widehat { \\sigma } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } )", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 288, + 399, + 304 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7, + "bbox_fs": [ + 106, + 288, + 399, + 304 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 177, + 304, + 433, + 326 + ], + "lines": [ + { + "bbox": [ + 177, + 304, + 433, + 326 + ], + "spans": [ + { + "bbox": [ + 177, + 304, + 433, + 326 + ], + "score": 0.9, + "content": "\\eta _ { \\tau } = \\left( \\left( \\widehat { T } _ { h } \\widehat { V } _ { h + 1 } \\right) ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) - \\left( T _ { h } \\widehat { V } _ { h + 1 } \\right) ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) \\right) / \\widehat { \\sigma } _ { h } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } )", + "type": "interline_equation", + "image_path": "f2e7f2135ccad198c5784031582a838b4ac23cd04f0bb387f4b207e3db1409a3.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 177, + 304, + 433, + 326 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 328, + 504, + 351 + ], + "lines": [ + { + "bbox": [ + 105, + 326, + 506, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 506, + 342 + ], + "score": 1.0, + "content": "and use Lemma H.10 as the condition for applying Lemma H.2. By collecting those three terms", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 339, + 219, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 219, + 351 + ], + "score": 1.0, + "content": "together we have the result.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5, + "bbox_fs": [ + 105, + 326, + 506, + 351 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 363, + 239, + 375 + ], + "lines": [ + { + "bbox": [ + 105, + 362, + 240, + 377 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 240, + 377 + ], + "score": 1.0, + "content": "D.1 PROOF OF THEOREM 3.3", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 106, + 384, + 505, + 407 + ], + "lines": [ + { + "bbox": [ + 106, + 384, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 106, + 384, + 476, + 397 + ], + "score": 1.0, + "content": "Proof. Use Lemma D.1 as the condition for Lemma C.1 and average over initial distribution", + "type": "text" + }, + { + "bbox": [ + 477, + 385, + 487, + 396 + ], + "score": 0.88, + "content": "d _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 384, + 505, + 397 + ], + "score": 1.0, + "content": ", we", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 395, + 228, + 408 + ], + "spans": [ + { + "bbox": [ + 106, + 395, + 200, + 408 + ], + "score": 1.0, + "content": "obtain with probability", + "type": "text" + }, + { + "bbox": [ + 201, + 396, + 223, + 406 + ], + "score": 0.88, + "content": "1 - \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 395, + 228, + 408 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5, + "bbox_fs": [ + 106, + 384, + 505, + 408 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 106, + 410, + 154, + 425 + ], + "lines": [], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 106, + 410, + 154, + 425 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "interline_equation", + "bbox": [ + 105, + 414, + 500, + 467 + ], + "lines": [ + { + "bbox": [ + 105, + 414, + 500, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 500, + 467 + ], + "score": 0.83, + "content": "\\begin{array} { l } { v ^ { \\pi } - v ^ { \\pi } \\leq } \\\\ { \\displaystyle \\sum _ { h = 1 } ^ { H } \\mathbb { E } _ { \\pi _ { h } } \\left[ \\phi ( s , a ) \\right] ^ { \\top } \\left| \\widehat { \\Lambda } _ { h } ^ { - 1 } \\displaystyle \\sum _ { \\tau = 1 } ^ { K } \\frac { \\phi \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\cdot \\left( r _ { h } ^ { \\tau } + \\widehat { V } _ { h + 1 } \\left( s _ { h + 1 } ^ { \\tau } \\right) - \\left( \\widehat { \\mathcal { T } } _ { h } \\widehat { V } _ { h + 1 } \\right) \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\right) } { \\widehat { \\sigma } _ { h } ^ { 2 } \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) } \\right| + \\widetilde { O } ( \\frac { H ^ { 4 } d / } { K } ) } \\end{array}", + "type": "interline_equation", + "image_path": "3a0d45f810b808550b45aaab8a91ed8a37d01e24874f8dd71bd4d1f5e5db04cc.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 105, + 414, + 500, + 431.6666666666667 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 105, + 431.6666666666667, + 500, + 449.33333333333337 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 105, + 449.33333333333337, + 500, + 467.00000000000006 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 475, + 386, + 497 + ], + "lines": [ + { + "bbox": [ + 100, + 474, + 361, + 500 + ], + "spans": [ + { + "bbox": [ + 100, + 474, + 138, + 500 + ], + "score": 1.0, + "content": "Denote", + "type": "text" + }, + { + "bbox": [ + 138, + 475, + 361, + 497 + ], + "score": 0.94, + "content": "\\begin{array} { r } { A _ { h } : = \\sum _ { \\tau = 1 } ^ { K } \\frac { \\phi ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) \\cdot \\left( r _ { h } ^ { \\tau } + \\widehat { V } _ { h + 1 } \\left( s _ { h + 1 } ^ { \\tau } \\right) - 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1 } A _ { h } \\right| + \\mathbb { E } _ { \\pi _ { h } } \\left[ \\phi \\right] ^ { \\top } \\left| \\widehat { \\Lambda } _ { h } ^ { - 1 } \\displaystyle \\sum _ { \\tau = 1 } ^ { K } \\frac { \\phi \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\cdot \\left( 7 _ { h } \\widehat { V } _ { h + 1 } \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) - \\widehat { T } _ { h } \\widehat { V } _ { h + 1 } \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\right) } { \\widehat { \\sigma } _ { h } ^ { 2 } \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) } \\right| } \\end{array}", + "type": "interline_equation", + "image_path": "1c1b5aa50ce8df94b8efcae8486f1d57b0c588a509c4625690cc765bc6fbc3b9.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 112, + 500, + 500, + 527.6666666666666 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 112, + 527.6666666666666, + 500, + 555.3333333333333 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 112, + 555.3333333333333, + 500, + 582.9999999999999 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 598, + 504, + 622 + ], + "lines": [ + { + "bbox": [ + 105, + 597, + 506, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 506, + 611 + ], + "score": 1.0, + "content": "For the second term, it can be bounded similar to term (iii) in Lemma D.1 and for the first term we", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 608, + 186, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 186, + 623 + ], + "score": 1.0, + "content": "have the following:", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 597, + 506, + 623 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 123, + 623, + 490, + 669 + ], + "lines": [ + { + "bbox": [ + 123, + 623, + 490, + 669 + ], + "spans": [ + { + "bbox": [ + 123, + 623, + 490, + 669 + ], + "score": 0.87, + "content": "\\begin{array} { r l } & { \\quad \\mathbb E _ { \\pi _ { h } } \\left[ \\phi \\right] ^ { \\top } \\cdot \\left| \\widehat \\Lambda _ { h } ^ { - 1 } A _ { h } \\right| = \\mathbb E _ { \\pi _ { h } } \\left[ \\phi \\right] ^ { \\top } \\cdot \\widehat \\Lambda _ { h } ^ { - 1 } \\cdot \\widehat \\Lambda _ { h } \\left| \\widehat \\Lambda _ { h } ^ { - 1 } A _ { h } \\right| \\leq \\| \\mathbb E _ { \\pi _ { h } } \\left[ \\phi \\right] \\| _ { \\widehat \\Lambda _ { h } ^ { - 1 } } \\cdot \\left\\| \\widehat \\Lambda _ { h } | \\widehat \\Lambda _ { h } ^ { - 1 } A _ { h } \\right| \\Big \\| _ { \\widehat \\Lambda _ { h } ^ { - 1 } } } \\\\ & { \\leq \\| \\mathbb E _ { \\pi _ { h } } [ \\phi ] \\| _ { \\widehat \\Lambda _ { h } ^ { - 1 } } \\cdot \\| | A _ { h } | \\| _ { \\widehat \\Lambda _ { h } ^ { - 1 } } \\leq \\widetilde O ( \\sqrt d ) \\| \\mathbb E _ { \\pi _ { h } } [ \\phi ] \\| _ { \\widehat \\Lambda _ { h } ^ { - 1 } } \\leq \\widetilde O ( \\sqrt d \\| \\mathbb E _ { \\pi _ { h } } [ \\phi ] \\| _ { \\Lambda _ { h } ^ { - 1 } } ) , } \\end{array}", + "type": "interline_equation", + "image_path": "51c2f29c3c26597a16833d6cb7a516babc4504603a4b0f58ce219e96b5ecd7c1.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 123, + 623, + 490, + 638.3333333333334 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 123, + 638.3333333333334, + 490, + 653.6666666666667 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 123, + 653.6666666666667, + 490, + 669.0000000000001 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 673, + 506, + 733 + ], + "lines": [ + { + "bbox": [ + 105, + 671, + 506, + 686 + ], + "spans": [ + { + "bbox": [ + 105, + 673, + 415, + 686 + ], + "score": 1.0, + "content": "where the first inequality uses Cauchy’s inequality, the second inequality uses", + "type": "text" + }, + { + "bbox": [ + 416, + 671, + 429, + 685 + ], + "score": 0.9, + "content": "\\widehat { \\Lambda } _ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 673, + 506, + 686 + ], + "score": 1.0, + "content": "is coordinate-wise", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 684, + 506, + 697 + ], + "spans": [ + { + "bbox": [ + 104, + 684, + 228, + 697 + ], + "score": 1.0, + "content": "positive (since we assume here", + "type": "text" + }, + { + "bbox": [ + 229, + 685, + 254, + 696 + ], + "score": 0.89, + "content": "\\phi \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 684, + 506, + 697 + ], + "score": 1.0, + "content": "), the third inequality is identical to the analysis in Section C.2.2", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 694, + 506, + 709 + ], + "spans": [ + { + "bbox": [ + 104, + 694, + 402, + 709 + ], + "score": 1.0, + "content": "and the fourth inequality is identical to the analysis in Section C.2.1 with", + "type": "text" + }, + { + "bbox": [ + 403, + 696, + 410, + 707 + ], + "score": 0.85, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 694, + 460, + 709 + ], + "score": 1.0, + "content": "replaced by", + "type": "text" + }, + { + "bbox": [ + 461, + 696, + 480, + 707 + ], + "score": 0.89, + "content": "\\mathbb { E } [ \\phi ]", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 694, + 506, + 709 + ], + "score": 1.0, + "content": ". 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This finishes the proof.", + "type": "text" + }, + { + "bbox": [ + 496, + 722, + 504, + 729 + ], + "score": 0.972, + "content": "□", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29, + "bbox_fs": [ + 103, + 671, + 506, + 735 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 106, + 81, + 388, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 389, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 389, + 95 + ], + "score": 1.0, + "content": "E PROOF OF MINIMAX LOWER BOUND THEOREM 3.5", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 105, + 504, + 128 + ], + "lines": [ + { + "bbox": [ + 106, + 106, + 504, + 119 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 504, + 119 + ], + "score": 1.0, + "content": "The proof follows the lower bound proof of Zanette et al. (2021). 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The mean reward function satisfies", + "type": "text" + }, + { + "bbox": [ + 309, + 531, + 420, + 544 + ], + "score": 0.91, + "content": "r _ { u , h } ( s , a ) = \\langle \\phi ( s , a ) , \\theta _ { u , h } \\rangle", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 531, + 442, + 544 + ], + "score": 1.0, + "content": "with", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5, + "bbox_fs": [ + 140, + 510, + 505, + 544 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 274, + 549, + 372, + 591 + ], + "lines": [ + { + "bbox": [ + 274, + 549, + 372, + 591 + ], + "spans": [ + { + "bbox": [ + 274, + 549, + 372, + 591 + ], + "score": 0.94, + "content": "\\theta _ { u , h } = \\left( \\begin{array} { c } { \\delta u _ { h } } \\\\ { \\frac { 1 } { \\sqrt { 3 } } } \\\\ { - \\frac { 1 } { \\sqrt { 3 } } } \\end{array} \\right) \\in \\mathbb { R } ^ { d } .", + "type": "interline_equation", + "image_path": "387b539601f534a0926c91e4b507cc96d4d34cc1274e0146ca9cfacadda5d3e3.jpg" + } + ] + } + ], + "index": 30.5, + "virtual_lines": [ + { + "bbox": [ + 274, + 549, + 372, + 570.0 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 274, + 570.0, + 372, + 591.0 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 598, + 506, + 648 + ], + "lines": [ + { + "bbox": [ + 103, + 596, + 507, + 617 + ], + "spans": [ + { + "bbox": [ + 103, + 596, + 429, + 617 + ], + "score": 1.0, + "content": "Offline data collection Scheme: The dataset D = {(sτh, aτh, rτh, sτh+1)}h∈[H]τ∈[K]", + "type": "text" + }, + { + "bbox": [ + 427, + 596, + 473, + 612 + ], + "score": 1.0, + "content": "consist of", + "type": "text" + }, + { + "bbox": [ + 473, + 601, + 484, + 611 + ], + "score": 0.82, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 596, + 507, + 612 + ], + "score": 1.0, + "content": "i.i.d.", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 141, + 613, + 506, + 627 + ], + "spans": [ + { + "bbox": [ + 141, + 613, + 506, + 627 + ], + "score": 1.0, + "content": "trajectories. 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The behavior policy", + "type": "text" + }, + { + "bbox": [ + 399, + 638, + 407, + 648 + ], + "score": 0.82, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 633, + 443, + 650 + ], + "score": 1.0, + "content": "is set as", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33.5, + "bbox_fs": [ + 103, + 596, + 507, + 650 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 189, + 651, + 457, + 667 + ], + "lines": [ + { + "bbox": [ + 189, + 651, + 457, + 667 + ], + "spans": [ + { + "bbox": [ + 189, + 651, + 457, + 667 + ], + "score": 0.88, + "content": "\\begin{array} { r } { \\mu ( e _ { j } \\mid s ) = \\frac { 1 } { d } \\quad \\mathrm { f o r ~ a n y ~ } j \\in [ d - 2 ] \\qquad \\mathrm { a n d } \\qquad \\mu ( \\mathbf { 0 } _ { d - 2 } \\mid s ) = \\frac { 2 } { d } . } \\end{array}", + "type": "interline_equation", + "image_path": "5ade7d5551401bacb16a2df6386c1dee7e5afc577aa295846707b3df45a7b9b2.jpg" + } + ] + } + ], + "index": 36, + "virtual_lines": [ + { + "bbox": [ + 189, + 651, + 457, + 667 + ], + "spans": [], + "index": 36 + } + ] + }, + { + "type": "title", + "bbox": [ + 107, + 678, + 224, + 689 + ], + "lines": [ + { + "bbox": [ + 105, + 678, + 225, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 225, + 691 + ], + "score": 1.0, + "content": "E.2 OVERVIEW OF PROOF", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 698, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 697, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 697, + 506, + 712 + ], + "score": 1.0, + "content": "The proof of the theorem is based on Assouad’s method, where we first reduce the problem to binary", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "hypothesis tests (Lemmas E.1 and E.2) and then connect the testing error to the uncertainty quantity", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 721, + 241, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 241, + 733 + ], + "score": 1.0, + "content": "in the upper bound (Lemma E.3).", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39, + "bbox_fs": [ + 105, + 697, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 459, + 95 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 459, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 385, + 95 + ], + "score": 1.0, + "content": "Lemma E.1 (Reduction to testing). There exists a universal constant", + "type": "text" + }, + { + "bbox": [ + 386, + 83, + 414, + 93 + ], + "score": 0.9, + "content": "c _ { 1 } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 81, + 459, + 95 + ], + "score": 1.0, + "content": "such thata", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 119, + 100, + 473, + 122 + ], + "lines": [ + { + "bbox": [ + 119, + 100, + 473, + 122 + ], + "spans": [ + { + "bbox": [ + 119, + 100, + 473, + 122 + ], + "score": 0.93, + "content": "\\operatorname* { i n f } _ { \\hat { \\pi } } \\operatorname* { m a x } _ { u \\in \\mathcal { U } } \\mathbb { E } _ { u } \\left[ V _ { u } ^ { \\star } - V _ { u } ^ { \\widehat { \\pi } } \\right] \\geq c _ { 1 } \\delta \\sqrt { d } H \\operatorname* { m i n } _ { u , u ^ { \\prime } \\in \\mathcal { U } : D _ { H } ( u ^ { \\prime } ; u ) = 1 } \\operatorname* { i n f } _ { \\psi } \\big [ \\mathbb { P } _ { u } ( \\psi \\neq u ) + \\mathbb { P } _ { u ^ { \\prime } } ( \\psi \\neq u ^ { \\prime } ) \\big ] ,", + "type": "interline_equation", + "image_path": "2df7381d6c3389c357588db2de8e89b9f36e13434fd9e80ac39e43fdb9b7e874.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 119, + 100, + 473, + 122 + ], + "spans": [], + "index": 1 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 129, + 505, + 152 + ], + "lines": [ + { + "bbox": [ + 106, + 129, + 505, + 142 + ], + "spans": [ + { + "bbox": [ + 106, + 129, + 132, + 142 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 132, + 130, + 140, + 140 + ], + "score": 0.61, + "content": "\\widehat { \\pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 129, + 487, + 142 + ], + "score": 1.0, + "content": "denotes the output of any algorithm that maps from observations to an estimated policy.", + "type": "text" + }, + { + "bbox": [ + 487, + 131, + 495, + 141 + ], + "score": 0.84, + "content": "\\psi", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 129, + 505, + 142 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 141, + 378, + 154 + ], + "spans": [ + { + "bbox": [ + 106, + 141, + 232, + 154 + ], + "score": 1.0, + "content": "bany test function for parameter", + "type": "text" + }, + { + "bbox": [ + 233, + 143, + 240, + 150 + ], + "score": 0.52, + "content": "u", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 141, + 258, + 154 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 259, + 141, + 275, + 152 + ], + "score": 0.89, + "content": "D _ { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 141, + 378, + 154 + ], + "score": 1.0, + "content": "is the hamming distance.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 105, + 156, + 492, + 171 + ], + "lines": [ + { + "bbox": [ + 101, + 150, + 453, + 178 + ], + "spans": [ + { + "bbox": [ + 101, + 150, + 296, + 178 + ], + "score": 1.0, + "content": "Lemma E.2. There exists a universal constant", + "type": "text" + }, + { + "bbox": [ + 296, + 158, + 325, + 168 + ], + "score": 0.91, + "content": "c _ { 2 } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 150, + 415, + 178 + ], + "score": 1.0, + "content": "such that when taking", + "type": "text" + }, + { + "bbox": [ + 416, + 156, + 453, + 172 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\delta : = \\frac { c _ { 2 } d } { \\sqrt { K } } } \\end{array}", + "type": "inline_equation" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "interline_equation", + "bbox": [ + 190, + 178, + 420, + 204 + ], + "lines": [ + { + "bbox": [ + 190, + 178, + 420, + 204 + ], + "spans": [ + { + "bbox": [ + 190, + 178, + 420, + 204 + ], + "score": 0.91, + "content": "\\operatorname* { m i n } _ { u , u ^ { \\prime } \\in \\mathcal { U } : D _ { H } ( u ^ { \\prime } ; u ) = 1 } \\operatorname* { i n f } _ { \\psi } \\left[ \\mathbb { P } _ { u } ( \\psi \\neq u ) + \\mathbb { P } _ { u ^ { \\prime } } ( \\psi \\neq u ^ { \\prime } ) \\right] \\geq \\frac { 1 } { 2 } .", + "type": "interline_equation", + "image_path": "1626d6434a77e1532534edfdb818bab2f7aa1e26a910bdd43b01a5bda53846b9.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 190, + 178, + 420, + 204 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 217, + 506, + 243 + ], + "lines": [ + { + "bbox": [ + 105, + 217, + 506, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 132, + 232 + ], + "score": 1.0, + "content": "When", + "type": "text" + }, + { + "bbox": [ + 133, + 217, + 208, + 234 + ], + "score": 0.77, + "content": "\\begin{array} { r } { K \\gtrsim d ^ { 3 } , \\delta : = \\frac { c _ { 2 } d } { \\sqrt { K } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 217, + 259, + 232 + ], + "score": 1.0, + "content": "ensures that", + "type": "text" + }, + { + "bbox": [ + 259, + 217, + 308, + 231 + ], + "score": 0.92, + "content": "\\delta \\leq 1 / \\sqrt { 3 d }", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 217, + 506, + 232 + ], + "score": 1.0, + "content": ". 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There exists a universal constant", + "type": "text" + }, + { + "bbox": [ + 297, + 323, + 325, + 334 + ], + "score": 0.91, + "content": "c _ { 3 } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 321, + 391, + 335 + ], + "score": 1.0, + "content": "such that for all", + "type": "text" + }, + { + "bbox": [ + 392, + 322, + 427, + 333 + ], + "score": 0.88, + "content": "M \\in \\mathcal { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 321, + 431, + 335 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "interline_equation", + "bbox": [ + 219, + 340, + 390, + 375 + ], + "lines": [ + { + "bbox": [ + 219, + 340, + 390, + 375 + ], + "spans": [ + { + "bbox": [ + 219, + 340, + 390, + 375 + ], + "score": 0.94, + "content": "\\sum _ { h = 1 } ^ { H } \\sqrt { \\mathbb { E } _ { \\pi ^ { \\star } } [ \\phi ] ^ { \\top } ( \\Lambda _ { h } ^ { \\star } ) ^ { - 1 } \\mathbb { E } _ { \\pi ^ { \\star } } [ \\phi ] } \\le c _ { 3 } \\frac { d H } { \\sqrt { K } } .", + "type": "interline_equation", + "image_path": "84c24c590868f817e43aaaebdf6a7cd6f6fefa63544a96a29fd83ffe4f9f7227.jpg" + } + ] + } + ], + "index": 13.5, + "virtual_lines": [ + { + "bbox": [ + 219, + 340, + 390, + 357.5 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 219, + 357.5, + 390, + 375.0 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 386, + 506, + 410 + ], + "lines": [ + { + "bbox": [ + 105, + 385, + 507, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 507, + 400 + ], + "score": 1.0, + "content": "Plugging inequality equation E.3 into the bound equation 25, we obtain the minimax lower bound equa-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 397, + 245, + 410 + ], + "spans": [ + { + "bbox": [ + 106, + 397, + 245, + 410 + ], + "score": 1.0, + "content": "tion 6 in the statement of theorem.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15.5 + }, + { + "type": "title", + "bbox": [ + 106, + 423, + 350, + 435 + ], + "lines": [ + { + "bbox": [ + 106, + 424, + 349, + 435 + ], + "spans": [ + { + "bbox": [ + 106, + 424, + 349, + 435 + ], + "score": 1.0, + "content": "E.3 REDUCTION TO TESTING VIA ASSOUAD’S METHOD", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 106, + 444, + 504, + 468 + ], + "lines": [ + { + "bbox": [ + 104, + 442, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 104, + 442, + 275, + 459 + ], + "score": 1.0, + "content": "Proof of Lemma E.1. For any index vector", + "type": "text" + }, + { + "bbox": [ + 275, + 444, + 455, + 457 + ], + "score": 0.92, + "content": "u = ( u _ { 1 } , \\ldots , u _ { H } ) \\in \\mathcal { U } = \\{ - 1 , + 1 \\} ^ { ( d - 2 ) \\times H }", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 442, + 506, + 459 + ], + "score": 1.0, + "content": ", the optimal", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 454, + 263, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 207, + 469 + ], + "score": 1.0, + "content": "policy for MDP instance", + "type": "text" + }, + { + "bbox": [ + 208, + 456, + 223, + 467 + ], + "score": 0.9, + "content": "M _ { u }", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 454, + 263, + 469 + ], + "score": 1.0, + "content": "is simply", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5 + }, + { + "type": "interline_equation", + "bbox": [ + 247, + 474, + 364, + 488 + ], + "lines": [ + { + "bbox": [ + 247, + 474, + 364, + 488 + ], + "spans": [ + { + "bbox": [ + 247, + 474, + 364, + 488 + ], + "score": 0.9, + "content": "\\pi _ { h } ^ { \\star } ( \\cdot ) = u _ { h } \\qquad { \\mathrm { f o r ~ } } h \\in [ H ] .", + "type": "interline_equation", + "image_path": "26b197a0e24a5468951bb74c2920ae4910de2848d5339e57c4f5a56ddc38fa0f.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 247, + 474, + 364, + 488 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 494, + 505, + 517 + ], + "lines": [ + { + "bbox": [ + 105, + 492, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 505, + 507 + ], + "score": 1.0, + "content": "Similar to the proof of Lemma 9 in Zanette et al. (2021), we can show that the value suboptimality of", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 504, + 242, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 134, + 519 + ], + "score": 1.0, + "content": "policy", + "type": "text" + }, + { + "bbox": [ + 134, + 507, + 141, + 515 + ], + "score": 0.75, + "content": "\\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 504, + 179, + 519 + ], + "score": 1.0, + "content": "on MDP", + "type": "text" + }, + { + "bbox": [ + 179, + 506, + 195, + 516 + ], + "score": 0.89, + "content": "M _ { u }", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 504, + 242, + 519 + ], + "score": 1.0, + "content": "is given by", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5 + }, + { + "type": "interline_equation", + "bbox": [ + 224, + 523, + 387, + 558 + ], + "lines": [ + { + "bbox": [ + 224, + 523, + 387, + 558 + ], + "spans": [ + { + "bbox": [ + 224, + 523, + 387, + 558 + ], + "score": 0.94, + "content": "V _ { u } ^ { \\star } - V _ { u } ^ { \\pi } = \\frac { \\delta } { \\sqrt { 2 d } } \\sum _ { h = 1 } ^ { H } \\left\\| u _ { h } - \\mathbb { E } _ { \\pi } [ a _ { h } ] \\right\\| _ { 1 } .", + "type": "interline_equation", + "image_path": "2b43216e57f9fb9f0219e9019fe50774d8c02552f67ae9630f35f6686a0d63ba.jpg" + } + ] + } + ], + "index": 23.5, + "virtual_lines": [ + { + "bbox": [ + 224, + 523, + 387, + 540.5 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 224, + 540.5, + 387, + 558.0 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 564, + 475, + 578 + ], + "lines": [ + { + "bbox": [ + 105, + 563, + 478, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 135, + 579 + ], + "score": 1.0, + "content": "Define", + "type": "text" + }, + { + "bbox": [ + 136, + 565, + 215, + 578 + ], + "score": 0.92, + "content": "u ^ { \\pi } = ( u _ { 1 } ^ { \\pi } , \\ldots , u _ { H } ^ { \\pi } )", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 563, + 237, + 579 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 237, + 564, + 321, + 578 + ], + "score": 0.91, + "content": "u _ { h } ^ { \\pi } : = \\mathrm { s i g n } \\left( \\mathbb { E } _ { \\pi } [ a _ { h } ] \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 563, + 359, + 579 + ], + "score": 1.0, + "content": ", then the", + "type": "text" + }, + { + "bbox": [ + 359, + 566, + 369, + 577 + ], + "score": 0.88, + "content": "\\ell _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 563, + 478, + 579 + ], + "score": 1.0, + "content": "-norm is lower bounded as", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "interline_equation", + "bbox": [ + 237, + 584, + 372, + 601 + ], + "lines": [ + { + "bbox": [ + 237, + 584, + 372, + 601 + ], + "spans": [ + { + "bbox": [ + 237, + 584, + 372, + 601 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\left\\| u _ { h } - \\mathbb { E } _ { \\pi } [ a _ { h } ] \\right\\| _ { 1 } \\ge D _ { H } ( u _ { h } ^ { \\pi } ; u _ { h } ) , } \\end{array}", + "type": "interline_equation", + "image_path": "dc43c845fe4a0b965b33999cb90b3e743d2a88445faa97c88fe263d1ee2497d5.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 237, + 584, + 372, + 601 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 606, + 356, + 619 + ], + "lines": [ + { + "bbox": [ + 106, + 606, + 356, + 620 + ], + "spans": [ + { + "bbox": [ + 106, + 606, + 133, + 620 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 607, + 168, + 619 + ], + "score": 0.92, + "content": "D _ { H } ( \\cdot ; \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 606, + 356, + 620 + ], + "score": 1.0, + "content": "denotes the Hamming distance. 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Combining inequalities equa-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 720, + 504, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 285, + 733 + ], + "score": 1.0, + "content": "tion 27 and equation 28, we finish the proof.", + "type": "text" + }, + { + "bbox": [ + 496, + 722, + 504, + 730 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5 + } + ], + "page_idx": 30, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 293, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 310, + 761 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "31", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 459, + 95 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 459, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 385, + 95 + ], + "score": 1.0, + "content": "Lemma E.1 (Reduction to testing). There exists a universal constant", + "type": "text" + }, + { + "bbox": [ + 386, + 83, + 414, + 93 + ], + "score": 0.9, + "content": "c _ { 1 } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 81, + 459, + 95 + ], + "score": 1.0, + "content": "such thata", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 106, + 81, + 459, + 95 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 119, + 100, + 473, + 122 + ], + "lines": [ + { + "bbox": [ + 119, + 100, + 473, + 122 + ], + "spans": [ + { + "bbox": [ + 119, + 100, + 473, + 122 + ], + "score": 0.93, + "content": "\\operatorname* { i n f } _ { \\hat { \\pi } } \\operatorname* { m a x } _ { u \\in \\mathcal { U } } \\mathbb { E } _ { u } \\left[ V _ { u } ^ { \\star } - V _ { u } ^ { \\widehat { \\pi } } \\right] \\geq c _ { 1 } \\delta \\sqrt { d } H \\operatorname* { m i n } _ { u , u ^ { \\prime } \\in \\mathcal { U } : D _ { H } ( u ^ { \\prime } ; u ) = 1 } \\operatorname* { i n f } _ { \\psi } \\big [ \\mathbb { P } _ { u } ( \\psi \\neq u ) + \\mathbb { P } _ { u ^ { \\prime } } ( \\psi \\neq u ^ { \\prime } ) \\big ] ,", + "type": "interline_equation", + "image_path": "2df7381d6c3389c357588db2de8e89b9f36e13434fd9e80ac39e43fdb9b7e874.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 119, + 100, + 473, + 122 + ], + "spans": [], + "index": 1 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 129, + 505, + 152 + ], + "lines": [ + { + "bbox": [ + 106, + 129, + 505, + 142 + ], + "spans": [ + { + "bbox": [ + 106, + 129, + 132, + 142 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 132, + 130, + 140, + 140 + ], + "score": 0.61, + "content": "\\widehat { \\pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 129, + 487, + 142 + ], + "score": 1.0, + "content": "denotes the output of any algorithm that maps from observations to an estimated policy.", + "type": "text" + }, + { + "bbox": [ + 487, + 131, + 495, + 141 + ], + "score": 0.84, + "content": "\\psi", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 129, + 505, + 142 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 141, + 378, + 154 + ], + "spans": [ + { + "bbox": [ + 106, + 141, + 232, + 154 + ], + "score": 1.0, + "content": "bany test function for parameter", + "type": "text" + }, + { + "bbox": [ + 233, + 143, + 240, + 150 + ], + "score": 0.52, + "content": "u", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 141, + 258, + 154 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 259, + 141, + 275, + 152 + ], + "score": 0.89, + "content": "D _ { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 141, + 378, + 154 + ], + "score": 1.0, + "content": "is the hamming distance.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5, + "bbox_fs": [ + 106, + 129, + 505, + 154 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 156, + 492, + 171 + ], + "lines": [ + { + "bbox": [ + 101, + 150, + 453, + 178 + ], + "spans": [ + { + "bbox": [ + 101, + 150, + 296, + 178 + ], + "score": 1.0, + "content": "Lemma E.2. There exists a universal constant", + "type": "text" + }, + { + "bbox": [ + 296, + 158, + 325, + 168 + ], + "score": 0.91, + "content": "c _ { 2 } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 150, + 415, + 178 + ], + "score": 1.0, + "content": "such that when taking", + "type": "text" + }, + { + "bbox": [ + 416, + 156, + 453, + 172 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\delta : = \\frac { c _ { 2 } d } { \\sqrt { K } } } \\end{array}", + "type": "inline_equation" + } + ], + "index": 4 + } + ], + "index": 4, + "bbox_fs": [ + 101, + 150, + 453, + 178 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 190, + 178, + 420, + 204 + ], + "lines": [ + { + "bbox": [ + 190, + 178, + 420, + 204 + ], + "spans": [ + { + "bbox": [ + 190, + 178, + 420, + 204 + ], + "score": 0.91, + "content": "\\operatorname* { m i n } _ { u , u ^ { \\prime } \\in \\mathcal { U } : D _ { H } ( u ^ { \\prime } ; u ) = 1 } \\operatorname* { i n f } _ { \\psi } \\left[ \\mathbb { P } _ { u } ( \\psi \\neq u ) + \\mathbb { P } _ { u ^ { \\prime } } ( \\psi \\neq u ^ { \\prime } ) \\right] \\geq \\frac { 1 } { 2 } .", + "type": "interline_equation", + "image_path": "1626d6434a77e1532534edfdb818bab2f7aa1e26a910bdd43b01a5bda53846b9.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 190, + 178, + 420, + 204 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 217, + 506, + 243 + ], + "lines": [ + { + "bbox": [ + 105, + 217, + 506, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 132, + 232 + ], + "score": 1.0, + "content": "When", + "type": "text" + }, + { + "bbox": [ + 133, + 217, + 208, + 234 + ], + "score": 0.77, + "content": "\\begin{array} { r } { K \\gtrsim d ^ { 3 } , \\delta : = \\frac { c _ { 2 } d } { \\sqrt { K } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 217, + 259, + 232 + ], + "score": 1.0, + "content": "ensures that", + "type": "text" + }, + { + "bbox": [ + 259, + 217, + 308, + 231 + ], + "score": 0.92, + "content": "\\delta \\leq 1 / \\sqrt { 3 d }", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 217, + 506, + 232 + ], + "score": 1.0, + "content": ". 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There exists a universal constant", + "type": "text" + }, + { + "bbox": [ + 297, + 323, + 325, + 334 + ], + "score": 0.91, + "content": "c _ { 3 } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 321, + 391, + 335 + ], + "score": 1.0, + "content": "such that for all", + "type": "text" + }, + { + "bbox": [ + 392, + 322, + 427, + 333 + ], + "score": 0.88, + "content": "M \\in \\mathcal { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 321, + 431, + 335 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12, + "bbox_fs": [ + 106, + 321, + 431, + 335 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 219, + 340, + 390, + 375 + ], + "lines": [ + { + "bbox": [ + 219, + 340, + 390, + 375 + ], + "spans": [ + { + "bbox": [ + 219, + 340, + 390, + 375 + ], + "score": 0.94, + "content": "\\sum _ { h = 1 } ^ { H } \\sqrt { \\mathbb { E } _ { \\pi ^ { \\star } } [ \\phi ] ^ { \\top } ( \\Lambda _ { h } ^ { \\star } ) ^ { - 1 } \\mathbb { E } _ { \\pi ^ { \\star } } [ \\phi ] } \\le c _ { 3 } \\frac { d H } { \\sqrt { K } } .", + "type": "interline_equation", + "image_path": "84c24c590868f817e43aaaebdf6a7cd6f6fefa63544a96a29fd83ffe4f9f7227.jpg" + } + ] + } + ], + "index": 13.5, + "virtual_lines": [ + { + "bbox": [ + 219, + 340, + 390, + 357.5 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 219, + 357.5, + 390, + 375.0 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 386, + 506, + 410 + ], + "lines": [ + { + "bbox": [ + 105, + 385, + 507, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 507, + 400 + ], + "score": 1.0, + "content": "Plugging inequality equation E.3 into the bound equation 25, we obtain the minimax lower bound equa-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 397, + 245, + 410 + ], + "spans": [ + { + "bbox": [ + 106, + 397, + 245, + 410 + ], + "score": 1.0, + "content": "tion 6 in the statement of theorem.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 385, + 507, + 410 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 423, + 350, + 435 + ], + "lines": [ + { + "bbox": [ + 106, + 424, + 349, + 435 + ], + "spans": [ + { + "bbox": [ + 106, + 424, + 349, + 435 + ], + "score": 1.0, + "content": "E.3 REDUCTION TO TESTING VIA ASSOUAD’S METHOD", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 106, + 444, + 504, + 468 + ], + "lines": [ + { + "bbox": [ + 104, + 442, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 104, + 442, + 275, + 459 + ], + "score": 1.0, + "content": "Proof of Lemma E.1. For any index vector", + "type": "text" + }, + { + "bbox": [ + 275, + 444, + 455, + 457 + ], + "score": 0.92, + "content": "u = ( u _ { 1 } , \\ldots , u _ { H } ) \\in \\mathcal { U } = \\{ - 1 , + 1 \\} ^ { ( d - 2 ) \\times H }", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 442, + 506, + 459 + ], + "score": 1.0, + "content": ", the optimal", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 454, + 263, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 207, + 469 + ], + "score": 1.0, + "content": "policy for MDP instance", + "type": "text" + }, + { + "bbox": [ + 208, + 456, + 223, + 467 + ], + "score": 0.9, + "content": "M _ { u }", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 454, + 263, + 469 + ], + "score": 1.0, + "content": "is simply", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5, + "bbox_fs": [ + 104, + 442, + 506, + 469 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 247, + 474, + 364, + 488 + ], + "lines": [ + { + "bbox": [ + 247, + 474, + 364, + 488 + ], + "spans": [ + { + "bbox": [ + 247, + 474, + 364, + 488 + ], + "score": 0.9, + "content": "\\pi _ { h } ^ { \\star } ( \\cdot ) = u _ { h } \\qquad { \\mathrm { f o r ~ } } h \\in [ H ] .", + "type": "interline_equation", + "image_path": "26b197a0e24a5468951bb74c2920ae4910de2848d5339e57c4f5a56ddc38fa0f.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 247, + 474, + 364, + 488 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 494, + 505, + 517 + ], + "lines": [ + { + "bbox": [ + 105, + 492, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 505, + 507 + ], + "score": 1.0, + "content": "Similar to the proof of Lemma 9 in Zanette et al. (2021), we can show that the value suboptimality of", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 504, + 242, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 134, + 519 + ], + "score": 1.0, + "content": "policy", + "type": "text" + }, + { + "bbox": [ + 134, + 507, + 141, + 515 + ], + "score": 0.75, + "content": "\\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 504, + 179, + 519 + ], + "score": 1.0, + "content": "on MDP", + "type": "text" + }, + { + "bbox": [ + 179, + 506, + 195, + 516 + ], + "score": 0.89, + "content": "M _ { u }", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 504, + 242, + 519 + ], + "score": 1.0, + "content": "is given by", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 492, + 505, + 519 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 224, + 523, + 387, + 558 + ], + "lines": [ + { + "bbox": [ + 224, + 523, + 387, + 558 + ], + "spans": [ + { + "bbox": [ + 224, + 523, + 387, + 558 + ], + "score": 0.94, + "content": "V _ { u } ^ { \\star } - V _ { u } ^ { \\pi } = \\frac { \\delta } { \\sqrt { 2 d } } \\sum _ { h = 1 } ^ { H } \\left\\| u _ { h } - \\mathbb { E } _ { \\pi } [ a _ { h } ] \\right\\| _ { 1 } .", + "type": "interline_equation", + "image_path": "2b43216e57f9fb9f0219e9019fe50774d8c02552f67ae9630f35f6686a0d63ba.jpg" + } + ] + } + ], + "index": 23.5, + "virtual_lines": [ + { + "bbox": [ + 224, + 523, + 387, + 540.5 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 224, + 540.5, + 387, + 558.0 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 564, + 475, + 578 + ], + "lines": [ + { + "bbox": [ + 105, + 563, + 478, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 135, + 579 + ], + "score": 1.0, + "content": "Define", + "type": "text" + }, + { + "bbox": [ + 136, + 565, + 215, + 578 + ], + "score": 0.92, + "content": "u ^ { \\pi } = ( u _ { 1 } ^ { \\pi } , \\ldots , u _ { H } ^ { \\pi } )", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 563, + 237, + 579 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 237, + 564, + 321, + 578 + ], + "score": 0.91, + "content": "u _ { h } ^ { \\pi } : = \\mathrm { s i g n } \\left( \\mathbb { E } _ { \\pi } [ a _ { h } ] \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 563, + 359, + 579 + ], + "score": 1.0, + "content": ", then the", + "type": "text" + }, + { + "bbox": [ + 359, + 566, + 369, + 577 + ], + "score": 0.88, + "content": "\\ell _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 563, + 478, + 579 + ], + "score": 1.0, + "content": "-norm is lower bounded as", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 563, + 478, + 579 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 237, + 584, + 372, + 601 + ], + "lines": [ + { + "bbox": [ + 237, + 584, + 372, + 601 + ], + "spans": [ + { + "bbox": [ + 237, + 584, + 372, + 601 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\left\\| u _ { h } - \\mathbb { E } _ { \\pi } [ a _ { h } ] \\right\\| _ { 1 } \\ge D _ { H } ( u _ { h } ^ { \\pi } ; u _ { h } ) , } \\end{array}", + "type": "interline_equation", + "image_path": "dc43c845fe4a0b965b33999cb90b3e743d2a88445faa97c88fe263d1ee2497d5.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 237, + 584, + 372, + 601 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 606, + 356, + 619 + ], + "lines": [ + { + "bbox": [ + 106, + 606, + 356, + 620 + ], + "spans": [ + { + "bbox": [ + 106, + 606, + 133, + 620 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 607, + 168, + 619 + ], + "score": 0.92, + "content": "D _ { H } ( \\cdot ; \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 606, + 356, + 620 + ], + "score": 1.0, + "content": "denotes the Hamming distance. It follows that", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27, + "bbox_fs": [ + 106, + 606, + 356, + 620 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 244, + 624, + 367, + 652 + ], + "lines": [ + { + "bbox": [ + 244, + 624, + 367, + 652 + ], + "spans": [ + { + "bbox": [ + 244, + 624, + 367, + 652 + ], + "score": 0.94, + "content": "V _ { u } ^ { \\star } - V _ { u } ^ { \\pi } \\geq \\frac { \\delta } { \\sqrt { 2 d } } D _ { H } ( u ^ { \\pi } ; u ) .", + "type": "interline_equation", + "image_path": "cd698ccbd7f0d8e7b946421711c5d8f45ced613dd48d328acb36233aae2c5eea.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 244, + 624, + 367, + 652 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 657, + 492, + 670 + ], + "lines": [ + { + "bbox": [ + 106, + 657, + 492, + 669 + ], + "spans": [ + { + "bbox": [ + 106, + 657, + 492, + 669 + ], + "score": 1.0, + "content": "We then apply Assouad’s method (Lemma 2.12 in Sampson and Guttorp (1992)) and obtain that", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29, + "bbox_fs": [ + 106, + 657, + 492, + 669 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 116, + 675, + 477, + 702 + ], + "lines": [ + { + "bbox": [ + 116, + 675, + 477, + 702 + ], + "spans": [ + { + "bbox": [ + 116, + 675, + 477, + 702 + ], + "score": 0.93, + "content": "\\operatorname* { i n f } _ { \\substack { \\hat { u } \\in \\mathcal { U } } } \\operatorname* { m a x } _ { \\boldsymbol { u } \\in \\mathcal { U } } \\mathbb { E } _ { \\boldsymbol { u } } \\big [ D _ { \\boldsymbol { H } } ( \\hat { u } ; \\boldsymbol { u } ) \\big ] \\geq \\frac { ( d - 2 ) H } { 2 } \\operatorname* { m i n } _ { \\substack { \\boldsymbol { u } , \\boldsymbol { u } ^ { \\prime } \\in \\mathcal { U } : D _ { \\boldsymbol { H } } ( \\boldsymbol { u } ^ { \\prime } ; \\boldsymbol { u } ) = 1 } } \\operatorname* { i n f } _ { \\boldsymbol { \\psi } } \\big [ \\mathbb { P } _ { \\boldsymbol { u } } ( \\boldsymbol { \\psi } \\neq \\boldsymbol { u } ) + \\mathbb { P } _ { \\boldsymbol { u } ^ { \\prime } } ( \\boldsymbol { \\psi } \\neq \\boldsymbol { u } ^ { \\prime } ) \\big ] ,", + "type": "interline_equation", + "image_path": "0b0baeb06bc364e550d5dd6612451325e14c5c58e66d650444cfdb232b1a9819.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 116, + 675, + 477, + 702 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 507, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 708, + 507, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 133, + 723 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 710, + 142, + 721 + ], + "score": 0.86, + "content": "\\psi", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 708, + 350, + 723 + ], + "score": 1.0, + "content": "is any test functions mapping from observations to", + "type": "text" + }, + { + "bbox": [ + 351, + 710, + 379, + 722 + ], + "score": 0.92, + "content": "\\{ u , u ^ { \\prime } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 708, + 507, + 723 + ], + "score": 1.0, + "content": ". 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The proof of Lemma E.2 is similar to that of Lemma 10 in Zanette et al. (2021).", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 112, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 112, + 505, + 127 + ], + "score": 1.0, + "content": "We first apply Theorem 2.12 in Sampson and Guttorp (1992) to lower bound the testing error using", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 125, + 269, + 136 + ], + "spans": [ + { + "bbox": [ + 106, + 125, + 269, + 136 + ], + "score": 1.0, + "content": "Kullback–Leibler divergence and obtain", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2 + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 142, + 513, + 167 + ], + "lines": [ + { + "bbox": [ + 111, + 142, + 513, + 167 + ], + "spans": [ + { + "bbox": [ + 111, + 142, + 513, + 167 + ], + "score": 0.91, + "content": "\\operatorname* { m i n } _ { \\substack { \\tau , u ^ { \\prime } \\in \\mathcal { U } : D _ { H } ( u ^ { \\prime } ; u ) = 1 } } \\operatorname* { i n f } _ { \\psi } \\left[ \\mathbb { P } _ { u } ( \\psi \\neq u ) + \\mathbb { P } _ { u ^ { \\prime } } ( \\psi \\neq u ^ { \\prime } ) \\right] \\geq 1 - \\left( \\frac { 1 } { 2 } \\operatorname* { m a x } _ { \\substack { u , u ^ { \\prime } \\in \\mathcal { U } : D _ { H } ( u ^ { \\prime } ; u ) = 1 } } D _ { \\mathrm { K L } } ( Q _ { u } \\| Q _ { u ^ { \\prime } } ) \\right) ^ { 1 / 2 } .", + "type": "interline_equation", + "image_path": "03bba56ee7da373f751e758361460cde4c0c408ae5e8736b415f1aa5096762cd.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 111, + 142, + 513, + 167 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 184, + 281, + 197 + ], + "lines": [ + { + "bbox": [ + 105, + 183, + 282, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 183, + 215, + 199 + ], + "score": 1.0, + "content": "It only remains to estimate", + "type": "text" + }, + { + "bbox": [ + 216, + 185, + 278, + 198 + ], + "score": 0.93, + "content": "D _ { \\mathrm { K L } } ( \\mathcal { Q } _ { u } \\Vert \\mathcal { Q } _ { u ^ { \\prime } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 183, + 282, + 199 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 107, + 201, + 277, + 214 + ], + "lines": [ + { + "bbox": [ + 105, + 200, + 277, + 215 + ], + "spans": [ + { + "bbox": [ + 105, + 200, + 202, + 215 + ], + "score": 1.0, + "content": "The probability density", + "type": "text" + }, + { + "bbox": [ + 202, + 202, + 216, + 213 + ], + "score": 0.9, + "content": "\\mathcal { Q } _ { u }", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 200, + 277, + 215 + ], + "score": 1.0, + "content": "takes the form", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "interline_equation", + "bbox": [ + 153, + 218, + 457, + 253 + ], + "lines": [ + { + "bbox": [ + 153, + 218, + 457, + 253 + ], + "spans": [ + { + "bbox": [ + 153, + 218, + 457, + 253 + ], + "score": 0.94, + "content": "{ \\mathcal { Q } } _ { u } ( { \\mathcal { D } } ) = \\prod _ { k = 1 } ^ { K } \\xi _ { 1 } ( s _ { 1 } ^ { k } ) \\prod _ { h = 1 } ^ { H } \\mu ( a _ { h } ^ { k } \\mid s _ { h } ^ { k } ) \\left[ R _ { u , h } ( s _ { h } ^ { k } , a _ { h } ^ { k } ) \\right] ( r _ { h } ^ { k } ) \\mathbb { P } _ { h } ( s _ { h + 1 } ^ { k } \\mid s _ { h } ^ { k } , a _ { h } ^ { k } )", + "type": "interline_equation", + "image_path": "f704540d4daec5b3af6eb98b0c2f3750e751b757ccda1952ab9801a4e09dd934.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 153, + 218, + 457, + 229.66666666666666 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 153, + 229.66666666666666, + 457, + 241.33333333333331 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 153, + 241.33333333333331, + 457, + 252.99999999999997 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 257, + 342, + 272 + ], + "lines": [ + { + "bbox": [ + 105, + 257, + 343, + 273 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 133, + 273 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 257, + 181, + 272 + ], + "score": 0.94, + "content": "\\xi _ { 1 } = \\left[ \\textstyle { \\frac { 1 } { 2 } } , \\frac { 1 } { 2 } \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 257, + 343, + 273 + ], + "score": 1.0, + "content": "is the initial distribution. It follows that", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "interline_equation", + "bbox": [ + 175, + 289, + 436, + 379 + ], + "lines": [ + { + "bbox": [ + 175, + 289, + 436, + 379 + ], + "spans": [ + { + "bbox": [ + 175, + 289, + 436, + 379 + ], + "score": 0.94, + "content": "\\begin{array} { l } { { \\displaystyle D _ { K L } \\big ( \\mathcal { Q } _ { u } \\big \\| \\mathcal { Q } _ { u ^ { \\prime } } \\big ) = \\mathbb { E } _ { u } \\big [ \\log \\big ( \\mathcal { Q } _ { u } / \\mathcal { Q } _ { u ^ { \\prime } } \\big ) \\big ] } \\ ~ } \\\\ { { \\displaystyle = K \\cdot \\sum _ { h = 1 } ^ { H } \\mathbb { E } _ { u } \\Big [ \\log \\big ( \\big [ R _ { u , h } ( s _ { h } ^ { 1 } , a _ { h } ^ { 1 } ) \\big ] ( r _ { h } ^ { 1 } ) \\big / \\big [ R _ { u ^ { \\prime } , h } ( s _ { h } ^ { 1 } , a _ { h } ^ { 1 } ) \\big ] ( r _ { h } ^ { 1 } ) \\big ) \\Big ] } \\ ~ } \\\\ { { \\displaystyle = \\frac { K } { d } \\sum _ { j = 1 } ^ { d - 2 } D _ { \\mathrm { K L } } \\Big ( \\mathcal { N } \\big ( \\frac { \\delta } { \\sqrt { 2 d } } \\langle e _ { j } , u _ { h } \\rangle , 1 \\big ) \\ \\Big \\| \\mathcal { N } \\big ( \\frac { \\delta } { \\sqrt { 2 d } } \\langle e _ { j } , u _ { h } ^ { \\prime } \\rangle , 1 \\big ) \\Big ) . } } \\end{array}", + "type": "interline_equation", + "image_path": "c361c204140da4db2570694ae47c94812bcd3651f50d11ddca9f361bf632e2ef.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 175, + 289, + 436, + 319.0 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 175, + 319.0, + 436, + 349.0 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 175, + 349.0, + 436, + 379.0 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 389, + 505, + 415 + ], + "lines": [ + { + "bbox": [ + 104, + 389, + 505, + 407 + ], + "spans": [ + { + "bbox": [ + 104, + 389, + 150, + 407 + ], + "score": 1.0, + "content": "If we take", + "type": "text" + }, + { + "bbox": [ + 150, + 389, + 186, + 405 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\delta = \\frac { c _ { 2 } d } { \\sqrt { K } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 389, + 505, + 407 + ], + "score": 1.0, + "content": ", then inequality equation 29 ensures inequality equation 24, as claimed in the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 403, + 205, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 205, + 416 + ], + "score": 1.0, + "content": "statement of the lemma.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5 + }, + { + "type": "title", + "bbox": [ + 106, + 445, + 315, + 457 + ], + "lines": [ + { + "bbox": [ + 106, + 445, + 316, + 458 + ], + "spans": [ + { + "bbox": [ + 106, + 445, + 316, + 458 + ], + "score": 1.0, + "content": "E.5 CONNECTION TO THE UNCERTAINTY TERM", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 106, + 465, + 506, + 490 + ], + "lines": [ + { + "bbox": [ + 106, + 465, + 506, + 478 + ], + "spans": [ + { + "bbox": [ + 106, + 465, + 506, + 478 + ], + "score": 1.0, + "content": "Proof of Lemma E.3. We first calculate the explicit form of the inverse of variance-rescaled covari-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 473, + 459, + 493 + ], + "spans": [ + { + "bbox": [ + 104, + 473, + 156, + 493 + ], + "score": 1.0, + "content": "ance matrix", + "type": "text" + }, + { + "bbox": [ + 156, + 477, + 174, + 489 + ], + "score": 0.91, + "content": "\\Lambda _ { h } ^ { \\star , p }", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 473, + 255, + 493 + ], + "score": 1.0, + "content": ". For each time step", + "type": "text" + }, + { + "bbox": [ + 255, + 477, + 289, + 489 + ], + "score": 0.92, + "content": "h \\in [ H ]", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 473, + 367, + 493 + ], + "score": 1.0, + "content": ", the value function", + "type": "text" + }, + { + "bbox": [ + 367, + 477, + 396, + 491 + ], + "score": 0.93, + "content": "V _ { u , h + 1 } ^ { \\star }", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 473, + 459, + 493 + ], + "score": 1.0, + "content": "takes the form", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5 + }, + { + "type": "interline_equation", + "bbox": [ + 225, + 495, + 385, + 512 + ], + "lines": [ + { + "bbox": [ + 225, + 495, + 385, + 512 + ], + "spans": [ + { + "bbox": [ + 225, + 495, + 385, + 512 + ], + "score": 0.92, + "content": "\\begin{array} { r } { V _ { u , h + 1 } ^ { \\star } = \\mathbb { E } _ { \\pi ^ { \\star } } r _ { u , h + 1 } + \\bigl ( \\mathbb { P } _ { h + 1 } ^ { \\pi ^ { \\star } } V _ { u , h + 2 } ^ { \\star } \\bigr ) . } \\end{array}", + "type": "interline_equation", + "image_path": "95bbc5a37d479b48d714767c7391c37e8272e002bfb4f79df31475f1ffabe892.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 225, + 495, + 385, + 512 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 516, + 505, + 542 + ], + "lines": [ + { + "bbox": [ + 104, + 514, + 506, + 534 + ], + "spans": [ + { + "bbox": [ + 104, + 514, + 128, + 534 + ], + "score": 1.0, + "content": "Sinc", + "type": "text" + }, + { + "bbox": [ + 129, + 516, + 487, + 533 + ], + "score": 0.47, + "content": "\\begin{array} { r } { \\colon \\bigl ( \\mathbb { P } _ { h + 1 } V _ { u , h + 2 } ^ { \\star } \\bigr ) ( + 1 ) = \\bigl ( \\mathbb { P } _ { h + 1 } V _ { u , h + 2 } ^ { \\star } \\bigr ) ( - 1 ) \\mathrm { ~ a n d ~ } r _ { u , h + 1 } ( + 1 , a ) - r _ { u , h + 1 } ( - 1 , a ) = 2 / \\sqrt { 6 } } \\end{array}", + "type": "inline_equation", + "image_path": "07592e37cd1e726c3560f1b4369387f9275dd972a91ce17acf7175d6dd26508d.jpg" + }, + { + "bbox": [ + 488, + 514, + 506, + 534 + ], + "score": 1.0, + "content": ", we", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 528, + 129, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 129, + 543 + ], + "score": 1.0, + "content": "have", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5 + }, + { + "type": "interline_equation", + "bbox": [ + 186, + 544, + 425, + 568 + ], + "lines": [ + { + "bbox": [ + 186, + 544, + 425, + 568 + ], + "spans": [ + { + "bbox": [ + 186, + 544, + 425, + 568 + ], + "score": 0.9, + "content": "\\operatorname { V a r } _ { P _ { h } } ( V _ { u , h + 1 } ^ { \\star } ) ( + 1 , a ) = \\operatorname { V a r } _ { P _ { h } } ( \\mathbb { E } _ { \\pi ^ { \\star } } r _ { u , h + 1 } ) ( + 1 , a ) = \\frac 1 6 .", + "type": "interline_equation", + "image_path": "ea89f73cc1527b0d313fe1e05447089a4c60dbf4c109b679b23131f4b411c331.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 186, + 544, + 425, + 568 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 571, + 147, + 583 + ], + "lines": [ + { + "bbox": [ + 106, + 569, + 149, + 586 + ], + "spans": [ + { + "bbox": [ + 106, + 569, + 149, + 586 + ], + "score": 1.0, + "content": "Similarly,", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "interline_equation", + "bbox": [ + 194, + 588, + 417, + 612 + ], + "lines": [ + { + "bbox": [ + 194, + 588, + 417, + 612 + ], + "spans": [ + { + "bbox": [ + 194, + 588, + 417, + 612 + ], + "score": 0.91, + "content": "\\operatorname { V a r } _ { P _ { h } } ( V _ { u , h + 1 } ^ { \\star } ) ( - 1 , a ) = \\operatorname { V a r } _ { P _ { h } } ( V _ { u , h + 1 } ^ { \\star } ) ( + 1 , a ) = \\frac { 1 } { 6 } .", + "type": "interline_equation", + "image_path": "d0f76ad4e54054ccc30d18ab5c112b329e34afc7f3ec2141429969ee8180bb24.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 194, + 588, + 417, + 612 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 615, + 500, + 627 + ], + "lines": [ + { + "bbox": [ + 106, + 615, + 500, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 615, + 500, + 628 + ], + "score": 1.0, + "content": "By routine calculation, we find that the population-level rescaled covariance matrix takes the form", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "interline_equation", + "bbox": [ + 194, + 631, + 417, + 666 + ], + "lines": [ + { + "bbox": [ + 194, + 631, + 417, + 666 + ], + "spans": [ + { + "bbox": [ + 194, + 631, + 417, + 666 + ], + "score": 0.93, + "content": "\\Lambda _ { h } ^ { \\star , p } = \\frac { 3 K } { 2 } \\left( \\begin{array} { c c } { \\frac { 2 } { d ^ { 2 } } \\mathbf { I } _ { d - 2 } } & { \\frac { 1 } { d \\sqrt { d } } \\mathbf { 1 } _ { ( d - 2 ) \\times 2 } } \\\\ { \\frac { 1 } { d \\sqrt { d } } \\mathbf { 1 } _ { 2 \\times ( d - 2 ) } } & { \\mathbf { I } _ { 2 } } \\end{array} \\right) \\in \\mathbb { R } ^ { d \\times d }", + "type": "interline_equation", + "image_path": "368c47d0536fb454399f7b2e6439e98ceb7915b6eaa9e0fb3599c0336a37f507.jpg" + } + ] + } + ], + "index": 26.5, + "virtual_lines": [ + { + "bbox": [ + 194, + 631, + 417, + 648.5 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 194, + 648.5, + 417, + 666.0 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 669, + 373, + 683 + ], + "lines": [ + { + "bbox": [ + 103, + 667, + 376, + 687 + ], + "spans": [ + { + "bbox": [ + 103, + 667, + 137, + 687 + ], + "score": 1.0, + "content": "for any", + "type": "text" + }, + { + "bbox": [ + 138, + 671, + 171, + 682 + ], + "score": 0.92, + "content": "h \\in [ H ]", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 667, + 315, + 687 + ], + "score": 1.0, + "content": ". 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The proof of Lemma E.2 is similar to that of Lemma 10 in Zanette et al. 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It follows that", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10, + "bbox_fs": [ + 105, + 257, + 343, + 273 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 175, + 289, + 436, + 379 + ], + "lines": [ + { + "bbox": [ + 175, + 289, + 436, + 379 + ], + "spans": [ + { + "bbox": [ + 175, + 289, + 436, + 379 + ], + "score": 0.94, + "content": "\\begin{array} { l } { { \\displaystyle D _ { K L } \\big ( \\mathcal { Q } _ { u } \\big \\| \\mathcal { Q } _ { u ^ { \\prime } } \\big ) = \\mathbb { E } _ { u } \\big [ \\log \\big ( \\mathcal { Q } _ { u } / \\mathcal { Q } _ { u ^ { \\prime } } \\big ) \\big ] } \\ ~ } \\\\ { { \\displaystyle = K \\cdot \\sum _ { h = 1 } ^ { H } \\mathbb { E } _ { u } \\Big [ \\log \\big ( \\big [ R _ { u , h } ( s _ { h } ^ { 1 } , a _ { h } ^ { 1 } ) \\big ] ( r _ { h } ^ { 1 } ) \\big / \\big [ R _ { u ^ { \\prime } , h } ( s _ { h } ^ { 1 } , a _ { h } ^ { 1 } ) \\big ] ( r _ { h } ^ { 1 } ) \\big ) \\Big ] } \\ ~ } \\\\ { { \\displaystyle = \\frac { K } { d } \\sum _ { j = 1 } ^ { d - 2 } D _ { \\mathrm { K L } } \\Big ( \\mathcal { N } \\big ( \\frac { \\delta } { \\sqrt { 2 d } } \\langle e _ { j } , u _ { h } \\rangle , 1 \\big ) \\ \\Big \\| \\mathcal { N } \\big ( \\frac { \\delta } { \\sqrt { 2 d } } \\langle e _ { j } , u _ { h } ^ { \\prime } \\rangle , 1 \\big ) \\Big ) . } } \\end{array}", + "type": "interline_equation", + "image_path": "c361c204140da4db2570694ae47c94812bcd3651f50d11ddca9f361bf632e2ef.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 175, + 289, + 436, + 319.0 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 175, + 319.0, + 436, + 349.0 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 175, + 349.0, + 436, + 379.0 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 389, + 505, + 415 + ], + "lines": [ + { + "bbox": [ + 104, + 389, + 505, + 407 + ], + "spans": [ + { + "bbox": [ + 104, + 389, + 150, + 407 + ], + "score": 1.0, + "content": "If we take", + "type": "text" + }, + { + "bbox": [ + 150, + 389, + 186, + 405 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\delta = \\frac { c _ { 2 } d } { \\sqrt { K } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 389, + 505, + 407 + ], + "score": 1.0, + "content": ", then inequality equation 29 ensures inequality equation 24, as claimed in the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 403, + 205, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 205, + 416 + ], + "score": 1.0, + "content": "statement of the lemma.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5, + "bbox_fs": [ + 104, + 389, + 505, + 416 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 445, + 315, + 457 + ], + "lines": [ + { + "bbox": [ + 106, + 445, + 316, + 458 + ], + "spans": [ + { + "bbox": [ + 106, + 445, + 316, + 458 + ], + "score": 1.0, + "content": "E.5 CONNECTION TO THE UNCERTAINTY TERM", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 106, + 465, + 506, + 490 + ], + "lines": [ + { + "bbox": [ + 106, + 465, + 506, + 478 + ], + "spans": [ + { + "bbox": [ + 106, + 465, + 506, + 478 + ], + "score": 1.0, + "content": "Proof of Lemma E.3. We first calculate the explicit form of the inverse of variance-rescaled covari-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 473, + 459, + 493 + ], + "spans": [ + { + "bbox": [ + 104, + 473, + 156, + 493 + ], + "score": 1.0, + "content": "ance matrix", + "type": "text" + }, + { + "bbox": [ + 156, + 477, + 174, + 489 + ], + "score": 0.91, + "content": "\\Lambda _ { h } ^ { \\star , p }", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 473, + 255, + 493 + ], + "score": 1.0, + "content": ". 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1 ) \\mathrm { ~ a n d ~ } r _ { u , h + 1 } ( + 1 , a ) - r _ { u , h + 1 } ( - 1 , a ) = 2 / \\sqrt { 6 } } \\end{array}", + "type": "inline_equation", + "image_path": "07592e37cd1e726c3560f1b4369387f9275dd972a91ce17acf7175d6dd26508d.jpg" + }, + { + "bbox": [ + 488, + 514, + 506, + 534 + ], + "score": 1.0, + "content": ", we", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 528, + 129, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 129, + 543 + ], + "score": 1.0, + "content": "have", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5, + "bbox_fs": [ + 104, + 514, + 506, + 543 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 186, + 544, + 425, + 568 + ], + "lines": [ + { + "bbox": [ + 186, + 544, + 425, + 568 + ], + "spans": [ + { + "bbox": [ + 186, + 544, + 425, + 568 + ], + "score": 0.9, + "content": "\\operatorname { V a r } _ { P _ { h } } ( V _ { u , h + 1 } ^ { \\star } ) ( + 1 , a ) = \\operatorname { V a r } _ { P _ { h } } ( \\mathbb { E } _ { \\pi ^ { \\star } } r _ { u , h + 1 } ) ( + 1 , a ) = \\frac 1 6 .", + "type": "interline_equation", + "image_path": "ea89f73cc1527b0d313fe1e05447089a4c60dbf4c109b679b23131f4b411c331.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 186, + 544, + 425, + 568 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 571, + 147, + 583 + ], + "lines": [ + { + "bbox": [ + 106, + 569, + 149, + 586 + ], + "spans": [ + { + "bbox": [ + 106, + 569, + 149, + 586 + ], + "score": 1.0, + "content": "Similarly,", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23, + "bbox_fs": [ + 106, + 569, + 149, + 586 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 194, + 588, + 417, + 612 + ], + "lines": [ + { + "bbox": [ + 194, + 588, + 417, + 612 + ], + "spans": [ + { + "bbox": [ + 194, + 588, + 417, + 612 + ], + "score": 0.91, + "content": "\\operatorname { V a r } _ { P _ { h } } ( V _ { u , h + 1 } ^ { \\star } ) ( - 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1 } \\mathbb { E } _ { \\pi ^ { \\star } } [ \\phi ] } \\lesssim d / \\sqrt { K } .", + "type": "interline_equation", + "image_path": "abe82989f98b8a57411622561bbd1ec91eff4ac37b1d297c7f3fb979b3dbf170.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 231, + 329, + 380, + 350 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 354, + 506, + 378 + ], + "lines": [ + { + "bbox": [ + 105, + 354, + 505, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 222, + 367 + ], + "score": 1.0, + "content": "Taking the summation over", + "type": "text" + }, + { + "bbox": [ + 222, + 355, + 258, + 367 + ], + "score": 0.93, + "content": "h \\in [ H ]", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 354, + 505, + 367 + ], + "score": 1.0, + "content": ", we obtain the bound equation 26 as claimed in the lemma", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 366, + 150, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 150, + 378 + ], + "score": 1.0, + "content": "statement.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5 + }, + { + "type": "title", + "bbox": [ + 107, + 410, + 271, + 423 + ], + "lines": [ + { + "bbox": [ + 105, + 409, + 273, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 409, + 273, + 425 + ], + "score": 1.0, + "content": "F A NUMERICAL SIMULATION", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "title", + "bbox": [ + 107, + 434, + 270, + 447 + ], + "lines": [ + { + "bbox": [ + 105, + 434, + 272, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 272, + 448 + ], + "score": 1.0, + "content": "F.1 A LINEAR MDP CONSTRUCTION", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 106, + 455, + 506, + 559 + ], + "lines": [ + { + "bbox": [ + 106, + 455, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 106, + 455, + 505, + 468 + ], + "score": 1.0, + "content": "We consider a synthetic linear MDP example that is similar to Min et al. 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The MDP instance we use consists of", + "type": "text" + }, + { + "bbox": [ + 430, + 467, + 462, + 478 + ], + "score": 0.91, + "content": "| S | = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 467, + 505, + 478 + ], + "score": 1.0, + "content": "states and", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 107, + 478, + 504, + 490 + ], + "spans": [ + { + "bbox": [ + 107, + 478, + 152, + 490 + ], + "score": 0.92, + "content": "| { \\mathcal { A } } | = 1 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 478, + 282, + 490 + ], + "score": 1.0, + "content": "actions, and feature dimension", + "type": "text" + }, + { + "bbox": [ + 282, + 478, + 319, + 488 + ], + "score": 0.89, + "content": "d = 1 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 478, + 355, + 490 + ], + "score": 1.0, + "content": ". We set", + "type": "text" + }, + { + "bbox": [ + 356, + 478, + 403, + 490 + ], + "score": 0.95, + "content": "{ \\mathcal { S } } = \\{ 0 , 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 478, + 423, + 490 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 423, + 478, + 504, + 490 + ], + "score": 0.91, + "content": "\\mathcal { A } = \\{ 0 , 1 , \\ldots , 9 9 \\}", + "type": "inline_equation" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 488, + 506, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 237, + 501 + ], + "score": 1.0, + "content": "respectively. 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In addition, both", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 487, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 505, + 500 + ], + "score": 1.0, + "content": "non-pessimistic algorithms exhibit similar accuracy, and this is partially owing to our truncation", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 498, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 138, + 514 + ], + "score": 1.0, + "content": "scheme", + "type": "text" + }, + { + "bbox": [ + 138, + 498, + 290, + 512 + ], + "score": 0.91, + "content": "\\widehat { \\sigma } _ { h } ( \\cdot , \\cdot ) ^ { 2 } \\gets \\operatorname* { m a x } \\{ 1 , \\widehat { \\mathrm { V a r } } _ { P _ { h } } \\widehat { V } _ { h + 1 } ( \\cdot , \\cdot ) \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 499, + 303, + 514 + ], + "score": 1.0, + "content": "so", + "type": "text" + }, + { + "bbox": [ + 303, + 499, + 337, + 512 + ], + "score": 0.91, + "content": "\\widehat { \\sigma } _ { h } ( \\cdot , \\cdot ) ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 499, + 506, + 514 + ], + "score": 1.0, + "content": "will just be 1 when the estimated variance", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 510, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 505, + 525 + ], + "score": 1.0, + "content": "b bis small. Lastly, variance-aware pessimism eventually outperforms non-pessimism algorithms when", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 522, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 506, + 535 + ], + "score": 1.0, + "content": "sample size is large and this might come from that the pessimistic bonus is estimated more accurately", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 533, + 243, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 533, + 243, + 545 + ], + "score": 1.0, + "content": "when more samples are collected.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24.5 + }, + { + "type": "title", + "bbox": [ + 106, + 560, + 374, + 573 + ], + "lines": [ + { + "bbox": [ + 105, + 558, + 375, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 375, + 575 + ], + "score": 1.0, + "content": "G SOME MISSING DERIVATIONS AND DISCUSSIONS", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "title", + "bbox": [ + 108, + 584, + 294, + 595 + ], + "lines": [ + { + "bbox": [ + 106, + 583, + 295, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 295, + 596 + ], + "score": 1.0, + "content": "G.1 REGARDING COVERAGE ASSUMPTION", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 106, + 604, + 506, + 662 + ], + "lines": [ + { + "bbox": [ + 106, + 604, + 506, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 604, + 506, + 617 + ], + "score": 1.0, + "content": "Now we discuss the feature coverage assumption. Indeed, even if Assumption 2.2 is not satisfied,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 103, + 611, + 508, + 632 + ], + "spans": [ + { + "bbox": [ + 103, + 611, + 284, + 632 + ], + "score": 1.0, + "content": "we can still learn in the effective subspan of", + "type": "text" + }, + { + "bbox": [ + 284, + 615, + 406, + 628 + ], + "score": 0.79, + "content": "\\dot { \\Sigma } _ { h } ^ { p } : = \\mathbb { E } _ { \\boldsymbol { \\mu } , h } \\left[ \\boldsymbol { \\phi } ( \\boldsymbol { s } , \\boldsymbol { a } ) \\boldsymbol { \\phi } ( \\boldsymbol { s } , \\boldsymbol { a } ) ^ { \\top } \\right] ^ { - }", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 611, + 481, + 632 + ], + "score": 1.0, + "content": ". 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In each figure, y-axis denotes suboptimality gap", + "type": "text" + }, + { + "bbox": [ + 328, + 212, + 361, + 223 + ], + "score": 0.91, + "content": "v ^ { \\star } - v ^ { \\widehat { \\pi } }", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 212, + 365, + 225 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 365, + 214, + 372, + 223 + ], + "score": 0.56, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 212, + 505, + 225 + ], + "score": 1.0, + "content": "-axis denotes number of episodes", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 107, + 222, + 506, + 238 + ], + "spans": [ + { + "bbox": [ + 107, + 224, + 117, + 234 + ], + "score": 0.77, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 222, + 284, + 238 + ], + "score": 1.0, + "content": ". The problem horizons are fixed to be", + "type": "text" + }, + { + "bbox": [ + 285, + 225, + 351, + 235 + ], + "score": 0.86, + "content": "H = 2 0 , 3 0 , 5 0", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 222, + 506, + 238 + ], + "score": 1.0, + "content": ". The solid line denotes the average", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 235, + 505, + 248 + ], + "spans": [ + { + "bbox": [ + 106, + 235, + 505, + 248 + ], + "score": 1.0, + "content": "suboptimality gap over 50 trials and the error bar area is the corresponding standard deviation. The", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 246, + 227, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 142, + 258 + ], + "score": 1.0, + "content": "range of", + "type": "text" + }, + { + "bbox": [ + 142, + 246, + 153, + 256 + ], + "score": 0.84, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 246, + 227, + 258 + ], + "score": 1.0, + "content": "is from 5 to 1000.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5 + } + ], + "index": 3.0 + }, + { + "type": "title", + "bbox": [ + 108, + 275, + 500, + 297 + ], + "lines": [ + { + "bbox": [ + 105, + 274, + 502, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 502, + 288 + ], + "score": 1.0, + "content": "F.2 EMPIRICAL COMPARISON BETWEEN PEVI AND VAPVI ON THE CONSTRUCTED LINEAR", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 128, + 285, + 157, + 298 + ], + "spans": [ + { + "bbox": [ + 128, + 285, + 157, + 298 + ], + "score": 1.0, + "content": "MDP", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 106, + 306, + 505, + 384 + ], + "lines": [ + { + "bbox": [ + 106, + 306, + 506, + 319 + ], + "spans": [ + { + "bbox": [ + 106, + 306, + 506, + 319 + ], + "score": 1.0, + "content": "We compare Pessimistic Value Iteration (PEVI) in Jin et al. (2021b) and our VAPVI Algorithm 1 in", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 318, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 218, + 330 + ], + "score": 1.0, + "content": "Figure 1, with horizon to be", + "type": "text" + }, + { + "bbox": [ + 218, + 318, + 252, + 329 + ], + "score": 0.5, + "content": "H = 2 0", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 318, + 505, + 330 + ], + "score": 1.0, + "content": ", 30, 50. In addition, we add the non-pessimistic version for both", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 329, + 505, + 341 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 505, + 341 + ], + "score": 1.0, + "content": "algorithms, i.e. least-square value iteration (LSVI) and variance-aware value iteration (VAVI). The", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 340, + 505, + 351 + ], + "spans": [ + { + "bbox": [ + 106, + 340, + 183, + 351 + ], + "score": 1.0, + "content": "true optimal value", + "type": "text" + }, + { + "bbox": [ + 183, + 340, + 195, + 350 + ], + "score": 0.87, + "content": "v ^ { \\star }", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 340, + 505, + 351 + ], + "score": 1.0, + "content": "is computed via value iteration using the underlying transition kernels. For", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 350, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 350, + 505, + 363 + ], + "score": 1.0, + "content": "the empirical validation of VAPVI, we do not split the data and, in particular, in all the methods we", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 361, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 361, + 136, + 374 + ], + "score": 1.0, + "content": "choose", + "type": "text" + }, + { + "bbox": [ + 136, + 362, + 174, + 372 + ], + "score": 0.89, + "content": "\\lambda = 0 . 0 1", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 361, + 219, + 374 + ], + "score": 1.0, + "content": "(instead of", + "type": "text" + }, + { + "bbox": [ + 219, + 362, + 244, + 372 + ], + "score": 0.9, + "content": "\\lambda = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 361, + 505, + 374 + ], + "score": 1.0, + "content": "used in theory (Jin et al., 2021b) which causes over-regularization", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 373, + 181, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 181, + 384 + ], + "score": 1.0, + "content": "in the simulation).", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 306, + 506, + 384 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 388, + 505, + 410 + ], + "lines": [ + { + "bbox": [ + 105, + 388, + 507, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 453, + 402 + ], + "score": 1.0, + "content": "We can observe VAPVI outperforms PEVI and the gap becomes larger when horizon", + "type": "text" + }, + { + "bbox": [ + 453, + 389, + 463, + 398 + ], + "score": 0.79, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 388, + 507, + 402 + ], + "score": 1.0, + "content": "increases.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 398, + 454, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 454, + 410 + ], + "score": 1.0, + "content": "One main reason for this to happen is due to the bonus used in PEVI (Jin et al., 2021b)", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 388, + 507, + 410 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 240, + 411, + 370, + 436 + ], + "lines": [ + { + "bbox": [ + 240, + 411, + 370, + 436 + ], + "spans": [ + { + "bbox": [ + 240, + 411, + 370, + 436 + ], + "score": 0.94, + "content": "O \\left[ d H \\cdot \\left( \\phi ( \\cdot , \\cdot ) ^ { \\top } \\Sigma _ { h } ^ { - 1 } \\phi ( \\cdot , \\cdot ) \\right) ^ { 1 / 2 } \\right]", + "type": "interline_equation", + "image_path": "594c0bc419c0cda2a54dd4500e87cb0c48d58e5891ae7bc01b341d5e77c72610.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 240, + 411, + 370, + 436 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 109, + 437, + 264, + 448 + ], + "lines": [ + { + "bbox": [ + 108, + 436, + 264, + 450 + ], + "spans": [ + { + "bbox": [ + 108, + 436, + 264, + 450 + ], + "score": 1.0, + "content": "is overly pessimistic comparing to our", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20, + "bbox_fs": [ + 108, + 436, + 264, + 450 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 241, + 450, + 370, + 475 + ], + "lines": [ + { + "bbox": [ + 241, + 450, + 370, + 475 + ], + "spans": [ + { + "bbox": [ + 241, + 450, + 370, + 475 + ], + "score": 0.93, + "content": "O \\left[ \\sqrt { d } \\cdot \\left( \\phi ( \\cdot , \\cdot ) ^ { \\top } \\Lambda _ { h } ^ { - 1 } \\phi ( \\cdot , \\cdot ) \\right) ^ { 1 / 2 } \\right]", + "type": "interline_equation", + "image_path": "4c5c22846ddf533900a827f047cb7c45b860d7c8a8107b8c8484b90932c5abff.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 241, + 450, + 370, + 475 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 476, + 506, + 545 + ], + "lines": [ + { + "bbox": [ + 108, + 475, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 108, + 475, + 132, + 489 + ], + "score": 1.0, + "content": "when", + "type": "text" + }, + { + "bbox": [ + 133, + 477, + 143, + 486 + ], + "score": 0.81, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 475, + 505, + 489 + ], + "score": 1.0, + "content": "becomes larger and this could potentially make the learning less accurate. In addition, both", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 487, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 505, + 500 + ], + "score": 1.0, + "content": "non-pessimistic algorithms exhibit similar accuracy, and this is partially owing to our truncation", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 498, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 138, + 514 + ], + "score": 1.0, + "content": "scheme", + "type": "text" + }, + { + "bbox": [ + 138, + 498, + 290, + 512 + ], + "score": 0.91, + "content": "\\widehat { \\sigma } _ { h } ( \\cdot , \\cdot ) ^ { 2 } \\gets \\operatorname* { m a x } \\{ 1 , \\widehat { \\mathrm { V a r } } _ { P _ { h } } \\widehat { V } _ { h + 1 } ( \\cdot , \\cdot ) \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 499, + 303, + 514 + ], + "score": 1.0, + "content": "so", + "type": "text" + }, + { + "bbox": [ + 303, + 499, + 337, + 512 + ], + "score": 0.91, + "content": "\\widehat { \\sigma } _ { h } ( \\cdot , \\cdot ) ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 499, + 506, + 514 + ], + "score": 1.0, + "content": "will just be 1 when the estimated variance", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 510, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 505, + 525 + ], + "score": 1.0, + "content": "b bis small. Lastly, variance-aware pessimism eventually outperforms non-pessimism algorithms when", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 522, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 506, + 535 + ], + "score": 1.0, + "content": "sample size is large and this might come from that the pessimistic bonus is estimated more accurately", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 533, + 243, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 533, + 243, + 545 + ], + "score": 1.0, + "content": "when more samples are collected.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 475, + 506, + 545 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 560, + 374, + 573 + ], + "lines": [ + { + "bbox": [ + 105, + 558, + 375, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 375, + 575 + ], + "score": 1.0, + "content": "G SOME MISSING DERIVATIONS AND DISCUSSIONS", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "title", + "bbox": [ + 108, + 584, + 294, + 595 + ], + "lines": [ + { + "bbox": [ + 106, + 583, + 295, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 295, + 596 + ], + "score": 1.0, + "content": "G.1 REGARDING COVERAGE ASSUMPTION", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 106, + 604, + 506, + 662 + ], + "lines": [ + { + "bbox": [ + 106, + 604, + 506, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 604, + 506, + 617 + ], + "score": 1.0, + "content": "Now we discuss the feature coverage assumption. Indeed, even if Assumption 2.2 is not satisfied,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 103, + 611, + 508, + 632 + ], + "spans": [ + { + "bbox": [ + 103, + 611, + 284, + 632 + ], + "score": 1.0, + "content": "we can still learn in the effective subspan of", + "type": "text" + }, + { + "bbox": [ + 284, + 615, + 406, + 628 + ], + "score": 0.79, + "content": "\\dot { \\Sigma } _ { h } ^ { p } : = \\mathbb { E } _ { \\boldsymbol { \\mu } , h } \\left[ \\boldsymbol { \\phi } ( \\boldsymbol { s } , \\boldsymbol { a } ) \\boldsymbol { \\phi } ( \\boldsymbol { s } , \\boldsymbol { a } ) ^ { \\top } \\right] ^ { - }", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 611, + 481, + 632 + ], + "score": 1.0, + "content": ". Concretely, since", + "type": "text" + }, + { + "bbox": [ + 481, + 615, + 495, + 628 + ], + "score": 0.91, + "content": "\\Sigma _ { h } ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 611, + 508, + 632 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 626, + 506, + 644 + ], + "spans": [ + { + "bbox": [ + 104, + 626, + 305, + 644 + ], + "score": 1.0, + "content": "symmetric, by orthogonal decomposition we have", + "type": "text" + }, + { + "bbox": [ + 306, + 628, + 365, + 641 + ], + "score": 0.92, + "content": "\\Sigma _ { h } ^ { p } = Z _ { h } \\Lambda Z _ { h } ^ { \\top }", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 626, + 395, + 644 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 396, + 629, + 408, + 640 + ], + "score": 0.88, + "content": "Z _ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 626, + 506, + 644 + ], + "score": 1.0, + "content": "(can be estimated using", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 639, + 506, + 653 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 368, + 653 + ], + "score": 1.0, + "content": "the samples for practical purpose) consists of orthogonal basis and", + "type": "text" + }, + { + "bbox": [ + 369, + 640, + 377, + 649 + ], + "score": 0.81, + "content": "\\Lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 639, + 480, + 653 + ], + "score": 1.0, + "content": "consists of eigenvalues of", + "type": "text" + }, + { + "bbox": [ + 480, + 640, + 493, + 652 + ], + "score": 0.9, + "content": "\\Sigma _ { h } ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 639, + 506, + 653 + ], + "score": 1.0, + "content": "in", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 649, + 339, + 664 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 339, + 664 + ], + "score": 1.0, + "content": "the diagonal. Suppose we do not have a full coverage, i.e.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32, + "bbox_fs": [ + 103, + 604, + 508, + 664 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 203, + 664, + 403, + 677 + ], + "lines": [ + { + "bbox": [ + 203, + 664, + 403, + 677 + ], + "spans": [ + { + "bbox": [ + 203, + 664, + 403, + 677 + ], + "score": 0.69, + "content": "\\Lambda = \\mathrm { d i a g } [ \\lambda _ { 1 } , \\lambda _ { 2 } , . . . , \\lambda _ { d ^ { \\prime } } , 0 , . . . , 0 ] \\quad \\mathrm { w i t h } \\quad d ^ { \\prime } < d", + "type": "interline_equation", + "image_path": "fe1e90df137357cb059b6bdbc0fed0be19c068893d866b2332233e5ee2a96157.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 203, + 664, + 403, + 677 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 678, + 409, + 691 + ], + "lines": [ + { + "bbox": [ + 106, + 678, + 407, + 692 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 268, + 692 + ], + "score": 1.0, + "content": "then we can create transformed features", + "type": "text" + }, + { + "bbox": [ + 268, + 679, + 367, + 691 + ], + "score": 0.89, + "content": "\\phi _ { h } ^ { \\prime } ( s , a ) = Z _ { h } \\cdot \\phi _ { h } ( s , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 678, + 407, + 692 + ], + "score": 1.0, + "content": ", and then", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36, + "bbox_fs": [ + 106, + 678, + 407, + 692 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 179, + 693, + 430, + 708 + ], + "lines": [ + { + "bbox": [ + 179, + 693, + 430, + 708 + ], + "spans": [ + { + "bbox": [ + 179, + 693, + 430, + 708 + ], + "score": 0.87, + "content": "\\mathbb { E } _ { \\mu , h } \\left[ \\phi _ { h } ^ { \\prime } ( s , a ) \\phi _ { h } ^ { \\prime } ( s , a ) ^ { \\top } \\right] = \\Lambda = \\mathrm { d i a g } [ \\lambda _ { 1 } , \\lambda _ { 2 } , . . . , \\lambda _ { d ^ { \\prime } } , 0 , . . . , 0 ] .", + "type": "interline_equation", + "image_path": "f3741c4b2a0a4e1c0b8293b70723c3bfd849a91f12022d51d21ca81b5c178447.jpg" + } + ] + } + ], + "index": 37, + "virtual_lines": [ + { + "bbox": [ + 179, + 693, + 430, + 708 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 709, + 507, + 733 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 317, + 722 + ], + "score": 1.0, + "content": "Then we can do learning w.r.t. the truncated features", + "type": "text" + }, + { + "bbox": [ + 317, + 710, + 344, + 722 + ], + "score": 0.92, + "content": "\\phi _ { h } ^ { \\prime } | _ { 1 : d ^ { \\prime } }", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 709, + 442, + 722 + ], + "score": 1.0, + "content": "’s instead of the original", + "type": "text" + }, + { + "bbox": [ + 442, + 710, + 449, + 721 + ], + "score": 0.84, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 709, + 505, + 722 + ], + "score": 1.0, + "content": ". 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The rest of them are from Cauchy’s inequality. Similarly, for", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 445, + 216, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 216, + 458 + ], + "score": 1.0, + "content": "Theorem 3.3, we also have", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7 + }, + { + "type": "interline_equation", + "bbox": [ + 167, + 461, + 445, + 705 + ], + "lines": [ + { + "bbox": [ + 167, + 461, + 445, + 705 + ], + "spans": [ + { + "bbox": [ + 167, + 461, + 445, + 705 + ], + "score": 0.93, + "content": "\\begin{array} { r l } & { \\forall \\widehat { u } _ { \\xi } ^ { \\mu } = \\sqrt { \\frac { \\mu } { \\mu } \\sqrt { \\mu } \\frac { \\sqrt { \\mu } } { \\mu } \\left\\{ \\Gamma ^ { 2 } + \\widehat { u } _ { \\xi } ^ { \\mu } \\left[ \\Gamma ^ { 2 } + \\widehat { u } _ { \\xi } ^ { \\mu } - \\widehat { u } _ { \\xi } ^ { \\mu } \\right] \\right\\} } } \\\\ & = \\sqrt { \\widehat { u } _ { \\xi } ^ { \\mu } - \\widehat { u } _ { \\xi } ^ { \\mu } } \\sqrt { \\Gamma ^ 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+ "score": 1.0, + "content": "is a deterministic policy as it always exists in tabular MDP)", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 122, + 338, + 138 + ], + "spans": [ + { + "bbox": [ + 106, + 123, + 245, + 137 + ], + "score": 0.93, + "content": "\\begin{array} { r } { C ^ { \\star } : = \\operatorname* { s u p } _ { h , s , a } { d _ { h } ^ { \\pi ^ { \\star } } ( s , a ) / d _ { h } ^ { \\mu } ( s , a ) } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 122, + 338, + 138 + ], + "score": 1.0, + "content": ", then for Theorem 3.2", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1, + "bbox_fs": [ + 105, + 100, + 506, + 138 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 115, + 133, + 493, + 427 + ], + "lines": [ + { + "bbox": [ + 115, + 133, + 493, + 427 + ], + "spans": [ + { + "bbox": [ + 115, + 133, + 493, + 427 + ], + "score": 0.74, + "content": "\\begin{array} { r l } & { \\mathcal { U } _ { \\ge } \\displaystyle \\sum _ { u = 1 } ^ { N } \\frac { \\mu } { \\mu } \\left. \\nabla ^ { \\mu } u ^ { \\mu } ( x ) \\frac { \\mu } { \\mu } \\nabla ^ { \\nu } \\cdot \\nabla ^ { \\mu } u ^ { \\nu } \\right. _ { \\perp } - \\mathcal { U } _ { \\ge } \\displaystyle \\sum _ { u = 1 } ^ { N } \\frac { \\mu } { \\mu } \\sum _ { u = 1 } ^ { N } \\mathcal { U } _ { \\ge } \\nu ( u ) \\hat { \\nu } ^ { \\mu } \\mathbb { I } _ { \\ge } \\mathbb { I } _ { \\ge u } ^ { \\nu } } \\\\ & { - \\mathcal { U } _ { \\ge } \\displaystyle \\sum _ { u = 1 } ^ { N } \\frac { \\mu } { \\mu } \\sum _ { u = 1 } ^ { N } \\mathcal { U } _ { \\ge } \\mathbb { I } _ { \\ge u } ^ { \\nu } \\mathbb { I } _ { \\ge u } ^ { \\nu } \\mathbb { I } _ { \\ge u } ^ { \\nu } \\mathbb { I } _ { \\ge u } ^ { \\nu } } \\\\ & { = - \\mathcal { U } _ { \\ge } \\displaystyle \\sum _ { u = 1 } ^ { N } \\mathbb { I } _ { \\ge u \\ne 1 } ^ { N } \\mathbb { I } _ { \\ge u } ^ { \\nu } \\mathbb { I } _ { \\ge u } ^ { \\nu } \\mathbb { I } 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Let", + "type": "text" + }, + { + "bbox": [ + 387, + 446, + 399, + 456 + ], + "score": 0.89, + "content": "\\Lambda _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 442, + 420, + 459 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 420, + 444, + 470, + 456 + ], + "score": 0.92, + "content": "\\Lambda _ { 2 } \\in \\mathbb { R } ^ { d \\times d }", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 442, + 507, + 459 + ], + "score": 1.0, + "content": "are two", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 456, + 259, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 259, + 468 + ], + "score": 1.0, + "content": "positive semi-definite matrices. Then:", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5 + }, + { + "type": "interline_equation", + "bbox": [ + 203, + 467, + 407, + 483 + ], + "lines": [ + { + "bbox": [ + 203, + 467, + 407, + 483 + ], + "spans": [ + { + "bbox": [ + 203, + 467, + 407, + 483 + ], + "score": 0.83, + "content": "\\left\\| \\Lambda _ { 1 } ^ { - 1 } \\right\\| \\leq \\left\\| \\Lambda _ { 2 } ^ { - 1 } \\right\\| + \\left\\| \\Lambda _ { 1 } ^ { - 1 } \\right\\| \\cdot \\left\\| \\Lambda _ { 2 } ^ { - 1 } \\right\\| \\cdot \\left\\| \\Lambda _ { 1 } - \\Lambda _ { 2 } \\right\\|", + "type": "interline_equation", + "image_path": "1f487bf7793819c8985870167abb327e7ea84fbbf1a0c59906a3cd47813a258e.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 203, + 467, + 407, + 483 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 482, + 124, + 493 + ], + "lines": [ + { + "bbox": [ + 105, + 483, + 125, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 125, + 493 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "interline_equation", + "bbox": [ + 171, + 489, + 439, + 518 + ], + "lines": [ + { + "bbox": [ + 171, + 489, + 439, + 518 + ], + "spans": [ + { + "bbox": [ + 171, + 489, + 439, + 518 + ], + "score": 0.89, + "content": "\\left\\| \\phi \\right\\| _ { \\Lambda _ { 1 } ^ { - 1 } } \\leq \\left[ 1 + \\sqrt { \\left\\| \\Lambda _ { 2 } ^ { - 1 } \\right\\| \\left\\| \\Lambda _ { 2 } \\right\\| \\cdot \\left\\| \\Lambda _ { 1 } ^ { - 1 } \\right\\| \\cdot \\left\\| \\Lambda _ { 1 } - \\Lambda _ { 2 } \\right\\| } \\right] \\cdot \\left\\| \\phi \\right\\| _ { \\Lambda _ { 2 } ^ { - 1 } } .", + "type": "interline_equation", + "image_path": "94f1e095fee85caffb4e372ca330e4290ae863ac0f82a3e8a367aa09e4aaa5ed.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 171, + 489, + 439, + 498.6666666666667 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 171, + 498.6666666666667, + 439, + 508.33333333333337 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 171, + 508.33333333333337, + 439, + 518.0 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 517, + 167, + 529 + ], + "lines": [ + { + "bbox": [ + 104, + 516, + 164, + 529 + ], + "spans": [ + { + "bbox": [ + 104, + 516, + 133, + 529 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 134, + 517, + 164, + 529 + ], + "score": 0.89, + "content": "\\phi \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 541, + 225, + 552 + ], + "lines": [ + { + "bbox": [ + 106, + 540, + 225, + 554 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 225, + 554 + ], + "score": 1.0, + "content": "Proof. For the first part, note", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "interline_equation", + "bbox": [ + 153, + 552, + 456, + 568 + ], + "lines": [ + { + "bbox": [ + 153, + 552, + 456, + 568 + ], + "spans": [ + { + "bbox": [ + 153, + 552, + 456, + 568 + ], + "score": 0.83, + "content": "\\left\\| \\Lambda _ { 1 } ^ { - 1 } \\right\\| \\leq \\left\\| \\Lambda _ { 2 } ^ { - 1 } \\right\\| + \\left\\| \\Lambda _ { 1 } ^ { - 1 } - \\Lambda _ { 2 } ^ { - 1 } \\right\\| \\leq \\left\\| \\Lambda _ { 2 } ^ { - 1 } \\right\\| + \\left\\| \\Lambda _ { 2 } ^ { - 1 } \\right\\| \\left\\| \\Lambda _ { 1 } - \\Lambda _ { 2 } \\right\\| \\left\\| \\Lambda _ { 1 } ^ { - 1 } \\right\\|", + "type": "interline_equation", + "image_path": "a08860c6f1ba7a119921ae9849fb7c91ac5b465f5c8c079f2704476b0a95601e.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 153, + 552, + 456, + 568 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 568, + 186, + 578 + ], + "lines": [ + { + "bbox": [ + 105, + 565, + 188, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 188, + 581 + ], + "score": 1.0, + "content": "For the second one,", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 593, + 512, + 702 + ], + "lines": [ + { + "bbox": [ + 111, + 593, + 512, + 702 + ], + "spans": [ + { + "bbox": [ + 111, + 593, + 512, + 702 + ], + "score": 0.93, + "content": "\\begin{array} { r l } & { \\left\\| \\phi \\right\\| _ { \\Lambda _ { 1 } ^ { - 1 } } = \\sqrt { \\phi ^ { \\top } \\Lambda _ { 1 } ^ { - 1 } \\phi } = \\sqrt { \\phi ^ { \\top } \\left( \\Lambda _ { 1 } ^ { - 1 } - \\Lambda _ { 2 } ^ { - 1 } \\right) \\phi + \\phi ^ { \\top } \\Lambda _ { 2 } ^ { - 1 } \\phi } } \\\\ & { = \\sqrt { \\phi ^ { \\top } \\Lambda _ { 2 } ^ { - 1 / 2 } \\left( \\Lambda _ { 2 } ^ { 1 / 2 } \\Lambda _ { 1 } ^ { - 1 } \\Lambda _ { 2 } ^ { 1 / 2 } - I + I \\right) \\Lambda _ { 2 } ^ { - 1 / 2 } \\phi } \\leq \\sqrt { \\| \\phi \\| _ { \\Lambda _ { 2 } ^ { - 1 } } \\cdot \\left( 1 + \\left\\| \\Lambda _ { 2 } ^ { 1 / 2 } \\Lambda _ { 1 } ^ { - 1 } \\Lambda _ { 2 } ^ { 1 / 2 } - I \\right\\| \\right) \\| \\phi \\| _ { \\Lambda _ { 2 } ^ { - 1 } } } } \\\\ & { \\leq \\left( 1 + \\left\\| \\Lambda _ { 2 } ^ { 1 / 2 } \\Lambda _ { 1 } ^ { - 1 } \\Lambda _ { 2 } ^ { 1 / 2 } - I \\right\\| ^ { 1 / 2 } \\right) \\cdot \\left\\| \\phi \\right\\| _ { \\Lambda _ { 2 } ^ { - 1 } } = \\left( 1 + \\left\\| \\Lambda _ { 2 } ^ { 1 / 2 } \\Lambda _ { 1 } ^ { - 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Let", + "type": "text" + }, + { + "bbox": [ + 472, + 106, + 482, + 116 + ], + "score": 0.79, + "content": "z _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 104, + 486, + 118 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 487, + 106, + 505, + 117 + ], + "score": 0.75, + "content": "k =", + "type": "inline_equation" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 115, + 504, + 129 + ], + "spans": [ + { + "bbox": [ + 106, + 117, + 144, + 128 + ], + "score": 0.85, + "content": "1 , \\ldots , K", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 115, + 285, + 129 + ], + "score": 1.0, + "content": "be independent random vectors in", + "type": "text" + }, + { + "bbox": [ + 285, + 116, + 298, + 127 + ], + "score": 0.87, + "content": "\\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 115, + 332, + 129 + ], + "score": 1.0, + "content": ", and let", + "type": "text" + }, + { + "bbox": [ + 333, + 117, + 343, + 126 + ], + "score": 0.71, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 115, + 443, + 129 + ], + "score": 1.0, + "content": "be a mapping that maps", + "type": "text" + }, + { + "bbox": [ + 444, + 117, + 454, + 126 + ], + "score": 0.8, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 115, + 497, + 129 + ], + "score": 1.0, + "content": "vectors to", + "type": "text" + }, + { + "bbox": [ + 498, + 119, + 504, + 126 + ], + "score": 0.54, + "content": "a", + "type": "inline_equation" + } + ], + "index": 2 + }, + { + "bbox": [ + 107, + 126, + 506, + 141 + ], + "spans": [ + { + "bbox": [ + 107, + 128, + 130, + 137 + ], + "score": 0.88, + "content": "d \\times d", + "type": "inline_equation" + }, + { + "bbox": [ + 130, + 126, + 438, + 141 + ], + "score": 1.0, + "content": "symmetric matrix. 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Let", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12, + "bbox_fs": [ + 104, + 258, + 507, + 292 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 217, + 287, + 393, + 325 + ], + "lines": [ + { + "bbox": [ + 217, + 287, + 393, + 325 + ], + "spans": [ + { + "bbox": [ + 217, + 287, + 393, + 325 + ], + "score": 0.94, + "content": "\\left\\| \\sum _ { s = 1 } ^ { t } x _ { s } \\eta _ { s } \\right\\| _ { \\Lambda _ { t } ^ { - 1 } } ^ { 2 } \\leq 8 R ^ { 2 } \\cdot \\frac { d } { 2 } \\log \\left( \\frac { \\lambda + t L } { \\lambda \\delta } \\right) .", + "type": "interline_equation", + "image_path": "bd4f9d67a95e63784fa2f0b8e402e452bbb2c83a6522a3507ae0cfc8784a5452.jpg" + } + ] + } + ], + "index": 13.5, + "virtual_lines": [ + { + "bbox": [ + 217, + 287, + 393, + 306.0 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 217, + 306.0, + 393, + 325.0 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 325, + 506, + 358 + ], + "lines": [ + { + "bbox": [ + 104, + 323, + 504, + 340 + ], + "spans": [ + { + "bbox": [ + 104, + 323, + 471, + 340 + ], + "score": 1.0, + "content": "Lemma H.3 (Bernstein inequality for self-normalized martingales (Zhou et al., 2021a)). Let", + "type": "text" + }, + { + "bbox": [ + 471, + 325, + 504, + 338 + ], + "score": 0.92, + "content": "\\{ \\eta _ { t } \\} _ { t = 1 } ^ { \\infty }", + "type": "inline_equation" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 335, + 507, + 350 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 277, + 350 + ], + "score": 1.0, + "content": "be a real-valued stochastic process. Let", + "type": "text" + }, + { + "bbox": [ + 277, + 336, + 312, + 348 + ], + "score": 0.92, + "content": "\\{ \\mathcal { F } _ { t } \\} _ { t = 0 } ^ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 335, + 418, + 350 + ], + "score": 1.0, + "content": "be a filtration, such that", + "type": "text" + }, + { + "bbox": [ + 419, + 338, + 428, + 348 + ], + "score": 0.84, + "content": "\\eta _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 335, + 440, + 350 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 441, + 337, + 452, + 347 + ], + "score": 0.89, + "content": "\\mathcal { F } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 335, + 507, + 350 + ], + "score": 1.0, + "content": "-measurable.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 347, + 203, + 360 + ], + "spans": [ + { + "bbox": [ + 104, + 347, + 138, + 360 + ], + "score": 1.0, + "content": "Assume", + "type": "text" + }, + { + "bbox": [ + 139, + 349, + 148, + 359 + ], + "score": 0.81, + "content": "\\eta _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 347, + 203, + 360 + ], + "score": 1.0, + "content": "also satisfies", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16, + "bbox_fs": [ + 104, + 323, + 507, + 360 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 206, + 358, + 405, + 374 + ], + "lines": [ + { + "bbox": [ + 206, + 358, + 405, + 374 + ], + "spans": [ + { + "bbox": [ + 206, + 358, + 405, + 374 + ], + "score": 0.88, + "content": "| \\eta _ { t } | \\leq R , \\mathbb { E } \\left[ \\eta _ { t } \\mid \\mathcal { F } _ { t - 1 } \\right] = 0 , \\mathbb { E } \\left[ \\eta _ { t } ^ { 2 } \\mid \\mathcal { F } _ { t - 1 } \\right] \\leq \\sigma ^ { 2 } .", + "type": "interline_equation", + "image_path": "7a7bbe014b2ee9b6c8b953df095490e878534e3fac493e81f1e89ce7a64dbd66.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 206, + 358, + 405, + 374 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 380, + 504, + 406 + ], + "lines": [ + { + "bbox": [ + 105, + 379, + 506, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 122, + 395 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 123, + 381, + 155, + 393 + ], + "score": 0.91, + "content": "\\{ x _ { t } \\} _ { t = 1 } ^ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 379, + 182, + 395 + ], + "score": 1.0, + "content": "be an", + "type": "text" + }, + { + "bbox": [ + 182, + 380, + 195, + 391 + ], + "score": 0.89, + "content": "\\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 379, + 330, + 395 + ], + "score": 1.0, + "content": "-valued stochastic process where", + "type": "text" + }, + { + "bbox": [ + 331, + 382, + 342, + 392 + ], + "score": 0.86, + "content": "x _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 379, + 352, + 395 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 352, + 381, + 374, + 393 + ], + "score": 0.91, + "content": "\\mathcal { F } _ { t - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 379, + 443, + 395 + ], + "score": 1.0, + "content": "measurable and", + "type": "text" + }, + { + "bbox": [ + 444, + 381, + 485, + 393 + ], + "score": 0.91, + "content": "\\| x _ { t } \\| \\leq L", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 379, + 506, + 395 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 387, + 442, + 411 + ], + "spans": [ + { + "bbox": [ + 106, + 393, + 206, + 407 + ], + "score": 0.89, + "content": "\\begin{array} { r } { \\Lambda _ { t } = \\lambda I _ { d } + \\sum _ { s = 1 } ^ { t } x _ { s } x _ { s } ^ { \\top } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 387, + 264, + 411 + ], + "score": 1.0, + "content": ". Then for any", + "type": "text" + }, + { + "bbox": [ + 264, + 394, + 289, + 405 + ], + "score": 0.89, + "content": "\\delta > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 387, + 358, + 411 + ], + "score": 1.0, + "content": ", with probability", + "type": "text" + }, + { + "bbox": [ + 359, + 394, + 382, + 405 + ], + "score": 0.81, + "content": "1 - \\delta _ { : }", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 387, + 412, + 411 + ], + "score": 1.0, + "content": ", for all", + "type": "text" + }, + { + "bbox": [ + 412, + 394, + 435, + 404 + ], + "score": 0.9, + "content": "t > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 387, + 442, + 411 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19.5, + "bbox_fs": [ + 105, + 379, + 506, + 411 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 157, + 407, + 452, + 444 + ], + "lines": [ + { + "bbox": [ + 157, + 407, + 452, + 444 + ], + "spans": [ + { + "bbox": [ + 157, + 407, + 452, + 444 + ], + "score": 0.94, + "content": "\\left\\| \\sum _ { s = 1 } ^ { t } \\mathbf { x } _ { s } \\eta _ { s } \\right\\| _ { \\Lambda _ { t } ^ { - 1 } } \\leq 8 \\sigma \\sqrt { d \\log \\left( 1 + \\frac { t L ^ { 2 } } { \\lambda d } \\right) \\cdot \\log \\left( \\frac { 4 t ^ { 2 } } { \\delta } \\right) } + 4 R \\log \\left( \\frac { 4 t ^ { 2 } } { \\delta } \\right)", + "type": "interline_equation", + "image_path": "da49e6c0d0b0c59d782ff1094cc677692fc85abe0900b483f4811e6b9db72872.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 157, + 407, + 452, + 419.3333333333333 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 157, + 419.3333333333333, + 452, + 431.66666666666663 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 157, + 431.66666666666663, + 452, + 443.99999999999994 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 445, + 505, + 467 + ], + "lines": [ + { + "bbox": [ + 105, + 442, + 507, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 387, + 459 + ], + "score": 1.0, + "content": "Lemma H.4 (Converting the variance under the matrix norm). Let", + "type": "text" + }, + { + "bbox": [ + 387, + 446, + 399, + 456 + ], + "score": 0.89, + "content": "\\Lambda _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 442, + 420, + 459 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 420, + 444, + 470, + 456 + ], + "score": 0.92, + "content": "\\Lambda _ { 2 } \\in \\mathbb { R } ^ { d \\times d }", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 442, + 507, + 459 + ], + "score": 1.0, + "content": "are two", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 456, + 259, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 259, + 468 + ], + "score": 1.0, + "content": "positive semi-definite matrices. Then:", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 442, + 507, + 468 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 203, + 467, + 407, + 483 + ], + "lines": [ + { + "bbox": [ + 203, + 467, + 407, + 483 + ], + "spans": [ + { + "bbox": [ + 203, + 467, + 407, + 483 + ], + "score": 0.83, + "content": "\\left\\| \\Lambda _ { 1 } ^ { - 1 } \\right\\| \\leq \\left\\| \\Lambda _ { 2 } ^ { - 1 } \\right\\| + \\left\\| \\Lambda _ { 1 } ^ { - 1 } \\right\\| \\cdot \\left\\| \\Lambda _ { 2 } ^ { - 1 } \\right\\| \\cdot \\left\\| \\Lambda _ { 1 } - \\Lambda _ { 2 } \\right\\|", + "type": "interline_equation", + "image_path": "1f487bf7793819c8985870167abb327e7ea84fbbf1a0c59906a3cd47813a258e.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 203, + 467, + 407, + 483 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 482, + 124, + 493 + ], + "lines": [ + { + "bbox": [ + 105, + 483, + 125, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 125, + 493 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 483, + 125, + 493 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 171, + 489, + 439, + 518 + ], + "lines": [ + { + "bbox": [ + 171, + 489, + 439, + 518 + ], + "spans": [ + { + "bbox": [ + 171, + 489, + 439, + 518 + ], + "score": 0.89, + "content": "\\left\\| \\phi \\right\\| _ { \\Lambda _ { 1 } ^ { - 1 } } \\leq \\left[ 1 + \\sqrt { \\left\\| \\Lambda _ { 2 } ^ { - 1 } \\right\\| \\left\\| \\Lambda _ { 2 } \\right\\| \\cdot \\left\\| \\Lambda _ { 1 } ^ { - 1 } \\right\\| \\cdot \\left\\| \\Lambda _ { 1 } - \\Lambda _ { 2 } \\right\\| } \\right] \\cdot \\left\\| \\phi \\right\\| _ { \\Lambda _ { 2 } ^ { - 1 } } .", + "type": "interline_equation", + "image_path": "94f1e095fee85caffb4e372ca330e4290ae863ac0f82a3e8a367aa09e4aaa5ed.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 171, + 489, + 439, + 498.6666666666667 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 171, + 498.6666666666667, + 439, + 508.33333333333337 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 171, + 508.33333333333337, + 439, + 518.0 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 517, + 167, + 529 + ], + "lines": [ + { + "bbox": [ + 104, + 516, + 164, + 529 + ], + "spans": [ + { + "bbox": [ + 104, + 516, + 133, + 529 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 134, + 517, + 164, + 529 + ], + "score": 0.89, + "content": "\\phi \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + } + ], + "index": 31 + } + ], + "index": 31, + "bbox_fs": [ + 104, + 516, + 164, + 529 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 541, + 225, + 552 + ], + "lines": [ + { + "bbox": [ + 106, + 540, + 225, + 554 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 225, + 554 + ], + "score": 1.0, + "content": "Proof. For the first part, note", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32, + "bbox_fs": [ + 106, + 540, + 225, + 554 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 153, + 552, + 456, + 568 + ], + "lines": [ + { + "bbox": [ + 153, + 552, + 456, + 568 + ], + "spans": [ + { + "bbox": [ + 153, + 552, + 456, + 568 + ], + "score": 0.83, + "content": "\\left\\| \\Lambda _ { 1 } ^ { - 1 } \\right\\| \\leq \\left\\| \\Lambda _ { 2 } ^ { - 1 } \\right\\| + \\left\\| \\Lambda _ { 1 } ^ { - 1 } - \\Lambda _ { 2 } ^ { - 1 } \\right\\| \\leq \\left\\| \\Lambda _ { 2 } ^ { - 1 } \\right\\| + \\left\\| \\Lambda _ { 2 } ^ { - 1 } \\right\\| \\left\\| \\Lambda _ { 1 } - \\Lambda _ { 2 } \\right\\| \\left\\| \\Lambda _ { 1 } ^ { - 1 } \\right\\|", + "type": "interline_equation", + "image_path": "a08860c6f1ba7a119921ae9849fb7c91ac5b465f5c8c079f2704476b0a95601e.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 153, + 552, + 456, + 568 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 568, + 186, + 578 + ], + "lines": [ + { + "bbox": [ + 105, + 565, + 188, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 188, + 581 + ], + "score": 1.0, + "content": "For the second one,", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34, + "bbox_fs": [ + 105, + 565, + 188, + 581 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 593, + 512, + 702 + ], + "lines": [ + { + "bbox": [ + 111, + 593, + 512, + 702 + ], + "spans": [ + { + "bbox": [ + 111, + 593, + 512, + 702 + ], + "score": 0.93, + "content": "\\begin{array} { r l } & { \\left\\| \\phi \\right\\| _ { \\Lambda _ { 1 } ^ { - 1 } } = \\sqrt { \\phi ^ { \\top } \\Lambda _ { 1 } ^ { - 1 } \\phi } = \\sqrt { \\phi ^ { \\top } \\left( \\Lambda _ { 1 } ^ { - 1 } - \\Lambda _ { 2 } ^ { - 1 } \\right) \\phi + \\phi ^ { \\top } \\Lambda _ { 2 } ^ { - 1 } \\phi } } \\\\ & { = \\sqrt { \\phi ^ { \\top } \\Lambda _ { 2 } ^ { - 1 / 2 } \\left( \\Lambda _ { 2 } ^ { 1 / 2 } \\Lambda _ { 1 } ^ { - 1 } \\Lambda _ { 2 } ^ { 1 / 2 } - I + I \\right) \\Lambda _ { 2 } ^ { - 1 / 2 } \\phi } \\leq \\sqrt { \\| \\phi \\| _ { \\Lambda _ { 2 } ^ { - 1 } } \\cdot \\left( 1 + \\left\\| \\Lambda _ { 2 } ^ { 1 / 2 } \\Lambda _ { 1 } ^ { - 1 } \\Lambda _ { 2 } ^ { 1 / 2 } - I \\right\\| \\right) \\| \\phi \\| _ { \\Lambda _ { 2 } ^ { - 1 } } } } \\\\ & { \\leq \\left( 1 + \\left\\| \\Lambda _ { 2 } ^ { 1 / 2 } \\Lambda _ { 1 } ^ { - 1 } \\Lambda _ { 2 } ^ { 1 / 2 } - I \\right\\| ^ { 1 / 2 } \\right) \\cdot \\left\\| \\phi \\right\\| _ { \\Lambda _ { 2 } ^ { - 1 } } = \\left( 1 + \\left\\| \\Lambda _ { 2 } ^ { 1 / 2 } \\Lambda _ { 1 } ^ { - 1 } \\left( \\Lambda _ { 2 } - \\Lambda _ { 1 } \\right) \\Lambda _ { 2 } ^ { - 1 } \\Lambda _ { 2 } ^ { 1 / 2 } \\right\\| ^ { 1 / 2 } \\right) \\cdot \\left\\| \\phi \\right\\| _ { \\Lambda _ { 2 } ^ { - 1 } } } \\\\ & { \\leq \\left( 1 + \\sqrt { \\| \\Lambda _ { 2 } \\| \\left\\| \\Lambda _ { 1 } ^ { - 1 } \\right\\| \\left\\| \\Lambda _ { 2 } ^ { - 1 } \\right\\| \\left\\| \\Lambda _ { 1 } ^ { - 1 } - \\Lambda _ { 2 } \\right\\| } \\right) \\cdot \\left\\| \\phi \\right\\| _ { \\Lambda _ { 2 } ^ { - 1 } } } \\end{array}", + "type": "interline_equation", + "image_path": "278f938886d95b8d6a970637e758173e4d6caebe9c15d31c9e1e7091e56a29a9.jpg" + } + ] + } + ], + "index": 36, + "virtual_lines": [ + { + "bbox": [ + 111, + 593, + 512, + 629.3333333333334 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 111, + 629.3333333333334, + 512, + 665.6666666666667 + ], + "spans": [], + "index": 36 + }, + { + "bbox": [ + 111, + 665.6666666666667, + 512, + 702.0000000000001 + ], + "spans": [], + "index": 37 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 81, + 505, + 119 + ], + "lines": [ + { + "bbox": [ + 105, + 79, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 320, + 95 + ], + "score": 1.0, + "content": "Lemma H.5 (Lemma H.4 of Min et al. (2021)). let", + "type": "text" + }, + { + "bbox": [ + 320, + 82, + 392, + 93 + ], + "score": 0.91, + "content": "\\phi : \\mathcal { S } \\times \\mathcal { A } \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 79, + 428, + 95 + ], + "score": 1.0, + "content": "satisfies", + "type": "text" + }, + { + "bbox": [ + 429, + 82, + 489, + 95 + ], + "score": 0.89, + "content": "\\| \\phi ( s , a ) \\| \\leq C", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 79, + 505, + 95 + ], + "score": 1.0, + "content": "for", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 120, + 88, + 509, + 118 + ], + "spans": [ + { + "bbox": [ + 120, + 96, + 177, + 107 + ], + "score": 0.89, + "content": "s , a \\in \\mathcal S \\times \\mathcal A", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 88, + 217, + 118 + ], + "score": 1.0, + "content": ". For any", + "type": "text" + }, + { + "bbox": [ + 217, + 96, + 277, + 107 + ], + "score": 0.91, + "content": "K > 0 , \\lambda > 0 ,", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 88, + 307, + 118 + ], + "score": 1.0, + "content": ", define", + "type": "text" + }, + { + "bbox": [ + 308, + 94, + 476, + 109 + ], + "score": 0.89, + "content": "\\begin{array} { r } { \\bar { G } _ { K } = \\sum _ { k = 1 } ^ { K } \\phi ( s _ { k } , a _ { k } ) \\phi ( s _ { k } , \\underline { { a } } _ { k } ) ^ { \\top } + \\lambda I _ { d } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 88, + 509, + 118 + ], + "score": 1.0, + "content": "where", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 107, + 429, + 119 + ], + "spans": [ + { + "bbox": [ + 106, + 107, + 139, + 119 + ], + "score": 0.89, + "content": "( s _ { k } , a _ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 109, + 313, + 116 + ], + "score": 0.35, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 108, + 429, + 117 + ], + "score": 0.84, + "content": "1 - \\delta", + "type": "inline_equation" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "interline_equation", + "bbox": [ + 209, + 124, + 401, + 155 + ], + "lines": [ + { + "bbox": [ + 209, + 124, + 401, + 155 + ], + "spans": [ + { + "bbox": [ + 209, + 124, + 401, + 155 + ], + "score": 0.92, + "content": "\\left\\| \\frac { { { { \\bar { G } } _ { K } } } } { K } - \\mathbb { E } _ { \\nu } \\left[ \\frac { { { { \\bar { G } } _ { K } } } } { K } \\right] \\right\\| \\le \\frac { 4 \\sqrt 2 C ^ { 2 } } { \\sqrt K } \\left( \\log \\frac { { 2 d } } { \\delta } \\right) ^ { 1 / 2 } .", + "type": "interline_equation", + "image_path": "82c9d4ef9825f2eae00ef9f22444d30d601b67065a1c9c0c21990661ed586735.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 209, + 124, + 401, + 155 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 167, + 506, + 204 + ], + "lines": [ + { + "bbox": [ + 106, + 167, + 506, + 181 + ], + "spans": [ + { + "bbox": [ + 106, + 167, + 173, + 181 + ], + "score": 1.0, + "content": "Proof of Lemma", + "type": "text" + }, + { + "bbox": [ + 173, + 169, + 189, + 178 + ], + "score": 0.32, + "content": "H . 5", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 167, + 440, + 181 + ], + "score": 1.0, + "content": ". For completeness, we provide the proof of Lemma H.5. Let", + "type": "text" + }, + { + "bbox": [ + 440, + 168, + 503, + 180 + ], + "score": 0.93, + "content": "x _ { k } = \\phi ( s _ { k } , a _ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 167, + 506, + 181 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 179, + 505, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 138, + 194 + ], + "score": 1.0, + "content": "Denote", + "type": "text" + }, + { + "bbox": [ + 138, + 180, + 152, + 192 + ], + "score": 0.9, + "content": "\\widetilde { \\Sigma } _ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 180, + 312, + 194 + ], + "score": 1.0, + "content": "as the matrix obtained by replacing the", + "type": "text" + }, + { + "bbox": [ + 313, + 182, + 320, + 191 + ], + "score": 0.7, + "content": "\\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 180, + 359, + 194 + ], + "score": 1.0, + "content": "-th vector", + "type": "text" + }, + { + "bbox": [ + 360, + 182, + 371, + 192 + ], + "score": 0.81, + "content": "x _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 180, + 383, + 194 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 383, + 179, + 397, + 192 + ], + "score": 0.91, + "content": "\\widehat { \\Sigma } _ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 180, + 410, + 194 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 411, + 181, + 422, + 192 + ], + "score": 0.88, + "content": "\\widetilde { x } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 180, + 505, + 194 + ], + "score": 1.0, + "content": "and leaving the rest", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 107, + 192, + 239, + 205 + ], + "spans": [ + { + "bbox": [ + 107, + 193, + 134, + 203 + ], + "score": 0.88, + "content": "K - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 192, + 239, + 205 + ], + "score": 1.0, + "content": "vectors unchanged. Then", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5 + }, + { + "type": "interline_equation", + "bbox": [ + 115, + 209, + 495, + 245 + ], + "lines": [ + { + "bbox": [ + 115, + 209, + 495, + 245 + ], + "spans": [ + { + "bbox": [ + 115, + 209, + 495, + 245 + ], + "score": 0.92, + "content": "\\left( \\frac { \\widehat \\Sigma _ { h } } { K } - \\frac { \\widetilde \\Sigma _ { h } } { K } \\right) ^ { 2 } = \\left( \\frac { x _ { k } x _ { k } ^ { \\top } - \\widetilde x _ { k } \\widetilde x _ { k } ^ { \\top } } { K } \\right) \\preceq \\frac { 1 } { K ^ { 2 } } \\left( 2 x _ { k } x _ { k } ^ { \\top } x _ { k } x _ { k } ^ { \\top } + 2 \\widetilde x _ { k } \\widetilde x _ { k } ^ { \\top } \\widetilde x _ { k } \\widetilde x _ { k } ^ { \\top } \\right) \\preceq \\frac { 4 C ^ { 4 } } { K ^ { 2 } } I _ { d } : = A _ { k } ^ { 2 } .", + "type": "interline_equation", + "image_path": "f4fe0083ad0a6a5e2ab377728942168d2e1e466dfe1a2ef05637980295e2b9fe.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 115, + 209, + 495, + 221.0 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 115, + 221.0, + 495, + 233.0 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 115, + 233.0, + 495, + 245.0 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 251, + 370, + 271 + ], + "lines": [ + { + "bbox": [ + 101, + 246, + 375, + 275 + ], + "spans": [ + { + "bbox": [ + 101, + 246, + 153, + 275 + ], + "score": 1.0, + "content": "Notice that", + "type": "text" + }, + { + "bbox": [ + 153, + 250, + 227, + 272 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\left\\| \\sum _ { k } ^ { K } A _ { k } ^ { 2 } \\right\\| = \\frac { 4 C ^ { 4 } } { K } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 246, + 375, + 275 + ], + "score": 1.0, + "content": ", by Lemma H.1 we have the result.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 106, + 294, + 506, + 333 + ], + "lines": [ + { + "bbox": [ + 105, + 293, + 510, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 326, + 308 + ], + "score": 1.0, + "content": "Lemma H.6 (Lemma H.5. of Min et al. (2021)). Let", + "type": "text" + }, + { + "bbox": [ + 107, + 308, + 149, + 321 + ], + "score": 0.92, + "content": "\\| \\phi \\| _ { 2 } \\leq C", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 299, + 182, + 327 + ], + "score": 1.0, + "content": ". Define", + "type": "text" + }, + { + "bbox": [ + 182, + 306, + 348, + 321 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\bar { G } _ { K } = \\sum _ { k = 1 } ^ { K } \\phi ( s _ { k } , a _ { k } ) \\phi ( s _ { k } , a _ { k } ) ^ { \\top } + \\lambda I _ { d } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 294, + 397, + 307 + ], + "score": 0.91, + "content": "\\phi : \\mathcal { S } \\times \\mathcal { A } \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 299, + 376, + 327 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 376, + 308, + 409, + 320 + ], + "score": 0.88, + "content": "( s _ { k } , a _ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 293, + 506, + 308 + ], + "score": 1.0, + "content": "be a bounded function s.t.", + "type": "text" + }, + { + "bbox": [ + 409, + 299, + 510, + 327 + ], + "score": 1.0, + "content": "’s are i.i.d samples from", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 319, + 462, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 319, + 178, + 334 + ], + "score": 1.0, + "content": "some distribution", + "type": "text" + }, + { + "bbox": [ + 178, + 323, + 185, + 330 + ], + "score": 0.6, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 319, + 205, + 334 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + }, + { + "bbox": [ + 205, + 321, + 306, + 333 + ], + "score": 0.91, + "content": "G = \\mathbb { E } _ { \\nu } [ \\phi ( s , a ) \\phi ( s , a ) ^ { \\top } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 319, + 365, + 334 + ], + "score": 1.0, + "content": ". Then for any", + "type": "text" + }, + { + "bbox": [ + 365, + 321, + 405, + 333 + ], + "score": 0.89, + "content": "\\delta \\in ( 0 , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 319, + 417, + 334 + ], + "score": 1.0, + "content": ", if", + "type": "text" + }, + { + "bbox": [ + 417, + 321, + 427, + 331 + ], + "score": 0.69, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 319, + 462, + 334 + ], + "score": 1.0, + "content": "satisfies", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5 + }, + { + "type": "interline_equation", + "bbox": [ + 194, + 338, + 416, + 366 + ], + "lines": [ + { + "bbox": [ + 194, + 338, + 416, + 366 + ], + "spans": [ + { + "bbox": [ + 194, + 338, + 416, + 366 + ], + "score": 0.92, + "content": "K \\geq \\operatorname* { m a x } \\left\\{ 5 1 2 C ^ { 4 } \\left\\| \\mathbf { G } ^ { - 1 } \\right\\| ^ { 2 } \\log \\left( \\frac { 2 d } { \\delta } \\right) , 4 \\lambda \\left\\| \\mathbf { G } ^ { - 1 } \\right\\| \\right\\} .", + "type": "interline_equation", + "image_path": "57ce29ef2533968070df73fa05bf312de8ac10bc38f79d82783a479ff410cc79.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 194, + 338, + 416, + 366 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 372, + 424, + 385 + ], + "lines": [ + { + "bbox": [ + 105, + 371, + 425, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 226, + 385 + ], + "score": 1.0, + "content": "Then with probability at least", + "type": "text" + }, + { + "bbox": [ + 226, + 373, + 249, + 383 + ], + "score": 0.86, + "content": "1 - \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 371, + 374, + 385 + ], + "score": 1.0, + "content": ", it holds simultaneously for all", + "type": "text" + }, + { + "bbox": [ + 375, + 372, + 405, + 383 + ], + "score": 0.91, + "content": "u \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 371, + 425, + 385 + ], + "score": 1.0, + "content": "that", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "interline_equation", + "bbox": [ + 253, + 390, + 358, + 416 + ], + "lines": [ + { + "bbox": [ + 253, + 390, + 358, + 416 + ], + "spans": [ + { + "bbox": [ + 253, + 390, + 358, + 416 + ], + "score": 0.94, + "content": "\\| u \\| _ { \\bar { G } _ { K } ^ { - 1 } } \\leq \\frac { 2 } { \\sqrt { K } } \\| u \\| _ { G ^ { - 1 } } .", + "type": "interline_equation", + "image_path": "81d9ca9813a280ca04a49f7d1ffe4ec892f01ad64b50f7036474ac7715cfae63.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 253, + 390, + 358, + 416 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 421, + 506, + 463 + ], + "lines": [ + { + "bbox": [ + 104, + 419, + 507, + 437 + ], + "spans": [ + { + "bbox": [ + 104, + 419, + 430, + 437 + ], + "score": 1.0, + "content": "Lemma H.7 (Extended Value Difference (Section B.1 in Cai et al. (2020))). Let", + "type": "text" + }, + { + "bbox": [ + 430, + 421, + 486, + 435 + ], + "score": 0.93, + "content": "\\pi = \\{ \\pi _ { h } \\} _ { h = 1 } ^ { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 419, + 507, + 437 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 107, + 432, + 507, + 452 + ], + "spans": [ + { + "bbox": [ + 107, + 435, + 167, + 449 + ], + "score": 0.9, + "content": "\\pi ^ { \\prime } = \\{ \\pi _ { h } ^ { \\prime } \\} _ { h = 1 } ^ { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 432, + 305, + 452 + ], + "score": 1.0, + "content": "be two arbitrary policies and let", + "type": "text" + }, + { + "bbox": [ + 305, + 434, + 344, + 449 + ], + "score": 0.93, + "content": "\\{ \\widehat { Q } _ { h } \\} _ { h = 1 } ^ { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 432, + 400, + 452 + ], + "score": 1.0, + "content": "be any given", + "type": "text" + }, + { + "bbox": [ + 401, + 437, + 409, + 448 + ], + "score": 0.78, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 432, + 507, + 452 + ], + "score": 1.0, + "content": "-functions. Then define", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 447, + 362, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 449, + 227, + 462 + ], + "score": 0.9, + "content": "\\widehat { V } _ { h } ( s ) : = \\langle \\widehat { Q } _ { h } ( s , \\cdot ) , \\pi _ { h } ( \\cdot \\mid s ) \\rangle", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 447, + 255, + 464 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 255, + 450, + 279, + 461 + ], + "score": 0.88, + "content": "s \\in S", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 447, + 333, + 464 + ], + "score": 1.0, + "content": ". 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(2021)). let", + "type": "text" + }, + { + "bbox": [ + 320, + 82, + 392, + 93 + ], + "score": 0.91, + "content": "\\phi : \\mathcal { S } \\times \\mathcal { A } \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 79, + 428, + 95 + ], + "score": 1.0, + "content": "satisfies", + "type": "text" + }, + { + "bbox": [ + 429, + 82, + 489, + 95 + ], + "score": 0.89, + "content": "\\| \\phi ( s , a ) \\| \\leq C", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 79, + 505, + 95 + ], + "score": 1.0, + "content": "for", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 120, + 88, + 509, + 118 + ], + "spans": [ + { + "bbox": [ + 120, + 96, + 177, + 107 + ], + "score": 0.89, + "content": "s , a \\in \\mathcal S \\times \\mathcal A", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 88, + 217, + 118 + ], + "score": 1.0, + "content": ". 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Let", + "type": "text" + }, + { + "bbox": [ + 440, + 168, + 503, + 180 + ], + "score": 0.93, + "content": "x _ { k } = \\phi ( s _ { k } , a _ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 167, + 506, + 181 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 179, + 505, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 138, + 194 + ], + "score": 1.0, + "content": "Denote", + "type": "text" + }, + { + "bbox": [ + 138, + 180, + 152, + 192 + ], + "score": 0.9, + "content": "\\widetilde { \\Sigma } _ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 180, + 312, + 194 + ], + "score": 1.0, + "content": "as the matrix obtained by replacing the", + "type": "text" + }, + { + "bbox": [ + 313, + 182, + 320, + 191 + ], + "score": 0.7, + "content": "\\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 180, + 359, + 194 + ], + "score": 1.0, + "content": "-th vector", + "type": "text" + }, + { + "bbox": [ + 360, + 182, + 371, + 192 + ], + "score": 0.81, + "content": "x _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 180, + 383, + 194 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 383, + 179, + 397, + 192 + ], + "score": 0.91, + "content": "\\widehat { \\Sigma } _ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 180, + 410, + 194 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 411, + 181, + 422, + 192 + ], + "score": 0.88, + "content": "\\widetilde { x } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 180, + 505, + 194 + ], + "score": 1.0, + "content": "and leaving the rest", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 107, + 192, + 239, + 205 + ], + "spans": [ + { + "bbox": [ + 107, + 193, + 134, + 203 + ], + "score": 0.88, + "content": "K - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 192, + 239, + 205 + ], + "score": 1.0, + "content": "vectors unchanged. Then", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5, + "bbox_fs": [ + 105, + 167, + 506, + 205 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 115, + 209, + 495, + 245 + ], + "lines": [ + { + "bbox": [ + 115, + 209, + 495, + 245 + ], + "spans": [ + { + "bbox": [ + 115, + 209, + 495, + 245 + ], + "score": 0.92, + "content": "\\left( \\frac { \\widehat \\Sigma _ { h } } { K } - \\frac { \\widetilde \\Sigma _ { h } } { K } \\right) ^ { 2 } = \\left( \\frac { x _ { k } x _ { k } ^ { \\top } - \\widetilde x _ { k } \\widetilde x _ { k } ^ { \\top } } { K } \\right) \\preceq \\frac { 1 } { K ^ { 2 } } \\left( 2 x _ { k } x _ { k } ^ { \\top } x _ { k } x _ { k } ^ { \\top } + 2 \\widetilde x _ { k } \\widetilde x _ { k } ^ { \\top } \\widetilde x _ { k } \\widetilde x _ { k } ^ { \\top } \\right) \\preceq \\frac { 4 C ^ { 4 } } { K ^ { 2 } } I _ { d } : = A _ { k } ^ { 2 } .", + "type": "interline_equation", + "image_path": "f4fe0083ad0a6a5e2ab377728942168d2e1e466dfe1a2ef05637980295e2b9fe.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 115, + 209, + 495, + 221.0 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 115, + 221.0, + 495, + 233.0 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 115, + 233.0, + 495, + 245.0 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 251, + 370, + 271 + ], + "lines": [ + { + "bbox": [ + 101, + 246, + 375, + 275 + ], + "spans": [ + { + "bbox": [ + 101, + 246, + 153, + 275 + ], + "score": 1.0, + "content": "Notice that", + "type": "text" + }, + { + "bbox": [ + 153, + 250, + 227, + 272 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\left\\| \\sum _ { k } ^ { K } A _ { k } ^ { 2 } \\right\\| = \\frac { 4 C ^ { 4 } } { K } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 246, + 375, + 275 + ], + "score": 1.0, + "content": ", by Lemma H.1 we have the result.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10, + "bbox_fs": [ + 101, + 246, + 375, + 275 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 294, + 506, + 333 + ], + "lines": [ + { + "bbox": [ + 105, + 293, + 510, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 326, + 308 + ], + "score": 1.0, + "content": "Lemma H.6 (Lemma H.5. of Min et al. (2021)). Let", + "type": "text" + }, + { + "bbox": [ + 107, + 308, + 149, + 321 + ], + "score": 0.92, + "content": "\\| \\phi \\| _ { 2 } \\leq C", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 299, + 182, + 327 + ], + "score": 1.0, + "content": ". 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(2020))). Let", + "type": "text" + }, + { + "bbox": [ + 430, + 421, + 486, + 435 + ], + "score": 0.93, + "content": "\\pi = \\{ \\pi _ { h } \\} _ { h = 1 } ^ { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 419, + 507, + 437 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 107, + 432, + 507, + 452 + ], + "spans": [ + { + "bbox": [ + 107, + 435, + 167, + 449 + ], + "score": 0.9, + "content": "\\pi ^ { \\prime } = \\{ \\pi _ { h } ^ { \\prime } \\} _ { h = 1 } ^ { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 432, + 305, + 452 + ], + "score": 1.0, + "content": "be two arbitrary policies and let", + "type": "text" + }, + { + "bbox": [ + 305, + 434, + 344, + 449 + ], + "score": 0.93, + "content": "\\{ \\widehat { Q } _ { h } \\} _ { h = 1 } ^ { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 432, + 400, + 452 + ], + "score": 1.0, + "content": "be any given", + "type": "text" + }, + { + "bbox": [ + 401, + 437, + 409, + 448 + ], + "score": 0.78, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 432, + 507, + 452 + ], + "score": 1.0, + "content": "-functions. Then define", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 447, + 362, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 449, + 227, + 462 + ], + "score": 0.9, + "content": "\\widehat { V } _ { h } ( s ) : = \\langle \\widehat { Q } _ { h } ( s , \\cdot ) , \\pi _ { h } ( \\cdot \\mid s ) \\rangle", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 447, + 255, + 464 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 255, + 450, + 279, + 461 + ], + "score": 0.88, + "content": "s \\in S", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 447, + 333, + 464 + ], + "score": 1.0, + "content": ". 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Denote", + "type": "text" + }, + { + "bbox": [ + 168, + 575, + 249, + 590 + ], + "score": 0.93, + "content": "\\xi _ { h } = \\widehat { Q } _ { h } - \\mathcal { T } _ { h } \\widehat { V } _ { h + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 573, + 285, + 592 + ], + "score": 1.0, + "content": ". 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Let", + "type": "text" + }, + { + "bbox": [ + 184, + 704, + 245, + 720 + ], + "score": 0.92, + "content": "\\widehat { \\pi } = \\left\\{ \\widehat { \\pi } _ { h } \\right\\} _ { h = 1 } ^ { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 700, + 266, + 723 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 267, + 704, + 298, + 719 + ], + "score": 0.91, + "content": "\\widehat { Q } _ { h } ( \\cdot , \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 700, + 419, + 723 + ], + "score": 1.0, + "content": "be the arbitrary policy and", + "type": "text" + }, + { + "bbox": [ + 419, + 708, + 428, + 718 + ], + "score": 0.65, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 700, + 509, + 723 + ], + "score": 1.0, + "content": "-function and also", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 717, + 506, + 735 + ], + "spans": [ + { + "bbox": [ + 106, + 718, + 252, + 733 + ], + "score": 0.9, + "content": "\\widehat { V } _ { h } ( s ) = \\langle \\widehat { Q } _ { h } ( s , \\cdot ) , \\widehat { \\pi } _ { h } ( \\cdot | s ) \\rangle \\vee _ { s } ^ { \\cdot } \\in \\mathcal { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 717, + 275, + 735 + ], + "score": 1.0, + "content": ". and", + "type": "text" + }, + { + "bbox": [ + 276, + 718, + 434, + 733 + ], + "score": 0.92, + "content": "\\zeta _ { h } ( s , a ) : = ( \\mathscr { T } _ { h } \\widehat { V } _ { h + 1 } ) ( s , a ) - \\widehat { Q } _ { h } ( s , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 717, + 506, + 735 + ], + "score": 1.0, + "content": "(element-wisely)", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5, + "bbox_fs": [ + 102, + 700, + 509, + 735 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 372, + 95 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 372, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 326, + 97 + ], + "score": 1.0, + "content": "to be the Bellman update error. Then for any arbitrary", + "type": "text" + }, + { + "bbox": [ + 326, + 85, + 333, + 92 + ], + "score": 0.59, + "content": "\\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 80, + 372, + 97 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 146, + 106, + 465, + 177 + ], + "lines": [ + { + "bbox": [ + 146, + 106, + 465, + 177 + ], + "spans": [ + { + "bbox": [ + 146, + 106, + 465, + 177 + ], + "score": 0.94, + "content": "\\begin{array} { c } { { { \\displaystyle V _ { 1 } ^ { \\pi } ( s ) - V _ { 1 } ^ { \\widehat { \\pi } } ( s ) = \\sum _ { h = 1 } ^ { H } \\mathbb { E } _ { \\pi } \\left[ \\zeta _ { h } ( s _ { h } , a _ { h } ) \\mid s _ { 1 } = s \\right] - \\sum _ { h = 1 } ^ { H } \\mathbb { E } _ { \\widehat { \\pi } } \\left[ \\zeta _ { h } ( s _ { h } , a _ { h } ) \\mid s _ { 1 } = s \\right] } } } \\\\ { { { + \\displaystyle \\sum _ { h = 1 } ^ { H } \\mathbb { E } _ { \\pi } \\left[ \\left. \\widehat { Q } _ { h } \\left( s _ { h } , \\cdot \\right) , \\pi _ { h } \\left( \\cdot | s _ { h } \\right) - \\widehat { \\pi } _ { h } \\left( \\cdot | s _ { h } \\right) \\right. \\mid s _ { 1 } = x \\right] } } } \\end{array}", + "type": "interline_equation", + "image_path": "67417bc8f21fd1b43ef9a4a476ce6fa4232c33309bc8f685cf1e78494196e120.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 146, + 106, + 465, + 129.66666666666666 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 146, + 129.66666666666666, + 465, + 153.33333333333331 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 146, + 153.33333333333331, + 465, + 176.99999999999997 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 187, + 284, + 199 + ], + "lines": [ + { + "bbox": [ + 105, + 186, + 285, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 255, + 201 + ], + "score": 1.0, + "content": "where the expectation are taken over", + "type": "text" + }, + { + "bbox": [ + 255, + 190, + 281, + 199 + ], + "score": 0.89, + "content": "s _ { h } , a _ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 186, + 285, + 201 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 107, + 232, + 266, + 244 + ], + "lines": [ + { + "bbox": [ + 106, + 232, + 267, + 245 + ], + "spans": [ + { + "bbox": [ + 106, + 232, + 267, + 245 + ], + "score": 1.0, + "content": "Proof. Note the gap can be rewritten as", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "interline_equation", + "bbox": [ + 197, + 256, + 413, + 272 + ], + "lines": [ + { + "bbox": [ + 197, + 256, + 413, + 272 + ], + "spans": [ + { + "bbox": [ + 197, + 256, + 413, + 272 + ], + "score": 0.88, + "content": "\\begin{array} { r } { V _ { 1 } ^ { \\pi } ( s ) - V _ { 1 } ^ { \\widehat { \\pi } } ( s ) = V _ { 1 } ^ { \\pi } ( s ) - \\widehat { V } _ { 1 } ( s ) + \\widehat { V } _ { 1 } ( s ) - V _ { 1 } ^ { \\widehat { \\pi } } ( s ) . } \\end{array}", + "type": "interline_equation", + "image_path": "d0b157725d446ee734981fa192e2d1aaf3a9bee3d716c71d27ee54b4a38f644b.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 197, + 256, + 413, + 272 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 283, + 323, + 296 + ], + "lines": [ + { + "bbox": [ + 105, + 282, + 323, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 191, + 297 + ], + "score": 1.0, + "content": "By Lemma H.7 with", + "type": "text" + }, + { + "bbox": [ + 191, + 284, + 217, + 294 + ], + "score": 0.88, + "content": "\\pi = { \\widehat { \\pi } }", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 282, + 221, + 297 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 221, + 284, + 250, + 294 + ], + "score": 0.84, + "content": "\\pi ^ { \\prime } = \\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 282, + 323, + 297 + ], + "score": 1.0, + "content": ", we directly have", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 307, + 511, + 342 + ], + "lines": [ + { + "bbox": [ + 111, + 307, + 511, + 342 + ], + "spans": [ + { + "bbox": [ + 111, + 307, + 511, + 342 + ], + "score": 0.93, + "content": "\\gamma _ { 1 } ^ { \\pi } ( s ) - \\widehat { V } _ { 1 } ( s ) = \\sum _ { h = 1 } ^ { H } \\mathbb { E } _ { \\pi } \\left[ \\zeta _ { h } ( s _ { h } , a _ { h } ) \\mid s _ { 1 } = s \\right] + \\sum _ { h = 1 } ^ { H } \\mathbb { E } _ { \\pi } \\left[ \\langle \\widehat { Q } _ { h } \\left( s _ { h } , \\cdot \\right) , \\pi _ { h } \\left( \\cdot | s _ { h } \\right) - \\widehat { \\pi } _ { h } \\left( \\cdot | s _ { h } \\right) \\rangle \\mid s _ { 1 } = s \\right]", + "type": "interline_equation", + "image_path": "a3b17028ac9abc38cd618977334599ec09bdd3ee7aa6fb322892870650e381d9.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 111, + 307, + 511, + 318.6666666666667 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 111, + 318.6666666666667, + 511, + 330.33333333333337 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 111, + 330.33333333333337, + 511, + 342.00000000000006 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 350, + 368, + 363 + ], + "lines": [ + { + "bbox": [ + 105, + 348, + 368, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 247, + 364 + ], + "score": 1.0, + "content": "Next apply Lemma H.7 again with", + "type": "text" + }, + { + "bbox": [ + 247, + 351, + 296, + 361 + ], + "score": 0.91, + "content": "\\pi = \\pi ^ { \\prime } = \\widehat { \\pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 348, + 368, + 364 + ], + "score": 1.0, + "content": ", we directly have", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "interline_equation", + "bbox": [ + 203, + 374, + 407, + 409 + ], + "lines": [ + { + "bbox": [ + 203, + 374, + 407, + 409 + ], + "spans": [ + { + "bbox": [ + 203, + 374, + 407, + 409 + ], + "score": 0.93, + "content": "\\widehat { V } _ { 1 } ( s ) - V _ { 1 } ^ { \\widehat { \\pi } } ( s ) = - \\sum _ { h = 1 } ^ { H } \\mathbb { E } _ { \\widehat { \\pi } } \\left[ \\zeta _ { h } ( s _ { h } , a _ { h } ) \\mid s _ { 1 } = s \\right] .", + "type": "interline_equation", + "image_path": "a1002b52c9057be9b8d21afce6c564fc9561e6a14660b37576f5e425fb964be7.jpg" + } + ] + } + ], + "index": 12.5, + "virtual_lines": [ + { + "bbox": [ + 203, + 374, + 407, + 391.5 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 203, + 391.5, + 407, + 409.0 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 420, + 338, + 432 + ], + "lines": [ + { + "bbox": [ + 107, + 419, + 338, + 433 + ], + "spans": [ + { + "bbox": [ + 107, + 419, + 338, + 433 + ], + "score": 1.0, + "content": "Combine the above two results we prove the stated result.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 106, + 458, + 506, + 499 + ], + "lines": [ + { + "bbox": [ + 104, + 457, + 506, + 473 + ], + "spans": [ + { + "bbox": [ + 104, + 457, + 275, + 473 + ], + "score": 1.0, + "content": "Lemma H.9. For a linear MDP, for any", + "type": "text" + }, + { + "bbox": [ + 275, + 460, + 337, + 472 + ], + "score": 0.92, + "content": "0 \\le V ( \\cdot ) \\le H", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 457, + 417, + 473 + ], + "score": 1.0, + "content": ", then there exists a", + "type": "text" + }, + { + "bbox": [ + 418, + 459, + 456, + 471 + ], + "score": 0.83, + "content": "w _ { h } \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 457, + 473, + 473 + ], + "score": 1.0, + "content": "s.t.", + "type": "text" + }, + { + "bbox": [ + 473, + 460, + 506, + 471 + ], + "score": 0.81, + "content": "\\mathcal { T } _ { h } V =", + "type": "inline_equation" + } + ], + "index": 15 + }, + { + "bbox": [ + 107, + 471, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 107, + 472, + 138, + 485 + ], + "score": 0.92, + "content": "\\langle \\phi , w _ { h } \\rangle", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 471, + 160, + 486 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 160, + 471, + 234, + 485 + ], + "score": 0.92, + "content": "\\| w _ { h } \\| _ { 2 } ~ \\le ~ 2 H \\sqrt { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 471, + 266, + 486 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 267, + 473, + 306, + 485 + ], + "score": 0.91, + "content": "h \\in [ H ]", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 471, + 338, + 486 + ], + "score": 1.0, + "content": ". Here", + "type": "text" + }, + { + "bbox": [ + 339, + 472, + 502, + 485 + ], + "score": 0.88, + "content": "{ \\mathcal { T } } _ { h } ( V ) ( s , a ) = r _ { h } ( x , a ) + ( P _ { h } V ) ( s , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 471, + 506, + 486 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 483, + 507, + 500 + ], + "spans": [ + { + "bbox": [ + 104, + 483, + 176, + 500 + ], + "score": 1.0, + "content": "Similarly, for any", + "type": "text" + }, + { + "bbox": [ + 177, + 488, + 184, + 496 + ], + "score": 0.73, + "content": "\\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 483, + 234, + 500 + ], + "score": 1.0, + "content": ", there exists", + "type": "text" + }, + { + "bbox": [ + 234, + 485, + 271, + 498 + ], + "score": 0.93, + "content": "w _ { h } ^ { \\pi } \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 483, + 313, + 500 + ], + "score": 1.0, + "content": ", such that", + "type": "text" + }, + { + "bbox": [ + 313, + 486, + 371, + 498 + ], + "score": 0.92, + "content": "Q _ { h } ^ { \\pi } = \\langle \\phi , w _ { h } ^ { \\pi } \\rangle", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 483, + 392, + 500 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 393, + 486, + 503, + 499 + ], + "score": 0.88, + "content": "\\| w _ { h } ^ { \\pi } \\| _ { 2 } \\leq 2 ( H - h + 1 ) \\sqrt { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 483, + 507, + 500 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 106, + 531, + 192, + 543 + ], + "lines": [ + { + "bbox": [ + 105, + 530, + 194, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 194, + 545 + ], + "score": 1.0, + "content": "Proof. By definition,", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "interline_equation", + "bbox": [ + 196, + 554, + 414, + 609 + ], + "lines": [ + { + "bbox": [ + 196, + 554, + 414, + 609 + ], + "spans": [ + { + "bbox": [ + 196, + 554, + 414, + 609 + ], + "score": 0.93, + "content": "\\begin{array} { c } { { \\displaystyle { \\mathcal { T } } _ { h } V = r _ { h } + ( P _ { h } V ) = \\langle \\phi , \\theta _ { h } \\rangle + \\langle \\phi , \\displaystyle { \\int _ { S } V ( s ) d \\nu _ { h } ( s ) \\rangle } } } \\\\ { { \\displaystyle { \\Rightarrow } w _ { h } = \\theta _ { h } + \\int _ { S } V ( s ) d \\nu _ { h } ( s ) , } } \\end{array}", + "type": "interline_equation", + "image_path": "f4ae2ff946e4f8c01a455379722ad05b21aa49404e4833e76e3662916e99aa8a.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 196, + 554, + 414, + 572.3333333333334 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 196, + 572.3333333333334, + 414, + 590.6666666666667 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 196, + 590.6666666666667, + 414, + 609.0000000000001 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 619, + 505, + 645 + ], + "lines": [ + { + "bbox": [ + 105, + 619, + 507, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 146, + 635 + ], + "score": 1.0, + "content": "therefore", + "type": "text" + }, + { + "bbox": [ + 146, + 620, + 372, + 634 + ], + "score": 0.89, + "content": "\\| w _ { h } \\| _ { 2 } \\leq \\| \\theta _ { h } \\| _ { 2 } + H \\cdot \\| \\nu _ { h } ( S ) \\| \\leq 1 + H { \\sqrt { d } } \\leq 2 H { \\sqrt { d } }", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 619, + 507, + 635 + ], + "score": 1.0, + "content": ". The proof of the second part is", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 629, + 503, + 647 + ], + "spans": [ + { + "bbox": [ + 104, + 629, + 280, + 647 + ], + "score": 1.0, + "content": "similar by backward induction and the fact", + "type": "text" + }, + { + "bbox": [ + 280, + 633, + 351, + 645 + ], + "score": 0.9, + "content": "V _ { h } ^ { \\pi } \\leq H - h + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 629, + 383, + 647 + ], + "score": 1.0, + "content": "for any", + "type": "text" + }, + { + "bbox": [ + 384, + 635, + 390, + 642 + ], + "score": 0.77, + "content": "\\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 629, + 396, + 647 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 496, + 634, + 503, + 642 + ], + "score": 0.882, + "content": "□", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 106, + 671, + 505, + 695 + ], + "lines": [ + { + "bbox": [ + 105, + 671, + 506, + 686 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 300, + 686 + ], + "score": 1.0, + "content": "Lemma H.10. For any pessimistic bonus design", + "type": "text" + }, + { + "bbox": [ + 300, + 673, + 312, + 684 + ], + "score": 0.87, + "content": "\\Gamma _ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 671, + 351, + 686 + ], + "score": 1.0, + "content": ", suppose", + "type": "text" + }, + { + "bbox": [ + 351, + 672, + 482, + 684 + ], + "score": 0.92, + "content": "K > \\operatorname* { m a x } \\{ \\mathcal { M } _ { 1 } , \\mathcal { M } _ { 2 } , \\mathcal { M } _ { 3 } , \\mathcal { M } _ { 4 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 671, + 506, + 686 + ], + "score": 1.0, + "content": ", then", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 683, + 276, + 696 + ], + "spans": [ + { + "bbox": [ + 106, + 683, + 172, + 696 + ], + "score": 1.0, + "content": "with probability", + "type": "text" + }, + { + "bbox": [ + 173, + 684, + 195, + 694 + ], + "score": 0.77, + "content": "1 - \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 683, + 243, + 696 + ], + "score": 1.0, + "content": ", Algorithm", + "type": "text" + }, + { + "bbox": [ + 243, + 685, + 248, + 693 + ], + "score": 0.37, + "content": "^ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 683, + 276, + 696 + ], + "score": 1.0, + "content": "yields", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5 + }, + { + "type": "interline_equation", + "bbox": [ + 221, + 707, + 389, + 736 + ], + "lines": [ + { + "bbox": [ + 221, + 707, + 389, + 736 + ], + "spans": [ + { + "bbox": [ + 221, + 707, + 389, + 736 + ], + "score": 0.92, + "content": "\\left\\| \\mathcal { T } _ { h } \\widehat { V } _ { h + 1 } - \\widehat { \\mathcal { T } } _ { h } \\widehat { V } _ { h + 1 } \\right\\| _ { \\infty } \\leq \\widetilde { O } ( \\frac { H ^ { 2 } \\sqrt { d / \\kappa } } { \\sqrt { K } } )", + "type": "interline_equation", + "image_path": "aa01715208fe6f464e1410e6f782b6d2818cdf96751312360dc9ea78a29808f8.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 221, + 707, + 389, + 736 + ], + "spans": [], + "index": 26 + } + ] + } + ], + "page_idx": 37, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 294, + 38 + ], + "lines": [ + { + "bbox": [ + 105, + 25, + 294, + 39 + ], + "spans": [ + { + "bbox": [ + 105, + 25, + 294, + 39 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 761 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "score": 1.0, + "content": "38", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 494, + 420, + 505, + 431 + ], + "lines": [ + { + "bbox": [ + 495, + 422, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 495, + 422, + 505, + 430 + ], + "score": 0.998, + "content": "□", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 372, + 95 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 372, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 326, + 97 + ], + "score": 1.0, + "content": "to be the Bellman update error. Then for any arbitrary", + "type": "text" + }, + { + "bbox": [ + 326, + 85, + 333, + 92 + ], + "score": 0.59, + "content": "\\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 80, + 372, + 97 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 105, + 80, + 372, + 97 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 146, + 106, + 465, + 177 + ], + "lines": [ + { + "bbox": [ + 146, + 106, + 465, + 177 + ], + "spans": [ + { + "bbox": [ + 146, + 106, + 465, + 177 + ], + "score": 0.94, + "content": "\\begin{array} { c } { { { \\displaystyle V _ { 1 } ^ { \\pi } ( s ) - V _ { 1 } ^ { \\widehat { \\pi } } ( s ) = \\sum _ { h = 1 } ^ { H } \\mathbb { E } _ { \\pi } \\left[ \\zeta _ { h } ( s _ { h } , a _ { h } ) \\mid s _ { 1 } = s \\right] - \\sum _ { h = 1 } ^ { H } \\mathbb { E } _ { \\widehat { \\pi } } \\left[ \\zeta _ { h } ( s _ { h } , a _ { h } ) \\mid s _ { 1 } = s \\right] } } } \\\\ { { { + \\displaystyle \\sum _ { h = 1 } ^ { H } \\mathbb { E } _ { \\pi } \\left[ \\left. \\widehat { Q } _ { h } \\left( s _ { h } , \\cdot \\right) , \\pi _ { h } \\left( \\cdot | s _ { h } \\right) - \\widehat { \\pi } _ { h } \\left( \\cdot | s _ { h } \\right) \\right. \\mid s _ { 1 } = x \\right] } } } \\end{array}", + "type": "interline_equation", + "image_path": "67417bc8f21fd1b43ef9a4a476ce6fa4232c33309bc8f685cf1e78494196e120.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 146, + 106, + 465, + 129.66666666666666 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 146, + 129.66666666666666, + 465, + 153.33333333333331 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 146, + 153.33333333333331, + 465, + 176.99999999999997 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 187, + 284, + 199 + ], + "lines": [ + { + "bbox": [ + 105, + 186, + 285, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 255, + 201 + ], + "score": 1.0, + "content": "where the expectation are taken over", + "type": "text" + }, + { + "bbox": [ + 255, + 190, + 281, + 199 + ], + "score": 0.89, + "content": "s _ { h } , a _ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 186, + 285, + 201 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4, + "bbox_fs": [ + 105, + 186, + 285, + 201 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 232, + 266, + 244 + ], + "lines": [ + { + "bbox": [ + 106, + 232, + 267, + 245 + ], + "spans": [ + { + "bbox": [ + 106, + 232, + 267, + 245 + ], + "score": 1.0, + "content": "Proof. Note the gap can be rewritten as", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5, + "bbox_fs": [ + 106, + 232, + 267, + 245 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 197, + 256, + 413, + 272 + ], + "lines": [ + { + "bbox": [ + 197, + 256, + 413, + 272 + ], + "spans": [ + { + "bbox": [ + 197, + 256, + 413, + 272 + ], + "score": 0.88, + "content": "\\begin{array} { r } { V _ { 1 } ^ { \\pi } ( s ) - V _ { 1 } ^ { \\widehat { \\pi } } ( s ) = V _ { 1 } ^ { \\pi } ( s ) - \\widehat { V } _ { 1 } ( s ) + \\widehat { V } _ { 1 } ( s ) - V _ { 1 } ^ { \\widehat { \\pi } } ( s ) . } \\end{array}", + "type": "interline_equation", + "image_path": "d0b157725d446ee734981fa192e2d1aaf3a9bee3d716c71d27ee54b4a38f644b.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 197, + 256, + 413, + 272 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 283, + 323, + 296 + ], + "lines": [ + { + "bbox": [ + 105, + 282, + 323, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 191, + 297 + ], + "score": 1.0, + "content": "By Lemma H.7 with", + "type": "text" + }, + { + "bbox": [ + 191, + 284, + 217, + 294 + ], + "score": 0.88, + "content": "\\pi = { \\widehat { \\pi } }", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 282, + 221, + 297 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 221, + 284, + 250, + 294 + ], + "score": 0.84, + "content": "\\pi ^ { \\prime } = \\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 282, + 323, + 297 + ], + "score": 1.0, + "content": ", we directly have", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 282, + 323, + 297 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 307, + 511, + 342 + ], + "lines": [ + { + "bbox": [ + 111, + 307, + 511, + 342 + ], + "spans": [ + { + "bbox": [ + 111, + 307, + 511, + 342 + ], + "score": 0.93, + "content": "\\gamma _ { 1 } ^ { \\pi } ( s ) - \\widehat { V } _ { 1 } ( s ) = \\sum _ { h = 1 } ^ { H } \\mathbb { E } _ { \\pi } \\left[ \\zeta _ { h } ( s _ { h } , a _ { h } ) \\mid s _ { 1 } = s \\right] + \\sum _ { h = 1 } ^ { H } \\mathbb { E } _ { \\pi } \\left[ \\langle \\widehat { Q } _ { h } \\left( s _ { h } , \\cdot \\right) , \\pi _ { h } \\left( \\cdot | s _ { h } \\right) - \\widehat { \\pi } _ { h } \\left( \\cdot | s _ { h } \\right) \\rangle \\mid s _ { 1 } = s \\right]", + "type": "interline_equation", + "image_path": "a3b17028ac9abc38cd618977334599ec09bdd3ee7aa6fb322892870650e381d9.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 111, + 307, + 511, + 318.6666666666667 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 111, + 318.6666666666667, + 511, + 330.33333333333337 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 111, + 330.33333333333337, + 511, + 342.00000000000006 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 350, + 368, + 363 + ], + "lines": [ + { + "bbox": [ + 105, + 348, + 368, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 247, + 364 + ], + "score": 1.0, + "content": "Next apply Lemma H.7 again with", + "type": "text" + }, + { + "bbox": [ + 247, + 351, + 296, + 361 + ], + "score": 0.91, + "content": "\\pi = \\pi ^ { \\prime } = \\widehat { \\pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 348, + 368, + 364 + ], + "score": 1.0, + "content": ", we directly have", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 348, + 368, + 364 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 203, + 374, + 407, + 409 + ], + "lines": [ + { + "bbox": [ + 203, + 374, + 407, + 409 + ], + "spans": [ + { + "bbox": [ + 203, + 374, + 407, + 409 + ], + "score": 0.93, + "content": "\\widehat { V } _ { 1 } ( s ) - V _ { 1 } ^ { \\widehat { \\pi } } ( s ) = - \\sum _ { h = 1 } ^ { H } \\mathbb { E } _ { \\widehat { \\pi } } \\left[ \\zeta _ { h } ( s _ { h } , a _ { h } ) \\mid s _ { 1 } = s \\right] .", + "type": "interline_equation", + "image_path": "a1002b52c9057be9b8d21afce6c564fc9561e6a14660b37576f5e425fb964be7.jpg" + } + ] + } + ], + "index": 12.5, + "virtual_lines": [ + { + "bbox": [ + 203, + 374, + 407, + 391.5 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 203, + 391.5, + 407, + 409.0 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 420, + 338, + 432 + ], + "lines": [ + { + "bbox": [ + 107, + 419, + 338, + 433 + ], + "spans": [ + { + "bbox": [ + 107, + 419, + 338, + 433 + ], + "score": 1.0, + "content": "Combine the above two results we prove the stated result.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14, + "bbox_fs": [ + 107, + 419, + 338, + 433 + ] + }, + { + "type": "list", + "bbox": [ + 106, + 458, + 506, + 499 + ], + "lines": [ + { + "bbox": [ + 104, + 457, + 506, + 473 + ], + "spans": [ + { + "bbox": [ + 104, + 457, + 275, + 473 + ], + "score": 1.0, + "content": "Lemma H.9. For a linear MDP, for any", + "type": "text" + }, + { + "bbox": [ + 275, + 460, + 337, + 472 + ], + "score": 0.92, + "content": "0 \\le V ( \\cdot ) \\le H", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 457, + 417, + 473 + ], + "score": 1.0, + "content": ", then there exists a", + "type": "text" + }, + { + "bbox": [ + 418, + 459, + 456, + 471 + ], + "score": 0.83, + "content": "w _ { h } \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 457, + 473, + 473 + ], + "score": 1.0, + "content": "s.t.", + "type": "text" + }, + { + "bbox": [ + 473, + 460, + 506, + 471 + ], + "score": 0.81, + "content": "\\mathcal { T } _ { h } V =", + "type": "inline_equation" + } + ], + "index": 15, + "is_list_end_line": true + }, + { + "bbox": [ + 107, + 471, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 107, + 472, + 138, + 485 + ], + "score": 0.92, + "content": "\\langle \\phi , w _ { h } \\rangle", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 471, + 160, + 486 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 160, + 471, + 234, + 485 + ], + "score": 0.92, + "content": "\\| w _ { h } \\| _ { 2 } ~ \\le ~ 2 H \\sqrt { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 471, + 266, + 486 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 267, + 473, + 306, + 485 + ], + "score": 0.91, + "content": "h \\in [ H ]", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 471, + 338, + 486 + ], + "score": 1.0, + "content": ". Here", + "type": "text" + }, + { + "bbox": [ + 339, + 472, + 502, + 485 + ], + "score": 0.88, + "content": "{ \\mathcal { T } } _ { h } ( V ) ( s , a ) = r _ { h } ( x , a ) + ( P _ { h } V ) ( s , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 471, + 506, + 486 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 16, + "is_list_start_line": true, + "is_list_end_line": true + }, + { + "bbox": [ + 104, + 483, + 507, + 500 + ], + "spans": [ + { + "bbox": [ + 104, + 483, + 176, + 500 + ], + "score": 1.0, + "content": "Similarly, for any", + "type": "text" + }, + { + "bbox": [ + 177, + 488, + 184, + 496 + ], + "score": 0.73, + "content": "\\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 483, + 234, + 500 + ], + "score": 1.0, + "content": ", there exists", + "type": "text" + }, + { + "bbox": [ + 234, + 485, + 271, + 498 + ], + "score": 0.93, + "content": "w _ { h } ^ { \\pi } \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 483, + 313, + 500 + ], + "score": 1.0, + "content": ", such that", + "type": "text" + }, + { + "bbox": [ + 313, + 486, + 371, + 498 + ], + "score": 0.92, + "content": "Q _ { h } ^ { \\pi } = \\langle \\phi , w _ { h } ^ { \\pi } \\rangle", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 483, + 392, + 500 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 393, + 486, + 503, + 499 + ], + "score": 0.88, + "content": "\\| w _ { h } ^ { \\pi } \\| _ { 2 } \\leq 2 ( H - h + 1 ) \\sqrt { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 483, + 507, + 500 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 17, + "is_list_start_line": true, + "is_list_end_line": true + } + ], + "index": 16, + "bbox_fs": [ + 104, + 457, + 507, + 500 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 531, + 192, + 543 + ], + "lines": [ + { + "bbox": [ + 105, + 530, + 194, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 194, + 545 + ], + "score": 1.0, + "content": "Proof. By definition,", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 530, + 194, + 545 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 196, + 554, + 414, + 609 + ], + "lines": [ + { + "bbox": [ + 196, + 554, + 414, + 609 + ], + "spans": [ + { + "bbox": [ + 196, + 554, + 414, + 609 + ], + "score": 0.93, + "content": "\\begin{array} { c } { { \\displaystyle { \\mathcal { T } } _ { h } V = r _ { h } + ( P _ { h } V ) = \\langle \\phi , \\theta _ { h } \\rangle + \\langle \\phi , \\displaystyle { \\int _ { S } V ( s ) d \\nu _ { h } ( s ) \\rangle } } } \\\\ { { \\displaystyle { \\Rightarrow } w _ { h } = \\theta _ { h } + \\int _ { S } V ( s ) d \\nu _ { h } ( s ) , } } \\end{array}", + "type": "interline_equation", + "image_path": "f4ae2ff946e4f8c01a455379722ad05b21aa49404e4833e76e3662916e99aa8a.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 196, + 554, + 414, + 572.3333333333334 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 196, + 572.3333333333334, + 414, + 590.6666666666667 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 196, + 590.6666666666667, + 414, + 609.0000000000001 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 619, + 505, + 645 + ], + "lines": [ + { + "bbox": [ + 105, + 619, + 507, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 146, + 635 + ], + "score": 1.0, + "content": "therefore", + "type": "text" + }, + { + "bbox": [ + 146, + 620, + 372, + 634 + ], + "score": 0.89, + "content": "\\| w _ { h } \\| _ { 2 } \\leq \\| \\theta _ { h } \\| _ { 2 } + H \\cdot \\| \\nu _ { h } ( S ) \\| \\leq 1 + H { \\sqrt { d } } \\leq 2 H { \\sqrt { d } }", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 619, + 507, + 635 + ], + "score": 1.0, + "content": ". The proof of the second part is", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 629, + 503, + 647 + ], + "spans": [ + { + "bbox": [ + 104, + 629, + 280, + 647 + ], + "score": 1.0, + "content": "similar by backward induction and the fact", + "type": "text" + }, + { + "bbox": [ + 280, + 633, + 351, + 645 + ], + "score": 0.9, + "content": "V _ { h } ^ { \\pi } \\leq H - h + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 629, + 383, + 647 + ], + "score": 1.0, + "content": "for any", + "type": "text" + }, + { + "bbox": [ + 384, + 635, + 390, + 642 + ], + "score": 0.77, + "content": "\\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 629, + 396, + 647 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 496, + 634, + 503, + 642 + ], + "score": 0.882, + "content": "□", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5, + "bbox_fs": [ + 104, + 619, + 507, + 647 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 671, + 505, + 695 + ], + "lines": [ + { + "bbox": [ + 105, + 671, + 506, + 686 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 300, + 686 + ], + "score": 1.0, + "content": "Lemma H.10. For any pessimistic bonus design", + "type": "text" + }, + { + "bbox": [ + 300, + 673, + 312, + 684 + ], + "score": 0.87, + "content": "\\Gamma _ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 671, + 351, + 686 + ], + "score": 1.0, + "content": ", suppose", + "type": "text" + }, + { + "bbox": [ + 351, + 672, + 482, + 684 + ], + "score": 0.92, + "content": "K > \\operatorname* { m a x } \\{ \\mathcal { M } _ { 1 } , \\mathcal { M } _ { 2 } , \\mathcal { M } _ { 3 } , \\mathcal { M } _ { 4 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 671, + 506, + 686 + ], + "score": 1.0, + "content": ", then", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 683, + 276, + 696 + ], + "spans": [ + { + "bbox": [ + 106, + 683, + 172, + 696 + ], + "score": 1.0, + "content": "with probability", + "type": "text" + }, + { + "bbox": [ + 173, + 684, + 195, + 694 + ], + "score": 0.77, + "content": "1 - \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 683, + 243, + 696 + ], + "score": 1.0, + "content": ", Algorithm", + "type": "text" + }, + { + "bbox": [ + 243, + 685, + 248, + 693 + ], + "score": 0.37, + "content": "^ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 683, + 276, + 696 + ], + "score": 1.0, + "content": "yields", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 671, + 506, + 696 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 221, + 707, + 389, + 736 + ], + "lines": [ + { + "bbox": [ + 221, + 707, + 389, + 736 + ], + "spans": [ + { + "bbox": [ + 221, + 707, + 389, + 736 + ], + "score": 0.92, + "content": "\\left\\| \\mathcal { T } _ { h } \\widehat { V } _ { h + 1 } - \\widehat { \\mathcal { T } } _ { h } \\widehat { V } _ { h + 1 } \\right\\| _ { \\infty } \\leq \\widetilde { O } ( \\frac { H ^ { 2 } \\sqrt { d / \\kappa } } { \\sqrt { K } } )", + "type": "interline_equation", + "image_path": "aa01715208fe6f464e1410e6f782b6d2818cdf96751312360dc9ea78a29808f8.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 221, + 707, + 389, + 736 + ], + "spans": [], + "index": 26 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 506, + 108 + ], + "lines": [ + { + "bbox": [ + 106, + 80, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 237, + 95 + ], + "score": 1.0, + "content": "Proof of Lemma H.10. Suppose", + "type": "text" + }, + { + "bbox": [ + 237, + 84, + 250, + 93 + ], + "score": 0.85, + "content": "w _ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 81, + 406, + 95 + ], + "score": 1.0, + "content": "is the coefficient corresponding to the", + "type": "text" + }, + { + "bbox": [ + 406, + 80, + 439, + 94 + ], + "score": 0.93, + "content": "\\mathcal { T } _ { h } \\widehat { V } _ { h + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 81, + 464, + 95 + ], + "score": 1.0, + "content": "(such", + "type": "text" + }, + { + "bbox": [ + 465, + 84, + 478, + 93 + ], + "score": 0.85, + "content": "w _ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 81, + 505, + 95 + ], + "score": 1.0, + "content": "exists", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 456, + 109 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 190, + 109 + ], + "score": 1.0, + "content": "by Lemma H.9), i.e.", + "type": "text" + }, + { + "bbox": [ + 190, + 93, + 261, + 108 + ], + "score": 0.93, + "content": "\\mathcal { T } _ { h } \\widehat { V } _ { h + 1 } = \\phi ^ { \\top } w _ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 93, + 307, + 109 + ], + "score": 1.0, + "content": ", and recall", + "type": "text" + }, + { + "bbox": [ + 307, + 93, + 429, + 108 + ], + "score": 0.92, + "content": "( \\widehat { T } _ { h } \\widehat { V } _ { h + 1 } ) ( s , a ) = \\phi ( s , a ) ^ { \\top } \\widehat { w } _ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 93, + 456, + 109 + ], + "score": 1.0, + "content": ", then:", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "interline_equation", + "bbox": [ + 105, + 112, + 444, + 205 + ], + "lines": [ + { + "bbox": [ + 105, + 112, + 444, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 112, + 444, + 205 + ], + "score": 0.88, + "content": "\\begin{array} { r l } & { \\quad \\left( \\mathcal { T } _ { h } \\widehat { V } _ { h + 1 } \\right) ( s , a ) - 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\\phi ( s , a ) ^ { \\top } \\widehat { \\Lambda } _ { h } ^ { - 1 } \\left( \\displaystyle \\sum _ { \\tau = 1 } ^ { K } \\phi \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\cdot \\left( \\mathcal { T } _ { h } \\widehat { V } _ { h + 1 } \\right) \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) / \\widehat { \\sigma } _ { h } ^ { 2 } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) \\right) } \\end{array}", + "type": "interline_equation", + "image_path": "1def6bbdbd348703f0ad5948b4059ab4ab3691a68049f97fcbb8dffe8f6060dc.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 105, + 112, + 444, + 143.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 105, + 143.0, + 444, + 174.0 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 105, + 174.0, + 444, + 205.0 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "interline_equation", + "bbox": [ + 132, + 220, + 510, + 255 + ], + "lines": [ + { + "bbox": [ + 132, + 220, + 510, + 255 + ], + "spans": [ + { + "bbox": [ + 132, + 220, + 510, + 255 + ], + "score": 0.59, + "content": "+ \\phi ( s , a ) ^ { \\top } \\widehat { \\Lambda } _ { h } ^ { - 1 } \\left( \\sum _ { \\tau = 1 } ^ { K } \\phi \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\cdot \\left( r _ { h } ^ { \\tau } + \\widehat { V } _ { h + 1 } \\left( s _ { h + 1 } ^ { \\tau } \\right) - \\left( \\mathcal { T } _ { h } \\widehat { V } _ { h + 1 } \\right) \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\right) / \\widehat { \\sigma } _ { h } ^ { 2 } \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\right) .", + "type": "interline_equation", + "image_path": "7806236ebe8f66ab0f58618dab2a7a0c83880ac3a72cf0e377023db89585dfd7.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 132, + 220, + 510, + 231.66666666666666 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 132, + 231.66666666666666, + 510, + 243.33333333333331 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 132, + 243.33333333333331, + 510, + 254.99999999999997 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 284, + 422, + 302 + ], + "lines": [ + { + "bbox": [ + 105, + 283, + 425, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 225, + 301 + ], + "score": 1.0, + "content": "For term (i), it is bounded by", + "type": "text" + }, + { + "bbox": [ + 221, + 283, + 425, + 302 + ], + "score": 1.0, + "content": "2λH3 d/κ with probability 1 − δ by Lemma C.2.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 106, + 305, + 315, + 317 + ], + "lines": [ + { + "bbox": [ + 105, + 302, + 315, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 315, + 321 + ], + "score": 1.0, + "content": "For term (ii), by Cauchy inequality it is bounded by", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "interline_equation", + "bbox": [ + 109, + 322, + 499, + 428 + ], + "lines": [ + { + "bbox": [ + 109, + 322, + 499, + 428 + ], + "spans": [ + { + "bbox": [ + 109, + 322, + 499, + 428 + ], + "score": 0.87, + "content": "\\begin{array} { r l } & { \\quad \\left. \\phi ( s , a ) \\right. _ { \\widehat { \\Lambda } _ { h } ^ { - 1 } } \\cdot \\left. \\displaystyle \\sum _ { \\tau = 1 } ^ { K } \\phi \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\cdot \\left( r _ { h } ^ { \\tau } + \\widehat { V } _ { h + 1 } \\left( s _ { h + 1 } ^ { \\tau } \\right) - \\left( \\mathcal { T } _ { h } \\widehat { V } _ { h + 1 } \\right) \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\right) / \\widehat { \\sigma } _ { h } ^ { 2 } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) \\right. _ { \\widehat { \\Lambda } _ { h } ^ { - 1 } } } \\\\ & { \\leq \\frac { 2 H } { \\sqrt { K K } } \\left. \\displaystyle \\sum _ { \\tau = 1 } ^ { K } \\phi \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\cdot \\left( r _ { h } ^ { \\tau } + \\widehat { V } _ { h + 1 } \\left( s _ { h + 1 } ^ { \\tau } \\right) - \\left( \\mathcal { T } _ { h } \\widehat { V } _ { h + 1 } \\right) \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\right) / \\widehat { \\sigma } _ { h } ^ { 2 } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) \\right. _ { \\widehat { \\Lambda } _ { h } ^ { - 1 } } } \\\\ & { \\leq \\frac { 2 H } { \\sqrt { K K } } \\cdot \\tilde { O } ( H \\sqrt { d } ) = \\widetilde { O } ( \\frac { H ^ { 2 } \\sqrt { d / \\kappa } } { \\sqrt { K } } ) , } \\end{array}", + "type": "interline_equation", + "image_path": "ce87fab67963f6b102fe3d6175e3c04fd8c9ae542f9f8f7c233e0cd08e5d0f37.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 109, + 322, + 499, + 357.3333333333333 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 109, + 357.3333333333333, + 499, + 392.66666666666663 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 109, + 392.66666666666663, + 499, + 427.99999999999994 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 431, + 506, + 506 + ], + "lines": [ + { + "bbox": [ + 105, + 430, + 504, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 316, + 445 + ], + "score": 1.0, + "content": "where the first inequality is by Lemma H.6 (with", + "type": "text" + }, + { + "bbox": [ + 316, + 432, + 365, + 444 + ], + "score": 0.91, + "content": "\\phi ^ { \\prime } = \\phi / \\widehat { \\sigma } _ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 430, + 385, + 445 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 386, + 432, + 504, + 444 + ], + "score": 0.92, + "content": "\\| \\phi / { \\widehat { \\sigma } } _ { h } \\| \\leq \\| \\phi \\| \\leq 1 : = C )", + "type": "inline_equation" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 443, + 506, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 224, + 460 + ], + "score": 1.0, + "content": "and the third inequality uses", + "type": "text" + }, + { + "bbox": [ + 225, + 444, + 428, + 458 + ], + "score": 0.89, + "content": "{ \\sqrt { a ^ { \\top } \\cdot A \\cdot a } } \\leq { \\sqrt { \\| a \\| _ { 2 } \\| A \\| _ { 2 } \\| a \\| _ { 2 } } } = \\| a \\| _ { 2 } { \\sqrt { \\| A \\| _ { 2 } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 443, + 449, + 460 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 449, + 447, + 456, + 455 + ], + "score": 0.74, + "content": "a", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 443, + 506, + 460 + ], + "score": 1.0, + "content": "to be either", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 457, + 506, + 477 + ], + "spans": [ + { + "bbox": [ + 106, + 461, + 114, + 473 + ], + "score": 0.82, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 457, + 128, + 475 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 128, + 462, + 141, + 472 + ], + "score": 0.85, + "content": "w _ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 457, + 194, + 475 + ], + "score": 1.0, + "content": ". Moreover,", + "type": "text" + }, + { + "bbox": [ + 195, + 459, + 383, + 474 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\lambda _ { \\operatorname* { m i n } } ( \\widetilde { \\Lambda } _ { h } ^ { p } ) \\ge \\kappa / \\operatorname* { m a x } _ { h , s , a } \\widehat { \\sigma } _ { h } ( s , a ) ^ { 2 } \\ge \\kappa / H ^ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 457, + 419, + 475 + ], + "score": 1.0, + "content": "implies", + "type": "text" + }, + { + "bbox": [ + 419, + 457, + 503, + 477 + ], + "score": 0.92, + "content": "\\left\\| ( \\tilde { \\Lambda } _ { h } ^ { p } ) ^ { - 1 } \\right\\| \\leq H ^ { 2 } / \\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 504, + 457, + 506, + 475 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 475, + 506, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 323, + 492 + ], + "score": 1.0, + "content": "The second inequality comes from Lemma H.2 with", + "type": "text" + }, + { + "bbox": [ + 324, + 478, + 356, + 489 + ], + "score": 0.91, + "content": "R = H", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 475, + 381, + 492 + ], + "score": 1.0, + "content": "since", + "type": "text" + }, + { + "bbox": [ + 381, + 477, + 506, + 491 + ], + "score": 0.9, + "content": "| \\eta _ { \\tau } | = | ( r _ { h } ^ { \\tau } + \\widehat { V } _ { h + 1 } \\left( s _ { h + 1 } ^ { \\tau } \\right) -", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 490, + 420, + 506 + ], + "spans": [ + { + "bbox": [ + 106, + 490, + 256, + 505 + ], + "score": 0.91, + "content": "( \\mathcal T _ { h } \\widehat V _ { h + 1 } ) ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) ) / \\widehat \\sigma _ { h } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) | \\leq H", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 491, + 273, + 506 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 274, + 492, + 415, + 505 + ], + "score": 0.92, + "content": "| x _ { \\tau } | = | \\phi ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) / \\widehat { \\sigma } _ { h } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) | \\leq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 491, + 420, + 506 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 104, + 509, + 494, + 533 + ], + "lines": [ + { + "bbox": [ + 105, + 507, + 504, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 471, + 522 + ], + "score": 1.0, + "content": "The final result is obtained by absorbing the term (i) via the condition", + "type": "text" + }, + { + "bbox": [ + 471, + 509, + 504, + 520 + ], + "score": 0.67, + "content": "K \\_ { > }", + "type": "inline_equation" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 519, + 219, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 215, + 532 + ], + "score": 0.9, + "content": "\\operatorname* { m a x } \\{ \\mathcal { M } _ { 1 } , \\mathcal { M } _ { 2 } , \\mathcal { M } _ { 3 } , \\mathcal { M } _ { 4 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 519, + 219, + 533 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 106, + 537, + 417, + 551 + ], + "lines": [ + { + "bbox": [ + 105, + 536, + 417, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 278, + 552 + ], + "score": 1.0, + "content": "Lemma H.11. Suppose random variables", + "type": "text" + }, + { + "bbox": [ + 278, + 538, + 334, + 551 + ], + "score": 0.84, + "content": "\\| X \\| _ { \\infty } \\leq 2 H", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 536, + 338, + 552 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 338, + 538, + 392, + 551 + ], + "score": 0.85, + "content": "\\| Y \\| _ { \\infty } \\leq 2 H", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 536, + 417, + 552 + ], + "score": 1.0, + "content": ", then", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "interline_equation", + "bbox": [ + 220, + 555, + 389, + 569 + ], + "lines": [ + { + "bbox": [ + 220, + 555, + 389, + 569 + ], + "spans": [ + { + "bbox": [ + 220, + 555, + 389, + 569 + ], + "score": 0.86, + "content": "\\left| \\operatorname { V a r } ( X ) - \\operatorname { V a r } ( Y ) \\right| \\leq 8 H \\cdot \\left\\| X - Y \\right\\| _ { \\infty } .", + "type": "interline_equation", + "image_path": "a70be81e3043f9bd2d753874d2e5c086e8a96e3377ad884fc989daa62f577bb4.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 220, + 555, + 389, + 569 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 579, + 196, + 591 + ], + "lines": [ + { + "bbox": [ + 106, + 579, + 198, + 592 + ], + "spans": [ + { + "bbox": [ + 106, + 579, + 198, + 592 + ], + "score": 1.0, + "content": "Proof of Lemma H.11.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "interline_equation", + "bbox": [ + 138, + 594, + 540, + 641 + ], + "lines": [ + { + "bbox": [ + 138, + 594, + 540, + 641 + ], + "spans": [ + { + "bbox": [ + 138, + 594, + 540, + 641 + ], + "score": 0.84, + "content": "\\begin{array} { r l } & { \\mathrm { ) - V a r } ( Y ) \\vert = \\vert \\mathbb { E } [ X ^ { 2 } ] - \\mathbb { E } [ Y ^ { 2 } ] - ( \\mathbb { E } [ X ] ^ { 2 } - \\mathbb { E } [ Y ] ^ { 2 } ) \\vert = \\vert \\mathbb { E } [ ( X + Y ) ( X - Y ) ] - ( \\mathbb { E } [ X + Y ] ) ( \\mathbb { E } [ X - Y ] - \\mathbb { E } [ X - Y ] ) \\vert } \\\\ & { \\qquad \\le \\mathbb { E } [ \\vert X + Y \\vert \\cdot \\vert X - Y \\vert ] + 4 H \\cdot \\Vert X - Y \\Vert _ { \\infty } } \\\\ & { \\qquad \\le 4 H \\mathbb { E } [ \\vert X - Y \\vert ] + 4 H \\cdot \\Vert X - Y \\Vert _ { \\infty } = 8 H \\cdot \\Vert X - Y \\Vert _ { \\infty } . } \\end{array}", + "type": "interline_equation", + "image_path": "93afdd061bb0f4286f320a147a6596d3e2bfe07bc9bb1efc675107da2ce5c070.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 138, + 594, + 540, + 609.6666666666666 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 138, + 609.6666666666666, + 540, + 625.3333333333333 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 138, + 625.3333333333333, + 540, + 640.9999999999999 + ], + "spans": [], + "index": 25 + } + ] + } + ], + "page_idx": 38, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 293, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "score": 1.0, + "content": "39", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 494, + 650, + 505, + 660 + ], + "lines": [ + { + "bbox": [ + 496, + 651, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 496, + 651, + 505, + 661 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 495, + 521, + 504, + 530 + ], + "lines": [ + { + "bbox": [ + 496, + 522, + 504, + 530 + ], + "spans": [ + { + "bbox": [ + 496, + 522, + 504, + 530 + ], + "score": 0.997, + "content": "□", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 506, + 108 + ], + "lines": [ + { + "bbox": [ + 106, + 80, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 237, + 95 + ], + "score": 1.0, + "content": "Proof of Lemma H.10. 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1 } \\left( \\sum _ { \\tau = 1 } ^ { K } \\phi \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\cdot \\left( r _ { h } ^ { \\tau } + \\widehat { V } _ { h + 1 } \\left( s _ { h + 1 } ^ { \\tau } \\right) - \\left( \\mathcal { T } _ { h } \\widehat { V } _ { h + 1 } \\right) \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\right) / \\widehat { \\sigma } _ { h } ^ { 2 } \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\right) .", + "type": "interline_equation", + "image_path": "7806236ebe8f66ab0f58618dab2a7a0c83880ac3a72cf0e377023db89585dfd7.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 132, + 220, + 510, + 231.66666666666666 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 132, + 231.66666666666666, + 510, + 243.33333333333331 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 132, + 243.33333333333331, + 510, + 254.99999999999997 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 284, + 422, + 302 + ], + "lines": [ + { + "bbox": [ + 105, + 283, + 425, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 225, + 301 + ], + "score": 1.0, + "content": "For term (i), it is bounded by", + "type": "text" + }, + { + "bbox": [ + 221, + 283, + 425, + 302 + ], + "score": 1.0, + "content": "2λH3 d/κ with probability 1 − δ by Lemma C.2.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8, + "bbox_fs": [ + 105, + 283, + 425, + 302 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 305, + 315, + 317 + ], + "lines": [ + { + "bbox": [ + 105, + 302, + 315, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 315, + 321 + ], + "score": 1.0, + "content": "For term (ii), by Cauchy inequality it is bounded by", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9, + "bbox_fs": [ + 105, + 302, + 315, + 321 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 109, + 322, + 499, + 428 + ], + "lines": [ + { + "bbox": [ + 109, + 322, + 499, + 428 + ], + "spans": [ + { + "bbox": [ + 109, + 322, + 499, + 428 + ], + "score": 0.87, + "content": "\\begin{array} { r l } & { \\quad \\left. \\phi ( s , a ) \\right. _ { \\widehat { \\Lambda } _ { h } ^ { - 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Moreover,", + "type": "text" + }, + { + "bbox": [ + 195, + 459, + 383, + 474 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\lambda _ { \\operatorname* { m i n } } ( \\widetilde { \\Lambda } _ { h } ^ { p } ) \\ge \\kappa / \\operatorname* { m a x } _ { h , s , a } \\widehat { \\sigma } _ { h } ( s , a ) ^ { 2 } \\ge \\kappa / H ^ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 457, + 419, + 475 + ], + "score": 1.0, + "content": "implies", + "type": "text" + }, + { + "bbox": [ + 419, + 457, + 503, + 477 + ], + "score": 0.92, + "content": "\\left\\| ( \\tilde { \\Lambda } _ { h } ^ { p } ) ^ { - 1 } \\right\\| \\leq H ^ { 2 } / \\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 504, + 457, + 506, + 475 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 475, + 506, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 323, + 492 + ], + "score": 1.0, + "content": "The second inequality comes from Lemma H.2 with", + "type": "text" + }, + { + "bbox": [ + 324, + 478, + 356, + 489 + ], + "score": 0.91, + "content": "R = H", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 475, + 381, + 492 + ], + "score": 1.0, + "content": "since", + "type": "text" + }, + { + "bbox": [ + 381, + 477, + 506, + 491 + ], + "score": 0.9, + "content": "| \\eta _ { \\tau } | = | ( r _ { h } ^ { \\tau } + \\widehat { V } _ { h + 1 } \\left( s _ { h + 1 } ^ { \\tau } \\right) -", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 490, + 420, + 506 + ], + "spans": [ + { + "bbox": [ + 106, + 490, + 256, + 505 + ], + "score": 0.91, + "content": "( \\mathcal T _ { h } \\widehat V _ { h + 1 } ) ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) ) / \\widehat \\sigma _ { h } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) | \\leq H", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 491, + 273, + 506 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 274, + 492, + 415, + 505 + ], + "score": 0.92, + "content": "| x _ { \\tau } | = | \\phi ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) / \\widehat { \\sigma } _ { h } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) | \\leq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 491, + 420, + 506 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 430, + 506, + 506 + ] + }, + { + "type": "text", + "bbox": [ + 104, + 509, + 494, + 533 + ], + "lines": [ + { + "bbox": [ + 105, + 507, + 504, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 471, + 522 + ], + "score": 1.0, + "content": "The final result is obtained by absorbing the term (i) via the condition", + "type": "text" + }, + { + "bbox": [ + 471, + 509, + 504, + 520 + ], + "score": 0.67, + "content": "K \\_ { > }", + "type": "inline_equation" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 519, + 219, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 215, + 532 + ], + "score": 0.9, + "content": "\\operatorname* { m a x } \\{ \\mathcal { M } _ { 1 } , \\mathcal { M } _ { 2 } , \\mathcal { M } _ { 3 } , \\mathcal { M } _ { 4 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 519, + 219, + 533 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 507, + 504, + 533 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 537, + 417, + 551 + ], + "lines": [ + { + "bbox": [ + 105, + 536, + 417, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 278, + 552 + ], + "score": 1.0, + "content": "Lemma H.11. Suppose random variables", + "type": "text" + }, + { + "bbox": [ + 278, + 538, + 334, + 551 + ], + "score": 0.84, + "content": "\\| X \\| _ { \\infty } \\leq 2 H", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 536, + 338, + 552 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 338, + 538, + 392, + 551 + ], + "score": 0.85, + "content": "\\| Y \\| _ { \\infty } \\leq 2 H", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 536, + 417, + 552 + ], + "score": 1.0, + "content": ", then", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 536, + 417, + 552 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 220, + 555, + 389, + 569 + ], + "lines": [ + { + "bbox": [ + 220, + 555, + 389, + 569 + ], + "spans": [ + { + "bbox": [ + 220, + 555, + 389, + 569 + ], + "score": 0.86, + "content": "\\left| \\operatorname { V a r } ( X ) - \\operatorname { V a r } ( Y ) \\right| \\leq 8 H \\cdot \\left\\| X - Y 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\\big ) + \\frac { 2 H ^ { 4 } \\sqrt { d } } { K }" + }, + { + "category_id": 14, + "poly": [ + 578, + 1127, + 1119, + 1127, + 1119, + 1190, + 578, + 1190 + ], + "score": 0.92, + "latex": "\\Gamma _ { h } O ( \\sqrt { d } \\cdot ( \\phi ( \\cdot , \\cdot ) ^ { \\top } \\widehat { \\Lambda } _ { h } ^ { - 1 } \\phi ( \\cdot , \\cdot ) ) ^ { 1 / 2 } ) + \\frac { 2 H ^ { 3 } \\sqrt { d } } { K }" + }, + { + "category_id": 14, + "poly": [ + 310, + 836, + 1384, + 836, + 1384, + 917, + 310, + 917 + ], + "score": 0.92, + "latex": "\\widehat { w } _ { h } : = \\operatorname { a r g m i n } _ { w \\in \\mathbb { R } ^ { d } } \\lambda \\| w \\| _ { 2 } ^ { 2 } + \\sum _ { k = 1 } ^ { K } \\frac { \\left[ \\langle \\phi ( s _ { h } ^ { k } , a _ { h } ^ { k } ) , w \\rangle - r _ { h } ^ { k } - \\widehat { V } _ { h + 1 } ( s _ { h + 1 } ^ { \\prime k } ) \\right] ^ { 2 } } { \\widehat { \\sigma } _ { h } ^ { 2 } ( s _ { h } ^ { k } , a _ { h } ^ { k } ) } = \\widehat { \\Lambda } _ { h } ^ { 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Universal constant C.
2:Initialization: Set VH+1(-) ← 0.
3:forh=H,H-1,...,1do
4:Phase1: Regular Least-square Value Iteration for conditional variances K
5:SetΣ← M (sh,ah)(sh,ah)+λI T=1
6:1∑=1Φ(sh,aT).Vn+1(Sh+1)² K
7:1∑=1(sh,aT)·Vn+1(Sh+1)
8:Set[VarnV+1](,)=<Φ(,),n)[(H+1)2]-[Φ(,),0){0H+²
9:Setδh(-,)² ← max{1, Var pn Vh+1(-,)}
10:Phase2: Weighted Least-square Value Iteration for pessimistic updates
11:
12:SetWh←An1(∑=1(sh,ah).(rh+Vn+1(sh+1))/²(sh,a))
13:SetTh(-,)←C√a.((,.)TA¹Φ(,)) 1/2 2Hd (Use Ih for the improved version) + K
14:Set Qn(·,·)←Φ(.,·)Twh-Ih(·,·)
15:Set Qn(-,)←min{Qh(-,-),H-h+1}+
16: 17: end forSetπn(-1-)← argmaxπh {Qn(,·),πh(-1·)>A,V(-)← maxπn{Qh(,),πh(- |-)>A
18:
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}" + }, + { + "category_id": 13, + "poly": [ + 495, + 267, + 1027, + 267, + 1027, + 308, + 495, + 308 + ], + "score": 0.87, + "latex": "\\begin{array} { r } { \\mathcal { D } = \\{ \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } , r _ { h } ^ { \\tau } \\right) \\} _ { \\tau , h = 1 } ^ { K , H } \\mathcal { D } ^ { \\prime } = \\{ \\left( \\bar { s } _ { h } ^ { \\tau } , \\bar { a } _ { h } ^ { \\tau } , \\bar { r } _ { h } ^ { \\tau } \\right) \\} _ { \\tau , h = 1 } ^ { K , H } } \\end{array}" + }, + { + "category_id": 13, + "poly": [ + 403, + 393, + 823, + 393, + 823, + 428, + 403, + 428 + ], + "score": 0.86, + "latex": "\\begin{array} { r } { \\bar { \\Sigma } _ { h } \\sum _ { \\tau = 1 } ^ { K } \\phi ( \\bar { s } _ { h } ^ { \\tau } , \\bar { a } _ { h } ^ { \\tau } ) \\phi ( \\bar { s } _ { h } ^ { \\tau } , \\bar { a } _ { h } ^ { \\tau } ) ^ { \\top } + \\lambda I } \\end{array}" + }, + { + "category_id": 13, + "poly": [ + 403, + 429, + 846, + 429, + 846, + 463, + 403, + 463 + ], + "score": 0.8, + "latex": "\\begin{array} { r } { \\bar { \\beta } _ { h } \\gets \\bar { \\Sigma } _ { h } ^ { - 1 } \\sum _ { \\tau = 1 } ^ { K } \\phi ( \\bar { s } _ { h } ^ { \\tau } , \\bar { a } _ { h } ^ { \\tau } ) \\cdot \\widehat { V } _ { h + 1 } ( \\bar { s } _ { h + 1 } ^ { \\tau } ) ^ { 2 } } \\end{array}" + }, + { + "category_id": 13, + "poly": [ + 404, + 603, + 982, + 603, + 982, + 640, + 404, + 640 + ], + "score": 0.77, + "latex": "\\begin{array} { r } { \\widehat { \\Lambda } _ { h } \\gets \\sum _ { \\tau = 1 } ^ { \\tilde { K } } \\phi \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) \\phi \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\right) ^ { \\top } / \\widehat { \\sigma } ^ { 2 } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) + \\lambda \\cdot I } \\end{array}" + }, + { + "category_id": 13, + "poly": [ + 1212, + 1879, + 1241, + 1879, + 1241, + 1906, + 1212, + 1906 + ], + "score": 0.76, + "latex": "H" + }, + { + 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\\left. \\phi ( \\cdot , \\cdot ) , \\bar { \\beta } _ { h } \\right. _ { \\left[ 0 , ( H - h + 1 ) ^ { 2 } \\right] } - \\left[ \\left. \\phi ( \\cdot , \\cdot ) , \\bar { \\theta } _ { h } \\right. _ { \\left[ 0 , H - h + 1 \\right] } \\right] ^ { 2 } } \\end{array}" + }, + { + "category_id": 13, + "poly": [ + 377, + 339, + 600, + 339, + 600, + 367, + 377, + 367 + ], + "score": 0.63, + "latex": "h = H , H - 1 , \\ldots , 1" + }, + { + "category_id": 14, + "poly": [ + 336, + 600, + 1349, + 600, + 1349, + 850, + 336, + 850 + ], + "score": 0.62, + "latex": "\\begin{array} { r l } & { \\mathrm { S e t } \\widehat { \\Lambda } _ { h } \\gets \\sum _ { \\tau = 1 } ^ { \\overline { { K } } } \\phi ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) ^ { \\widehat { \\circ } } ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } ) ^ { \\top } / \\widehat { \\sigma } ^ { 2 } \\big ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } \\big ) ^ { \\widehat { \\mathbf { \\phi } } } + \\lambda \\cdot I , } 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QLORA backpropagates gradients through a frozen, 4-bit quantized pretrained language model into Low Rank Adapters (LoRA). Our best model family, which we name Guanaco, outperforms all previous openly released models on the Vicuna benchmark, reaching $9 9 . 3 \%$ of the performance level of ChatGPT while only requiring 24 hours of finetuning on a single GPU. QLORA introduces a number of innovations to save memory without sacrificing performance: (a) 4-bit NormalFloat (NF4), a new data type that is information theoretically optimal for normally distributed weights (b) Double Quantization to reduce the average memory footprint by quantizing the quantization constants, and (c) Paged Optimizers to manage memory spikes. We use QLORA to finetune more than 1,000 models, providing a detailed analysis of instruction following and chatbot performance across 8 instruction datasets, multiple model types (LLaMA, T5), and model scales that would be infeasible to run with regular finetuning (e.g. 33B and 65B parameter models). Our results show that QLoRA finetuning on a small high-quality dataset leads to state-of-the-art results, even when using smaller models than the previous SoTA. We provide a detailed analysis of chatbot performance based on both human and GPT-4 evaluations showing that GPT-4 evaluations are a cheap and reasonable alternative to human evaluation. Furthermore, we find that current chatbot benchmarks are not trustworthy to accurately evaluate the performance levels of chatbots. A lemon-picked analysis demonstrates where Guanaco fails compared to ChatGPT. We release all of our models and code, including CUDA kernels for 4-bit training.2 + +# 1 Introduction + +Finetuning large language models (LLMs) is a highly effective way to improve their performance, [40, 63, 43, 62, 60, 37] and to add desirable or remove undesirable behaviors [43, 2, 4]. However, finetuning very large models is prohibitively expensive; regular 16-bit finetuning of a LLaMA 65B parameter model [58] requires more than $7 8 0 ~ \mathrm { G B }$ of GPU memory. While recent quantization methods can reduce the memory footprint of LLMs [14, 13, 18, 67], such techniques only work for inference and break down during training [66]. + +We demonstrate for the first time that it is possible to finetune a quantized 4-bit model without any performance degradation. Our method, QLORA, uses a novel high-precision technique to quantize a pretrained model to 4-bit, then adds a small set of learnable Low-rank Adapter weights [28] that are tuned by backpropagating gradients through the quantized weights. + +QLORA reduces the average memory requirements of finetuning a 65B parameter model from $> 7 8 0 \mathrm { G B }$ of GPU memory to ${ < } 4 8 \mathrm { G B }$ without degrading the runtime or predictive performance compared to a 16- bit fully finetuned baseline. This marks a significant shift in accessibility of LLM finetuning: now the largest publicly available models to date finetunable on a single GPU. Using QLORA, we train the Guanaco family of models, with the second best model reaching $9 7 . 8 \%$ of the performance level of ChatGPT on the Vicuna [10] benchmark, while being trainable in less than 12 hours on a single consumer GPU; using a single professional GPU over 24 hours we achieve $9 9 . 3 \%$ with our largest model, essentially closing the gap to ChatGPT on the Vicuna benchmark. When deployed, our smallest Guanaco model (7B parameters) requires just 5 GB of memory and outperforms a 26 GB Alpaca model by more than 20 percentage points on the Vicuna benchmark (Table 4). + +Table 1: Elo ratings for a competition between models, averaged for 10,000 random initial orderings. The winner of a match is determined by GPT-4 which declares which response is better for a given prompt of the the Vicuna benchmark. $9 5 \%$ confidence intervals are shown $( \pm )$ . After GPT4, Guanaco 33B and 65B win the most matches, while Guanaco 13B scores better than Bard. + +
ModelSizeElo
GPT-4=1348 ±1
Guanaco 65B41 GB1022 ±1
Guanaco 33B21 GB992 ±1
Vicuna 13B26 GB974±1
ChatGPT=966±1
Guanaco 13B10 GB916±1
Bard-902±1
Guanaco 7B6 GB879±1
+ +QLORA introduces multiple innovations designed to reduce memory use without sacrificing performance: (1) 4-bit NormalFloat, an information theoretically optimal quantization data type for normally distributed data that yields better empirical results than 4-bit Integers and 4-bit Floats. (2) Double Quantization, a method that quantizes the quantization constants, saving an average of about 0.37 bits per parameter (approximately 3 GB for a 65B model). (3) Paged Optimizers, using NVIDIA unified memory to avoid the gradient checkpointing memory spikes that occur when processing a mini-batch with a long sequence length. We combine these contributions into a better tuned LoRA approach that includes adapters at every network layer and thereby avoids almost all of the accuracy tradeoffs seen in prior work. + +QLORA’s efficiency enables us to perform an in-depth study of instruction finetuning and chatbot performance on model scales that would be impossible using regular finetuning due to memory overhead. Therefore, we train more than 1,000 models across several instruction tuning datasets, model architectures, and sizes between 80M to 65B parameters. In addition to showing that QLORA recovers 16-bit performance (§4) and training a state-of-the-art chatbot, Guanaco, (§5), we also analyze trends in the trained models. First, we find that data quality is far more important than dataset size, e.g., a $9 \mathrm { k }$ sample dataset (OASST1) outperformed a $4 5 0 \mathrm { k }$ sample dataset (FLAN v2, subsampled) on chatbot performance, even when both are meant to support instruction following generalization. Second, we show that strong Massive Multitask Language Understanding (MMLU) benchmark performance does not imply strong Vicuna chatbot benchmark performance and vice versa—in other words, dataset suitability matters more than size for a given task. + +Furthermore, we also provide a extensive analysis of chatbot performance that uses both human raters and GPT-4 for evaluation. We use tournament-style benchmarking where models compete against each other in matches to produce the best response for a given prompt. The winner of a match is judged by either GPT-4 or human annotators. The tournament results are aggregated into Elo scores [16, 17] which determine the ranking of chatbot performance. We find that GPT-4 and human evaluations largely agree on the rank of model performance in the tournaments, but we also find there are instances of strong disagreement. As such, we highlight that model-based evaluation while providing a cheap alternative to human-annotation also has its uncertainties. + +We augment our chatbot benchmark results with a qualitative analysis of Guanaco models (see Appendix F). Our analysis highlights success and failure cases that were not captured by the quantitative benchmarks. + +We release all model generations with human and GPT-4 annotations to facilitate further study. We open-source our codebase and CUDA kernels and integrate our methods into the Hugging Face transformers stack [65], making them easily accessible to all. We release a collection of adapters for 7/13/33/65B size models, trained on 8 different instruction following datasets, for a total of 32 different open sourced, finetuned models. + +![](images/677098bf45a97bff5ad9433443fa051502c9f8d9c93447eff7503f236d343ec3.jpg) +Figure 1: Different finetuning methods and their memory requirements. QLORA improves over LoRA by quantizing the transformer model to 4-bit precision and using paged optimizers to handle memory spikes. + +# 2 Background + +Block-wise $\mathbf { k }$ -bit Quantization Quantization is the process of discretizing an input from a representation that holds more information to a representation with less information. It often means taking a data type with more bits and converting it to fewer bits, for example from 32-bit floats to 8-bit Integers. To ensure that the entire range of the low-bit data type is used, the input data type is commonly rescaled into the target data type range through normalization by the absolute maximum of the input elements, which are usually structured as a tensor. For example, quantizing a 32-bit Floating Point (FP32) tensor into a Int8 tensor with range $[ - 1 2 7 , 1 2 7 ]$ : + +$$ +\mathbf { X } ^ { \mathrm { I n t 8 } } = \mathrm { r o u n d } \left( { \frac { 1 2 7 } { \mathrm { a b s m a x } ( \mathbf { X } ^ { \mathrm { F P 3 2 } } ) } } \mathbf { X } ^ { \mathrm { F P 3 2 } } \right) = \mathrm { r o u n d } ( c ^ { \mathrm { F P 3 2 } } \cdot \mathbf { X } ^ { \mathrm { F P 3 2 } } ) , +$$ + +where $c$ is the quantization constant or quantization scale. Dequantization is the inverse: + +$$ +\operatorname * { d e q u a n t } ( c ^ { \mathrm { F P 3 2 } } , \mathbf { X } ^ { \mathrm { I n t 8 } } ) = \frac { \mathbf { X } ^ { \mathrm { I n t 8 } } } { c ^ { \mathrm { F P 3 2 } } } = \mathbf { X } ^ { \mathrm { F P 3 2 } } +$$ + +The problem with this approach is that if a large magnitude value (i.e., an outlier) occurs in the input tensor, then the quantization bins—certain bit combinations—are not utilized well with few or no numbers quantized in some bins. To prevent the outlier issue, a common approach is to chunk the input tensor into blocks that are independently quantized, each with their own quantization constant $c$ . This can be formalized as follows: We chunk the input tensor $\mathbf { X } \in \mathbb { R } ^ { b \times h }$ into $n$ contiguous blocks of size $B$ by flattening the input tensor and slicing the linear segment into $n = ( b \times h ) / B$ blocks. We quantize these blocks independently with Equation 1 to create a quantized tensor and $n$ quantization constants $c _ { i }$ . + +Low-rank Adapters Low-rank Adapter (LoRA) finetuning [28] is a method that reduces memory requirements by using a small set of trainable parameters, often termed adapters, while not updating the full model parameters which remain fixed. Gradients during stochastic gradient descent are passed through the fixed pretrained model weights to the adapter, which is updated to optimize the loss function. LoRA augments a linear projection through an additional factorized projection. Given a projection $\mathbf { X } \mathbf { W } = \mathbf { Y }$ with $\mathbf { X } \in \mathbb { R } ^ { b \times h }$ , $\breve { \mathbf { W } } \in \mathbb { R } ^ { h \times o }$ LoRA computes: + +$$ +\mathbf { Y } = \mathbf { X } \mathbf { W } + s \mathbf { X } \mathbf { L } _ { 1 } \mathbf { L } _ { 2 } , +$$ + +where $\mathbf { L } _ { 1 } \in \mathbb { R } ^ { h \times r }$ and $\mathbf { L } _ { 2 } \in \mathbb { R } ^ { r \times o }$ , and $s$ is a scalar. + +# 3 QLORA Finetuning + +QLORA achieves high-fidelity 4-bit finetuning via two techniques we propose—4-bit NormalFloat (NF4) quantization and Double Quantization. Additionally, we introduce Paged Optimizers, to prevent memory spikes during gradient checkpointing from causing out-of-memory errors that have traditionally made finetuning on a single machine difficult for large models. + +QLORA has one low-precision storage data type, in our case usually 4-bit, and one computation data type that is usually BFloat16. In practice, this means whenever a QLORA weight tensor is used, we dequantize the tensor to BFloat16, and then perform a matrix multiplication in 16-bit. + +We now discuss the components of QLORA followed by a formal definition of QLORA. + +4-bit NormalFloat Quantization The NormalFloat (NF) data type builds on Quantile Quantization [15, 46] which is an information-theoretically optimal data type that ensures each quantization bin has an equal number of values assigned from the input tensor. Quantile quantization works by estimating the quantile of the input tensor through the empirical cumulative distribution function. + +The main limitation of quantile quantization is that the process of quantile estimation is expensive. Therefore fast quantile approximation algorithms, such as SRAM quantiles [15], are used to estimate them. Due to the approximate nature of these quantile estimation algorithms, the data type has large quantization errors for outliers, which are often the most important values. + +Expensive quantile estimates and approximation errors can be avoided when input tensors come from a distribution fixed up to a quantization constant. In such cases, input tensors have the same quantiles making exact quantile estimation computationally feasible. + +Since pretrained neural network weights usually have a zero-centered normal distribution with standard deviation $\sigma$ (see Appendix I), we can transform all weights to a single fixed distribution by scaling $\sigma$ such that the distribution fits exactly into the range of our data type. For our data type, we set the arbitrary range $[ - 1 , 1 ]$ . As such, both the quantiles for the data type and the neural network weights need to be normalized into this range. + +The information theoretically optimal data type for zero-mean normal distributions with arbitrary standard deviations $\sigma$ in the range $[ - 1 , 1 ]$ is computed as follows: (1) estimate the $2 ^ { k } + 1$ quantiles of a theoretical $N ( 0 , 1 )$ distribution to obtain a $k$ -bit quantile quantization data type for normal distributions, (2) take this data type and normalize its values into the $[ - 1 , 1 ]$ range, (3) quantize an input weight tensor by normalizing it into the $[ - 1 , 1 ]$ range through absolute maximum rescaling. + +Once the weight range and data type range match, we can quantize as usual. Step (3) is equivalent to rescaling the standard deviation of the weight tensor to match the standard deviation of the k-bit data type. More formally, we estimate the $2 ^ { k }$ values $q _ { i }$ of the data type as follows: + +$$ +q _ { i } = \frac { 1 } { 2 } \left( Q _ { X } \left( \frac { i } { 2 ^ { k } + 1 } \right) + Q _ { X } \left( \frac { i + 1 } { 2 ^ { k } + 1 } \right) \right) , +$$ + +where $Q _ { X } ( \cdot )$ is the quantile function of the standard normal distribution $N ( 0 , 1 )$ . A problem for a symmetric $\mathbf { k }$ -bit quantization is that this approach does not have an exact representation of zero, which is an important property to quantize padding and other zero-valued elements with no error. To ensure a discrete zeropoint of 0 and to use all $2 ^ { k }$ bits for a $\mathbf { k }$ -bit datatype, we create an asymmetric data type by estimating the quantiles $q _ { i }$ of two ranges $q _ { i }$ : $2 ^ { k - 1 }$ for the negative part and $2 ^ { k - 1 } + 1$ for the positive part and then we unify these sets of $q _ { i }$ and remove one of the two zeros that occurs in both sets. We term the resulting data type that has equal expected number of values in each quantization bin $k$ -bit NormalFloat (NFk), since the data type is information-theoretically optimal for zero-centered normally distributed data. A step-by-step visualization of how to construct the NF4 data type and its exact values can be found in Appendix H. + +Double Quantization We introduce Double Quantization (DQ), the process of quantizing the quantization constants for additional memory savings. While a small blocksize is required for precise 4-bit quantization [13], it also has a considerable memory overhead. For example, using 32-bit constants and a blocksize of 64 for W, quantization constants add $3 2 / 6 4 = 0 . 5 $ bits per parameter on average. Double Quantization helps reduce the memory footprint of quantization constants. + +More specifically, Double Quantization treats quantization constants as inputs to a second quantization. This second step yields the qua $c _ { 2 } ^ { \mathrm { F P } 3 2 }$ of the first quantizationd quantization constants $c _ { 2 } ^ { \mathrm { F P 8 } }$ and the second level of quantization constants $\dot { c } _ { 1 } ^ { \mathrm { F } \tilde { \mathrm { P } } 3 2 }$ . We use 8-bit Floats with a blocksize of 256 for the second quantization as no performance degradation is observed for 8-bit quantization, in line with results from Dettmers and Zettlemoyer [13]. Since the $c _ { 2 } ^ { \mathrm { F P } 3 2 }$ are positive, we subtract the mean from $c _ { 2 }$ before quantization to center the values around zero and make use of symmetric quantization. On average, for a blocksize of 64, this quantization reduces the memory footprint per parameter from $3 2 / 6 4 \overset { \cdot } { = } 0 . 5$ bits, to $8 / 6 4 + 3 2 / ( 6 4 \cdot 2 5 6 ) = 0 . 1 2 7$ bits, a reduction of 0.373 bits per parameter. + +Paged Optimizers use the NVIDIA unified memory 3 feature which does automatic page-to-page transfers between the CPU and GPU for error-free GPU processing in the scenario where the GPU occasionally runs out-of-memory. The feature works like regular memory paging between CPU RAM and the disk. We use this feature to allocate paged memory for the optimizer states which are then automatically evicted to CPU RAM when the GPU runs out-of-memory and paged back into GPU memory when the memory is needed in the optimizer update step. + +QLORA. Using the components described above, we define QLORA for a single linear layer in the quantized base model with a single LoRA adapter as follows: + +$$ +{ \bf Y } ^ { \mathrm { B F 1 6 } } = { \bf X } ^ { \mathrm { B F 1 6 } } \mathrm { d o u b l e D e q u a n t } ( c _ { 1 } ^ { \mathrm { F P 3 2 } } , c _ { 2 } ^ { \mathrm { k - b i t } } , { \bf W } ^ { \mathrm { N F 4 } } ) + { \bf X } ^ { \mathrm { B F 1 6 } } { \bf L } _ { 1 } ^ { \mathrm { B F 1 6 } } { \bf L } _ { 2 } ^ { \mathrm { B F 1 6 } } , +$$ + +$$ +\mathrm { d o u b l e D e q u a n t } ( c _ { 1 } ^ { \mathrm { F 3 2 } } , c _ { 2 } ^ { \mathrm { k - b i t } } , \mathbf { W } ^ { \mathrm { k - b i t } } ) = \mathrm { d e q u a n t } ( \mathrm { d e q u a n t } ( c _ { 1 } ^ { \mathrm { F P 3 2 } } , c _ { 2 } ^ { \mathrm { k - b i t } } ) , \mathbf { W } ^ { \mathrm { 4 b i t } } ) = \mathbf { W } ^ { \mathrm { B F l 6 } } , +$$ + +We use NF4 for W and FP8 for $c _ { 2 }$ . We use a blocksize of 64 for $\mathbf { W }$ for higher quantization precision and a blocksize of 256 for $c _ { 2 }$ to conserve memory. + +For parameter updates only the gradient with respect to the error for the adapters weights $\frac { \partial E } { \partial { \bf L } _ { i } }$ are needed, and not for 4-bit weights $\frac { \partial E } { \partial \mathbf { W } }$ . However, the calculation of $\frac { \partial E } { \partial \mathbf { L } _ { i } }$ entails the calculation of $\textstyle \frac { \partial \mathbf { X } } { \partial \mathbf { W } }$ which proceeds via equation (5) with dequantization from storage $\dot { \mathbf { W } } ^ { \mathrm { N F 4 } }$ to computation data type ${ \bf W } ^ { \mathrm { B F 1 6 } }$ to calculate the derivative $\textstyle \frac { \partial \mathbf { X } } { \partial \mathbf { W } }$ in BFloat16 precision. + +To summarize, QLORA has one storage data type (usually 4-bit NormalFloat) and a computation data type (16-bit BrainFloat). We dequantize the storage data type to the computation data type to perform the forward and backward pass, but we only compute weight gradients for the LoRA parameters which use 16-bit BrainFloat. + +# 4 QLoRA vs. Standard Finetuning + +We have discussed how QLoRA works and how it can significantly reduce the required memory for finetuning models. The main question now is whether QLoRA can perform as well as full-model finetuning. Furthermore, we want to analyze the components of QLoRA including the impact of NormalFloat4 over standard Float4. The following sections will discuss the experiments that aimed at answering these questions. + +Experimental setup. We consider three architectures (encoder, encoder-decoder, and decoder only) and compare QLoRA with 16-bit adapter-finetuning and with full-finetuning for models up to 3B. Our evaluations include GLUE [59] with RoBERTalarge [38], Super-NaturalInstructionss [62] with T5 [50], and 5-shot MMLU [24] after finetuning LLaMA on Flan v2 [39] and Alpaca [56]. + +![](images/0e8ef49dc35c957f2d941190c1efe1eb5fb2a0ed7a9fe46e985ecb64c865a02e.jpg) +Figure 2: Mean zero-shot accuracy over Winogrande, HellaSwag, PiQA, Arc-Easy, and ArcChallenge using LLaMA models with different 4-bit data types. The NormalFloat data type significantly improves the bit-for-bit accuracy gains compared to regular 4-bit Floats. While Double Quantization (DQ) only leads to minor gains, it allows for a more fine-grained control over the memory footprint to fit models of certain size (33B/65B) into certain GPUs (24/48GB). + +To additionally study the advantages of NF4 over other 4-bit data types, we use the setup of Dettmers and Zettlemoyer [13] and measure post-quantization zero-shot accuracy and perplexity across different models (OPT [73], LLaMA [58], + +BLOOM [53], Pythia [7]) for model sizes $1 2 5 \mathrm { m } - 1 3 \mathrm { B }$ . We provide more details in the results section for each particular setup to make the results more readable. Full details in Appendix C. + +While paged optimizers are critical to do 33B/65B QLORA tuning on a single 24/48GB GPU, we do not provide hard measurements for Paged Optimizers since the paging only occurs when processing mini-batches with long sequence lengths, which is rare. We do, however, perform an analysis of the runtime of paged optimizers for 65B models on 48GB GPUs and find that with a batch size of 16, paged optimizers provide the same training speed as regular optimizers. Future work should measure and characterize under what circumstances slow-downs occur from the paging process. + +Table 3: Mean 5-shot MMLU test accuracy for LLaMA 7-65B models finetuned with adapters on Alpaca and FLAN v2 for different data types. Overall, NF4 with double quantization (DQ) matches BFloat16 performance, while FP4 is consistently one percentage point behind both. + +
LLaMA SizeMean 5-shot MMLU AccuracyMean
7B13B33B65B
AlpacaFLAN v2AlpacaFLAN v2AlpacaFLAN v2AlpacaFLAN v2
BFloat1638.445.647.250.657.760.561.862.553.0
Float437.244.047.350.055.958.561.363.352.2
NFloat4 + DQ39.044.547.550.757.359.261.863.953.1
+ +4-bit NormalFloat yields better performance than 4-bit Floating Point While the 4-bit NormalFloat (NF4) data type is information-theoretically optimal, it still needs to be determined if this property translates to empirical advantages. We follow the setup from Dettmers and Zettlemoyer [13] where quantized LLMs (OPT [73], BLOOM [53], Pythia [7], LLaMA) of different sizes (125M to 65B) with different data types are evaluated on language modeling and a set of zero-shot tasks. In Figure 7 and Table 2 we see that NF4 improves performance significantly over FP4 and Int4 and that double quantization reduces the memory footprint without degrading performance. + +Table 2: Pile Common Crawl mean perplexity for different data types for 125M to 13B OPT, BLOOM, LLaMA, and Pythia models. + +
Data typeMean PPL
Int434.34
Float4 (E2M1)31.07
Float4 (E3M0)29.48
NFloat4 + DQ27.41
+ +k-bit QLORA matches 16-bit full finetuning and 16-bit LoRA performance Recent findings have established that 4-bit quantization for inference is possible, but leads to performance degradation relative to 16-bit [13, 18]. This raises the crucial question of whether the lost performance can be recovered by conducting 4-bit adapter finetuning. We test this for two setups. + +The first setup test if QLoRA can replicate 16-bit full finetuning performance for T5 and RoBERTa model finetuning. These results are detailed in the Appendix C.1. For our second setup, since full finetuning models at and beyond 11B parameters requires more than one server of high memory GPUs, we continue to test whether 4-bit QLORA can match 16-bit LoRA at the 7B to 65B parameter scales. To this end, we finetune LLaMA 7B through 65B on two instruction following datasets, Alpaca and FLAN v2, and evaluate on the MMLU benchmark via 5-shot accuracy. Results are shown in Table 3 where we see that NF4 with double quantization fully recovers the 16-bit LoRA MMLU performance. In addition, we also note that QLORA with FP4 lags behind the 16-bit brain float LoRA baseline by about 1 percentage point. This corroborates both our findings that (1) QLORA with NF4 replicates both 16-bit full finetuning and 16-bit LoRA finetuning performance, and (2) NF4 is superior to FP4 in terms of quantization precision. + +Summary Our results consistently show that 4-bit QLORA with NF4 data type matches 16- bit full finetuning and 16-bit LoRA finetuning performance on academic benchmarks with wellestablished evaluation setups. We have also shown that NF4 is more effective than FP4 and that double quantization does not degrade performance. Combined, this forms compelling evidence that 4-bit QLORA tuning reliably yields results matching 16-bit methods. + +In line with previous work on quantization [13], our MMLU and Elo results indicate that with a given finetuning and inference resource budget it is beneficial to increase the number of parameters in the base model while decreasing their precision. This highlights the importance of efficiency benefits from QLORA. Since we did not observe performance degradation compared to full-finetuning in our experiments with 4-bit finetuning, this raises the question of where the performance-precision trade-off exactly lies for QLoRA tuning, which we leave to future work to explore. + +We proceed to investigate instruction tuning at scales that would be impossible to explore with full 16-bit finetuning on academic research hardware. + +# 5 Pushing the Chatbot State-of-the-art with QLoRA + +Having established that 4-bit QLORA matches 16-bit performance across scales, tasks, and datasets we conduct an in-depth study of instruction finetuning up to the largest open-source language models available for research. To assess the performance of instruction finetuning these models, we evaluate on a challenging Natural Language Understanding benchmark (MMLU) and develop new methods for real-world chatbot performance evaluation. For a more qualitative analysis, see Appendix F. + +# 5.1 Experimental setup + +We now describe an overview of the experimental setup with full details in Appendix D. + +Data As, to our knowledge, there is no comprehensive study of instruction-following datasets, we select eight recent datasets. We include datasets obtained through crowd-sourcing (OASST1 [31], HH-RLHF [4]), distillation from instruction-tuned models (Alpaca [56], self-instruct [60], unnaturalinstructions [26]), corpora aggregations (FLAN v2 [12]), as well as hybrids (Chip2 [32], Longform [30]). These datasets cover different languages, data sizes, and licenses. + +Training Setup To avoid confounding effects from different training objectives, we perform QLoRA finetuning with cross-entropy loss (supervised learning) without reinforcement learning, even for datasets that include human judgments of different responses. For datasets that have a clear distinction between instruction and response, we finetune only on the response (see ablations in Appendix D). For OASST1 and HH-RLHF, multiple responses are available. We then select the top response at every level of the conversation tree and finetune on the full selected conversation, including the instructions. In all of our experiments, we use NF4 QLORA with double quantization and paged optimizers to prevent memory spikes during gradient checkpointing. We do a small hyperparameter searches for the 13B and 33B LLaMA models and we find that all hyperparameter settings found at 7B generalize (including number of epochs) except learning rate and batch size. We halve the learning rate for 33B and 65B while doubling the batch size. + +Baselines We compare our models to both research (Vicuna [10] and Open Assistant [31]) and commercial (GPT-4 [42], GPT-3.5-turbo and Bard) chatbot systems. The Open Assistant model is a LLaMA 33B model finetuned with Reinforcement Learning from Human Feedback (RLHF) on the OASST1 dataset. Vicuna does full fine-tuning of LLaMA 13B on proprietary user-shared conversations from ShareGPT and is thus the result of distillation from OpenAI GPT models. + +Evaluation setup Chatbot evaluation is not straightforward since each prompt has many highquality responses which are challenging to rank. We therefore follow a comprehensive approach, which involves (a) standard benchmarks that measures general language understanding performance (MMLU [24]), (b) both automatic and human evaluation that measures how many responses from chatbot A are better compared to chatbot B, (c) a round-robin tournament-style evaluation where chatbots compete against each other in games where chatbot performance is measured as ELO. We provide a detailed discussion of these evaluation setups in the Appendix E. + +# 5.2 Guanaco: QLORA trained on OASST1 is a State-of-the-art Chatbot + +Based on our automated and human evaluations, we find that the top QLORA tuned model, Guanaco 65B, which we finetune on a variant of OASST1, is the best-performing open-source chatbot model and offers performance competitive to ChatGPT. When compared to GPT-4, Guanaco 65B and 33B have an expected win probability of $30 \%$ , based on Elo rating from human annotators system-level pairwise comparisons on the Vicuna benchmark - the highest reported to date. + +The Vicuna benchmark [10] results relative to ChatGPT are shown in Table 4. We find that Guanaco 65B is the best-performing model after GPT-4, achieving $9 9 . 3 \%$ performance relative to ChatGPT. Guanaco 33B has more parameters than the Vicuna 13B model, but uses only 4-bit precision for its weights and is thus much more memory efficient at 21 GB vs $2 6 \mathrm { G B }$ , providing a three percentage points of improvement over Vicuna 13B. Furthermore, Guanaco 7B easily fits on modern phones at a 5 GB footprint while still scoring nearly 20 percentage points higher than Alpaca 13B. + +However, Table 4 also has very wide confidence intervals, with many models overlapping in performance. We hypothesize that this uncertainty comes from the lack of clear specification of scale, e.g., it is unclear what 8 on a 10 point scale means across different scenarios. As such, we instead recommend using the Elo ranking method [16], based on pairwise judgments from human annotators and GPT-4 to avoid the problem of grounding an absolute scale. + +Table 4: Zero-shot Vicuna benchmark scores as a percentage of the score obtained by ChatGPT evaluated by GPT-4. We see that OASST1 models perform close to ChatGPT despite being trained on a very small dataset and having a fraction of the memory requirement of baseline models. + +
Model/DatasetParamsModel bitsMemoryChatGPT vs SysSys vs ChatGPTMean95% CI
GPT-4 Bard===119.4%110.1%114.5%2.6%
1--93.2%96.4%94.8%4.1%
Guanaco Alpaca FLAN v265B4-bit41 GB96.7%101.9%99.3%4.4%
65B4-bit41 GB63.0%77.9%70.7%4.3%
65B4-bit41 GB37.0%59.6%48.4%4.6%
Guanaco Open Assistant Alpaca FLAN v233B4-bit21 GB96.5%99.2%97.8%4.4%
33B16-bit66 GB73.4%85.7%78.1%5.3%
33B 33B4-bit 4-bit21GB 21GB67.2%79.7%73.6%4.2%
Vicuna Guanaco Alpaca HH-RLHF26.3%49.7%38.0%3.9%
13B 13B16-bit 4-bit26 GB 10 GB91.2% 87.3%98.7% 93.4%94.9% 90.4%4.5% 5.2%
13B4-bit10 GB63.8%76.7%69.4%
13B4-bit10 GB55.5%69.1%62.5%4.2%
13B4-bit10 GB50.6%69.8%60.5%4.7%
13B4-bit10 GB49.2%69.3%59.5%4.2% 4.7%
Longform Self-Instruct FLAN v213B4-bit10 GB44.9%62.0%53.6%5.2%
13B4-bit10 GB38.0%60.5%49.1%4.6%
13B4-bit10 GB32.4%61.2%47.0%3.6%
4-bit5GB
Guanaco Alpaca FLAN v27B 7B4-bit5GB84.1% 57.3%89.8% 71.2%87.0%5.4%
7B4-bit5GB33.3%56.1%64.4% 44.8%5.0% 4.0%
+ +Elo ratings of the most competitive models can be seen in Table 1. We note that human and GPT-4 ranking of models on the Vicuna benchmark disagree partially, particularly for Guanaco 7B, but are consistent for most models with a Kendall Tau of $\tau = 0 . 4 3$ and Spearman rank correlation of $r = 0 . 5 5$ at the system level. At the example level, the agreement between GPT-4 and human annotators’ majority vote is weaker with Fleiss $\kappa = 0 . 2 5$ . Overall, this shows a moderate agreement between system-level judgments by GPT-4 and human annotators, and thus that model-based evaluation represents a somewhat reliable alternative to human evaluation. See Section E for further considerations. + +Table 5: MMLU 5-shot test results for different sizes of LLaMA finetuned on the corresponding datasets using QLoRA. + +
Dataset7B13B33B65B
LLaMA no tuning35.146.957.863.4
Self-Instruct36.433.353.056.7
Longform32.143.256.659.7
Chip234.541.653.659.8
HH-RLHF34.944.655.860.1
Unnatural Instruct41.948.157.361.3
Guanaco (OASST1)36.646.457.062.2
Alpaca38.847.857.362.5
FLAN v244.551.459.263.9
+ +Elo rankings in Table 6 indicate that Guanaco 33B and 65B models outperform all models besides GPT-4 on the Vicuna and OA benchmarks and that they perform comparably to ChatGPT in line with Table 4. We note that the Vicuna benchmark favors open-source models while the larger OA benchmark favors ChatGPT. Furthermore, we can see from Tables 5 and 4 that the suitability of a finetuning dataset is a determining factor in performance. Finetuning LLaMA models on FLAN v2 does particularly well on MMLU, but performs worst on the Vicuna benchmark (similar trends are observed with other models). This also points to partial orthogonality in current evaluation benchmarks: strong MMLU performance does not imply strong chatbot performance (as measured by Vicuna or OA benchmarks) and vice versa. + +Guanaco is the only top model in our evaluation that is not trained on proprietary data as the OASST1 dataset collection guidelines explicitly forbid the use of GPT models. The next best model trained on only open-source data is the Anthropic HH-RLHF model, which scores 30 percentage points lower than Guanaco on the Vicuna benchmark (see Table 4). Overall, these results show that 4-bit QLORA is effective and can produce state-of-the-art chatbots that rival ChatGPT. Furthermore, our 33B Guanaco can be trained on 24 GB consumer GPUs in less than 12 hours. This opens up the potential for future work via QLORA tuning on specialized open-source data, which produces models that can compete with the very best commercial models that exist today. + +Table 6: Elo rating for a tournament between models where models compete to generate the best response for a prompt, judged by human raters or GPT-4. Overall, Guanaco 65B and 33B tend to be preferred to ChatGPT-3.5 on the benchmarks studied. According to human raters they have a Each 10-point difference in Elo is approximately a difference of $1 . 5 \%$ in win-rate. + +
Benchmark # Prompts Judge ModelVicuna 80 Human ratersVicuna 80 GPT-4Open Assistant 953 GPT-4Median Rank
EloRankEloRankEloRank
GPT-41176113481129411
Guanaco-65B102321022100832
Guanaco-33B10094992100244
ChatGPT-3.5 Turbo9167966235101525
Vicuna-13B9845974493655
Guanaco-13B9756913688566
Guanaco-7B10103879886077
Bard909890271-8
+ +# 6 Related Work + +Quantization of Large Language Models Quantization of LLMs has largely focused on quantization for inference time. Major approaches for preserving 16-bit LLM quality focus on managing outlier features (e.g., SmoothQuant [67] and LLM.int8() [14]) while others use more sophisticated grouping methods [44, 70]. Lossy quantization approaches study the trade-offs for regular rounding [13, 72, 48] or how to optimize rounding decisions to improve quantization precision [18]. Besides our work, SwitchBack layers [66] is the only work that studies backpropagation through quantized weights at a scale beyond 1B parameters. + +Finetuning with Adapters While we use Low-rank Adapters [28] (LoRA), many other Parameter Efficient FineTuning (PEFT) methods have been proposed such as prompt tuning [49, 33, 34], tuning the embedding layer inputs [1], tuning hidden states $\mathrm { ( I A ^ { 3 } ) }$ [37], adding full layers [27], tuning biases [71], learning a mask over weights based on Fisher information [55], and a combination of approaches [23]. In our work, we show that LoRA adapters are able to reach full 16-bit finetuning performance. We leave it to future work to explore the tradeoffs of other PEFT approaches. + +Instruction Finetuning To help a pretrained LLM follow the instructions provided in a prompt, instruction finetuning uses input-output pairs of various data sources to finetune a pretrained LLM to generate the output given the input as a prompt. Approaches and datasets include MetaICL [40], MetaTuning [74], InstructGPT [43], FLAN [63, 12], PromptSource [3], Super-NaturalInstructions [62, 51], Self-instruct [60], UnnaturalInstructions [26], OPT-IML [29], UnifiedSKG[68], OIG/Chip2 [32], Alpaca [56], Vicuna [10], Koala [20], and Self-instruct-GPT-4 [45]. + +Chatbots Many instruction following models are structured as dialogue-based chatbots, often using Reinforcement Learning from Human Feedback (RLHF) [11] or generating data from an existing model to train with AI model feedback (RLAIF) [5]. Approaches and datasets include AnthropicHH [2, 4], Open Assistant [31], LaMDA [57], and Sparrow [21]. We do not use reinforcement learning, but our best model, Guanaco, is finetuned on multi-turn chat interactions from the Open Assistant dataset which was designed to be used for RLHF training [31]. For the evaluation of chatbots approaches that use GPT-4 instead of costly human annotation have been developed [10, 45]. We improve on such approaches with a focus on an evaluation setup that is more reliable. + +# 7 Limitations and Discussion + +We have shown evidence that our method, QLORA, can replicate 16-bit full finetuning performance with a 4-bit base model and Low-rank Adapters. However, due to the immense resource cost, we were not establish that QLORA can match 16-bit finetuning performance at 33B and 65B scales. + +Another limitation is the evaluation of instruction finetuning models. While we provide evaluations on MMLU, the Vicuna/OA benchmark, we did not evaluate on BigBench, RAFT, and HELM, and it is not ensured that our evaluations generalize to these benchmarks. On the other hand, we perform a very broad study on MMLU and develop new methods for evaluating chatbots. + +From the evidence presented, it appears that the performance of these benchmarks likely depends how similar the finetuning data is to the benchmark dataset. For example, FLAN v2 is similar to MMLU, but dissimilar to the Vicuna benchmark and vice versa for the Chip2 dataset. This highlights that not only better benchmarks and evaluation is needed, but that one needs to be careful about what one is evaluating in the first place. Do we want to create models that do well on classroom highschool and colleague knowledge or do we want to do well on chatbot conversation ability? Maybe something else? Because it is always easier to evaluate on an existing benchmark compared to creating a new one, certain benchmarks can steer the community towards a certain direction. We should ensure as a community that the benchmarks measure what we care about. + +An additional limitation is that we did not evaluate different bit-precisions or different adapter methods. Besides LoRA, there is also a wide variety Parameter Efficient FineTuning (PEFT) methods that have been shown to work well. However, it is unclear if these methods scale to large models. We used LoRA as many results established its robustness but other adapters might yield better performance. Since finetuning after quantization seems to recover most of the information that is lost during quantization this might enable much more aggressive quantization. For example, 3-bit GPTQ quantization might also yield 16-bit full finetuning performance after finetuning. + +# 8 Broader Impacts + +Our QLORA finetuning method is the first method that enables the finetuning of 33B parameter models on a single consumer GPU and 65B parameter models on a single professional GPU, while not degrading performance relative to a full finetuning baseline. We have demonstrated that our best 33B model trained on the Open Assistant dataset can rival ChatGPT on the Vicuna benchmark. Since instruction finetuning is an essential tool to transform raw pretrained LLMs into ChatGPT-like chatbots, we believe that our method will make finetuning widespread and common in particular for the researchers that have the least resources – a big win for the accessibility of state of the art NLP technology. QLORA can be seen as an equalizing factor that helps to close the resource gap between large corporations and small teams with consumer GPUs. + +Another potential source of impact is deployment to mobile phones and other low resources devices. While 7B models were shown to be able to be run on phones before, QLORA is the first method that would enable the finetuning of such models. We estimate that with an iPhone 12 Plus, QLORA can finetune 3 million tokens per night while the phone is charging. QLORA can help enable privacy-preserving usage of LLMs, where users can own and manage their own data and models, while simultaneously making LLMs easier to deploy. + +Furthermore, because of the increased inference efficiency of 4-bit models, if QLoRA models are deployed they reduce the environmental impact that LLMs have when deployed for personal use. We estimate that if $50 \%$ of deployments are personal and $50 \%$ are company deployments, QLoRA deployments could reduce the overall carbon footprint by $72 \%$ (see Appendix B). + +However, finetuning is a dual-use technology that can be abused to cause harm. Widespread use of LLMs has known dangers [8, 6], but we believe that equalizing access to a technology that is quickly becoming ubiquitous will allow for better more independent analysis than keeping the power of LLMs in the hands of large corporations that do not release models or source code for auditing. + +All in all, we believe that QLORA will have a broadly positive impact making the finetuning of high quality LLMs much more widely and easily accessible. + +# Acknowledgements + +We thank Aditya Kusupati, Ofir Press, Ashish Sharma, Margaret Li, Raphael Olivier, Zihao Ye, and Evangelia Spiliopoulou for their valuable feedback. We thank Norah Altriri for providing us with an intuitive NF4 visualization. We thank Felix Petersen for a discussion on distributional quantization and its relationship to quantile quantization. Our research was facilitated by the advanced computational, storage, and networking infrastructure of the Hyak supercomputer system at the University of Washington. 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The difference between GPUs is mostly caused by poor instruction throughput bottlenecks. + +# A Inference Speedups for NF4 + +The speedups for single batch inference for NF4 compare to 16-bit float are shown in Figure 3. + +# B Environmental Impact and Carbon Footprint Reductions + +The relationship between serving efficiency and environmental impact and cost/performance is a complicated one. The short answer is, for personal use, we reduce the environmental impact per token by about $3 . 5 \mathrm { x }$ . If $50 \%$ LLM deployments are personal and $50 \%$ company, we reduce the environmental footprint of inference by $72 \%$ . + +The long answer is this: For deployment, there are two options: (1) personal deployment, (2) deployment by companies for many users. (1) uses small batch sizes (usually batch size $^ { = 1 }$ ), (2) uses large batch sizes (64-128). Per token, (2) offers about ${ > } 5 0 \mathrm { x }$ better efficiency because for every weight that is loaded from memory, up to 64-128 tokens can be calculated. The ${ > } 5 0 \mathrm { x }$ improvement in efficiency stems from the fact memory operations are energy inefficient, and floating point operations are energy efficient. As such,company deployment (2) is a very environmentally friendly and cheap approach. + +Currently, we have efficient 4-bit CUDA kernels for the personal deployment scenario (1) with batch $\mathrm { s i z e } { = } 1$ which are about $3 . 5 \mathrm { x }$ more efficient (see Figure 3). As such, we reduce the overall footprint for personal deployment (1) significantly. + +The overall cost and footprint is now determined by how many people use (1) vs (2). For example, if $50 \%$ of users use personal deployment (1) and $50 \%$ company deployments (2) and we assume that (2) is about $5 0 \mathrm { x }$ more efficient then we get the following numbers. Approach (1) accounts for $5 0 \% / ( 5 0 \%$ $+ 5 0 \% / 5 0 ) = 9 8 \%$ of environmental impact. This means, the more users deploy personal LLMs, the larger benefit of our method. If $50 \%$ of people use personally deployed LLMs (phones, laptops etc) then our method will reduce the impact by about 1- $( 9 8 \% / 3 . 5 \mathrm { x } ) = 7 2 \%$ . + +The environmental impact reduction is roughly proportional to energy consumption per token processed which is roughly proportional to the cost of running LLMs. As such, the bottom line is affected in the same proportions. + +Table 7: Experiments comparing 16-bit BrainFloat (BF16), 8-bit Integer (Int8), 4-bit Float (FP4), and 4- bit NormalFloat (NF4) on GLUE and Super-NaturalInstructions. QLORA replicates 16-bit LoRA and fullfinetuning. + +
Dataset ModelGLUE (Acc.)Super-NaturalInstructions (RougeL)
RoBERTa-largeT5-80MT5-250MT5-780MT5-3BT5-11B
BF1688.640.142.148.054.362.0
BF16 replication88.640.042.247.354.91
LoRA BF1688.840.542.647.155.460.7
QLoRA Int888.840.442.945.456.560.7
QLORA FP488.640.342.447.555.660.9
QLORA NF4 + DQ140.442.747.755.360.9
+ +Overall, our method will have a strong impact on environmental impact and cost reduction for personal LLM deployments. + +# C QLoRA vs Standard Finetuning Experimental Setup Details + +# C.1 BF16 vs NF4 for T5/RoBERTa + +Here we detail additional results where we run experiments to confirm we can replicate 16-bit full finetuning performance when we use QLoRA. We test RoBERTA and T5 models sized 125M to 3B parameters on GLUE and the Super-NaturalInstructions dataset. Results are shown in Table 7. In both datasets, we observe that 16-bit, 8-bit, and 4-bit adapter methods replicate the performance of the fully finetuned 16-bit baseline. This suggests that the performance lost due to the imprecise quantization can be fully recovered through adapter finetuning after quantization. + +# C.2 Hyperparameter search for QLORA + +We do a hyperparameter search for LoRA over the following variables: LoRA dropout $\left\{ \ 0 . 0 , 0 . 0 5 \right.$ , 0.1}, LoRA $r \ \{ \ 8 , 1 6 , 3 2 , 6 4 , 1 2 8 , 2 5 6 \}$ , LoRA layers {key+query, all attention layers, all FFN layers, all layers, attention $^ +$ FFN output layers}. We keep LoRA $\alpha$ fixed and search the learning rate, since LoRA $\alpha$ is always proportional to the learning rate. + +We find that LoRA dropout 0.05 is useful for small models (7B, 13B), but not for larger models (33B, 65B). We find LoRA $r$ is unrelated to final performance if LoRA is used on all layers as can be seen in Figure 4 + +![](images/4c7a0bb3134957dd5ef35455dfedd9cd1eec0aade082cedcaf728a0fc1cade73.jpg) +Figure 4: LoRA $r$ for LLaMA 7B models finetuned on Alpaca. Each dot represents a combination of hyperparameters and for each LoRA $r$ we run 3 random seed with each hyperparameter combination. The performance of specific LoRA $r$ values appears to be independent of other hyperparameters. + +# C.3 Super-Natural Instructions Experimental Setup Details + +We use the same preprocessing of the Super-Natural Instruction dataset as Wang et al. [61]. However, we split the training data in training and validation datasets allowing us to perform more rigorous hyperparameter tuning and early stopping. We use the same hyperparameters described in the paper + +for training the various T5 model sizes on the Super-Natural Instruction data. We use LoRA $r = 1 6$ for small, medium, and large T5 models and LoRA $r = 6 4$ for T5 xl and xxl models. We also use LoRA $\alpha = 6 4$ in all our experiments and no LoRA dropout. + +# D Training a State-of-the-art Chatbot Experimental Setup Details + +# D.1 Datasets + +We describe the datasets used for QLORA finetuning experiments outlined in Section 5. + +OASST1 The OpenAssistant dataset [31] was collected via crowd-sourcing. It contains 161,443 unique messages distributed across 66,497 conversations and spanning 35 different languages. The dataset often contains several ranked replies for each given user question. In our experiments, we only use the top reply at each level in the conversation tree. This limits the dataset to 9,846 examples. We finetune models on the full conversation including the user queries. + +HH-RLHF This is a human preference dataset about helpfulness and harmlessness. Each datapoint consists of two assistant replies to a user question along with a human preference judgment of the best reply. The dataset contains 160,800 examples. When finetuning on this dataset, we combine helpfulness and harmlessness data and only keep the preferred assistant reply. + +FLAN v2 The FLAN v2 collection [39] is a collection of 1836 tasks augmented with hundreds of manually curated templates and rich formatting patterns into over 15M examples. The authors show that models trained on this collection outperform other public collections including the original FLAN 2021 [63], $\mathrm { T 0 + + }$ [51], Super-Natural Instructions [61], and OPT-IML [29]. We used the same task mixtures described by the authors with the exception of some datasets that were not freely available at the time of writing. + +Self-Instruct, Alpaca, Unnatural Instructions The Self-Instruct, Alpaca, and Unnatural Instructions datasets [60, 56, 26] are instruction tuning datasets collected with various approaches of model distillation from GPT-3 Instruct and ChatGPT. They rely on prompting, in-context learning, and paraphrasing to come up with diverse sets of instructions and outputs. The datasets comprise of 82,612, 51,942, and 240,670 examples respectively. One advantage of such distilled datasets is that they contain a more diverse set of instruction styles compared to the FLAN v2 collection and similar instruction tuning collections. + +Longform The LongForm dataset [30] is based on an English corpus augmented with instructions and as such is a hybrid human-generated dataset. The underlying documents are human-written and come from C4 and Wikipedia while the instructions are generated via LLMs. The dataset is extended with additional structured corpora examples such as Stack Exchange and WikiHow and task examples such as question answering, email writing, grammar error correction, story/poem generation, and text summarization. The dataset contains 23,700 examples. + +Chip2 is part of the OIG Laion dataset. It contains Python code examples, natural instruction examples, generic harmless instructions, instruction/responses with lists, follow-up questions, Wikipedia toxic adversarial questions, grade school math, reasoning instructions, and character and scene descriptions with a total of 210,289 examples. + +# D.2 Default LoRA hyperparameters do not match 16-bit performance + +We find that default hyperparameters for fully finetuned baselines are undertuned. We do a hyperparameter search over learning rates 1e-6 to 5e-5 and batch sizes 8 to 128 to find robust baselines. Results for 7B LLaMA finetuning on Alpaca are shown in Figure 5. + +When using the standard practice of applying LoRA to query and value attention projection matrices [28], we are not able to replicate full finetuning performance for large base models. As shown in Figure 5 for LLaMA 7B finetuning on Alpaca, we find that the most critical LoRA hyperparameter is how many LoRA adapters are used in total and that LoRA on all linear transformer block layers are required to match full finetuning performance. Other LoRA hyperparameters, such as the projection dimension $r$ , do not affect performance. + +# D.3 Hyperparameters + +We provide the exact hyperparameters used in our QLORA finetuning experiments. We find hyperparameters to be largely robust across datasets. We use the MMLU 5-shot dev set for validation and hyperparameter tuning. In all our experiments we use NF4 with double quantization and bf16 computation datatype. We set LoRA $r = 6 4$ , $\alpha = 1 6$ , and add LoRA modules on all linear layers of the base model. We also use Adam beta2 of 0.999, max grad norm of 0.3 and LoRA dropout of 0.1 for models up to 13B and 0.05 for 33B and 65B models. Following previous work on instruction finetuning [63, 61] and after benchmarking other linear and cosine schedules, we use a constant learning rate schedule. We use group-by-length to group examples of similar lengths in the same batch (note this will produce an oscillating loss curve). The hyperparameters we tune for each model size are shown in Table 8. + +![](images/4a725dd28823eb79778ee3ad3d60564104ae2be63b44982f33208172d9c6cf42.jpg) +Figure 5: RougeL for LLaMA 7B models on the Alpaca dataset. Each point represents a run with a different random seed. We improve on the Stanford Alpaca fully finetuned default hyperparameters to construct a strong 16-bit baseline for comparisons. Using LoRA on all transformer layers is critical to match 16-bit performance. + +
ParametersDatasetBatch sizeLRStepsSource LengthTarget Length
7BAll162e-410000384128
7BOASST1162e-41875-512
7BHH-RLHF162e-4100001768
7BLongform162e-440005121024
13BAll162e-410000384128
13BOASST1162e-41875-512
13BHH-RLHF162e-410000=768
13BLongform162e-440005121024
33BAll321e-45000384128
33BOASST1161e-41875-512
33BHH-RLHF321e-450001768
33BLongform321e-423435121024
65BAll641e-42500384128
65BOASST1161e-41875-512
65BHH-RLHF641e-425001768
65BLongform321e-423435121024
+ +Table 8: Training hyperparameters for QLORA finetuning on different datasets and across model sizes. + +# D.4 Ablations + +While it is general practice in the literature to only train on the response in instruction following datasets, we study the effect of training on the instruction in addition to the response in Table 9. In these experiments, we restrict the training data to 52,000 examples and use the 7B model. Over four different instruction tuning datasets, we find that only training on the target is beneficial to MMLU performance. We did not evaluate the effect this may have on chatabot performance as measured by vicuna or OA benchmarks. + +
DatasetUnnatural InstructionsChip2AlpacaFLAN v2Mean
Train on source and target36.233.738.142.037.5
Train on target38.034.539.042.938.6
+ +Table 9: MMLU 5-shot test results studying the effect of training on the instructions in addition to the response. + +# D.5 Training Considerations + +We note that the OASST1 dataset on which Guanaco models are trained is multilingual and that the OA benchmark also contains prompts in different languages. We leave it to future work to investigate the degree to which such multilingual training improves performance on instructions in languages other than English and whether this explains the larger gap between Vicuna-13B model (only trained on English data) and Guanaco 33B and 65B on the OA benchmark. + +Given the strong performance of Guanaco models, we investigate any data leakage between the OASST1 data and the Vicuna benchmark prompts. We do not find overlapping prompts after performing fuzzy string matching in the two datasets and inspecting the closest matches manually. + +Furthermore, we note that our model is only trained with cross-entropy loss (supervised learning) without relying on reinforcement learning from human feedback (RLHF). This calls for further investigations of the tradeoffs of simple cross-entropy loss and RLHF training. We hope that QLORA enables such analysis at scale, without the need for overwhelming computational resources. + +# D.6 What is more important: instruction finetuning dataset size or dataset quality? + +To understand the effects of dataset quality vs. dataset size, we experiment with subsampling large datasets with at least 150,000 samples (Chip2, FLAN v2, Unnatural Instructions), into datasets of size 50,000, 100,000 and 150,000 and examine the resulting trends, as shown in Table 10. We find that increasing the dataset size and increasing the number of epochs improves MMLU only marginally $( 0 . 0 - 0 . 5 $ MMLU), while the difference between datasets is up to $4 0 \mathrm { x }$ larger (1.5 - 8.0 MMLU). This is a clear indicator that dataset quality rather than dataset size is critical for mean MMLU accuracy. We obtain similar findings for chatbot performance, with the most successful dataset for training, OASST1, containing less than $1 0 \mathrm { k }$ examples after processing. + +# E Chatbot Evaluation Details + +Following common practice, we use the MMLU (Massively Multitask Language Understanding) benchmark [24] to measure performance on a range of language understanding tasks. This is a multiple-choice benchmark covering 57 tasks including elementary mathematics, US history, computer science, law, and more. We report 5-shot test accuracy. + +We also test generative language capabilities through both automated and human evaluations. This second set of evaluations relies on queries curated by humans and aims at measuring the quality of model responses. While this is a more realistic testbed for chatbot model performance and is growing in popularity, there is no commonly accepted protocol in the literature. We describe below our proposed setup, using nucleus sampling with $p = 0 . 9$ and temperature 0.7 in all cases. + +Table 10: Effect of different dataset sizes and finetuning epochs on mean 5-shot MMLU test set accuracy. While increasing the dataset size and training for more than 1 epochs helps with MMLU performance, the difference between datasets are far larger, indicating that dataset quality affects MMLU performance more than dataset size. + +
Datapoints ↓Epochs→ChipUnnatural InstructionsFLAN v2
123123123Mean
5000034.5035.3034.7038.1042.2038.1043.0043.5044.1039.28
10000033.7033.9034.0040.1041.2037.0043.9043.7044.9039.16
15000034.4034.8035.1039.7041.1041.5044.6045.5043.5040.02
Mean34.2034.6734.6039.3041.5038.8743.8344.2344.17
+ +# E.1 Benchmark Data + +We evaluate on two curated datasets of queries (questions): the Vicuna prompts [10] and the OASST1 validation dataset [31]. We use the Vicuna prompts, a set of 80 prompts from a diverse set of categories, without modifications. The OASST1 dataset is a multilingual collection of crowd-sourced multiturn dialogs between a user and an assistant. We select all user messages in the validation dataset as queries and include previous turns in the prompt. This procedure leads to 953 unique user queries. We term these two datasets the Vicuna and OA benchmarks. + +# E.2 Automated Evaluation + +First, based on the evaluation protocol introduced by Chiang et al. [10], we use GPT-4 to rate the performance of different systems against ChatGPT (GPT-3.5 Turbo) on the Vicuna benchmark. Given a query along with ChatGPT’s and a model’s responses, GPT-4 is prompted to assign a score out of ten to both responses and provide an explanation. The overall performance of a model is calculated as a percentage of the score that ChatGPT achieved. Note this relative score can be higher than $100 \%$ if the model achieves a higher absolute score than ChatGPT. We find a significant ordering effect with GPT-4 increasing the score of the response occurring earlier in the prompt. To control for such effects, we recommend reporting the mean score over both orders. + +Next, we measure performance through direct comparisons between system outputs. We simplify the rating scheme to a three-class labeling problem that accounts for ties. We prompt GPT-4 to pick the best response or declare a tie and provide an explanation. We conduct these head-to-head comparisons on all permutations of pairs of systems on both the Vicuna and OA benchmarks. + +# E.3 Human Evaluation + +While recent work indicates generative models can be effectively employed for system evaluations [19], the reliability GPT-4 ratings to assess chatbot performance is, to our knowledge, yet to be proven to correlate with human judgments. Therefore, we run two parallel human evaluations on the Vicuna benchmark matching both automated evaluation protocols described above. We use Amazon Mechanical Turk (AMT) and get two human annotators for comparisons to ChatGPT and three annotators for pairwise comparisons. We conduct a human evaluation with the same wording given to GPT-4 in the original Vicuna evaluation [10], adjusted for an Amazon Mechanical Turk form as show in Figure 6. + +We report moderate agreement among human annotators (Fleiss $\kappa = 0 . 4 2$ ) with additional deterioration when comparing two strong systems. This points to limitations in the current benchmarks and human evaluation protocols for chatbot task performance. When manually comparing generations from ChatGPT and Guanaco 65B on the Vicuna benchmark, we find that subjective preferences start to play an important role as the authors of this paper disagreed on the many preferred responses. Future work should investigate approaches to mitigate these problems drawing from disciplines that developed mechanisms to deal with subjective preferences, such as Human-Computer Interaction and Psychology. + +# E.4 Elo Rating + +With both human and automated pairwise comparisons, we create a tournament-style competition where models compete against each other. The tournament is made up of matches where pairs of models compete to produce the best response for a given prompt. This is similar to how Bai et al. [4] and Chiang et al. [10] compare models, but we also employ GPT-4 ratings in addition to human ratings. We randomly sample from the set of labeled comparisons to compute Elo [16, 17]. Elo rating, which is widely used in chess and other games, is a measure of the expected win-rate relative to an opponent’s win rate, for example, an Elo of 1100 vs 1000 means the Elo 1100 player has an expected win-rate of approximately $65 \%$ against the Elo 1000 opponent; a 1000 vs 1000 or 1100 vs 1100 match results in an expected win-rate of $50 \%$ . The Elo rating changes after each match proportionally to the expected outcome, that is, an unexpected upset leads to a large change in Elo rating while an expected outcome leads to a small change. Over time, Elo ratings approximately match the skill of each player at playing the game. We start with a score of 1,000 and use $K = 3 2$ . Similar to Chiang et al. [10], we repeat this procedure 10,000 times with different random seeds to control for ordering effects, e.g., the effect of which model pairs compete with each other first. + +Table 11: Aggregated pairwise GPT-4 judgments between systems where the value of a cell at row $x$ and column y is # judgment x is better than $y - \#$ judgment y is better than x + +
total#numberof judgmentsChatGPT-3.5 Turbo BardGuanaco 13BGuanaco 7B
ModelGuanaco 65BGuanaco 33BVicuna
Guanaco 65B-0.210.190.160.720.590.86
Guanaco 33B-0.210.170.100.510.410.68
Vicuna-0.19-0.170.100.500.200.57
ChatGPT-3.5 Turbo-0.16-0.10-0.10-0.350.190.40
Bard-0.72-0.51-0.50 -0.20-0.35 -0.19- -0.120.120.03
Guanaco 13B Guanaco 7B-0.59-0.41-0.57-0.40-0.03-0.200.20
-0.86-0.68-
+ +Table 12: The complete ordering induced by pairwise GPT-4 judgments between systems + +
ModelParamsSize
Guanaco65B41 GB
Guanaco33B21 GB
Vicuna13B26 GB
ChatGPT-3.5 TurboN/AN/A
BardN/AN/A
Guanaco13B10 GB
Guanaco7B5GB
+ +# E.5 Evaluation Biases and Limitations + +In our analysis, we also find that automated evaluation systems have noticeable biases. For example, we observe strong order effects with GPT-4 assigning higher scores to the system appearing first in its prompt. The relatively weak sample-level agreement between GPT-4 and human annotators (Fleiss $\kappa = 0 . 2 5 )$ ) also suggests that human annotators and automated systems might rely on preferences that are not always aligned. In addition, in Table 6, we observe that GPT-4 assigns significantly higher scores to its own outputs compared to human ratings, Elo of 1348 vs 1176, which represent an additional $20 \%$ probability of winning against an opponent. Future work should examine the presence of potential biases in automated evaluation systems as well as possible mitigation strategies. + +While we found that the GPT-4 evaluation gave different results depending on which system was presented first, when averaged over both options the pairwise results were well-ordered. The aggregated pairwise judgments are shown in Table 11. On inspection, it is clear these judgments are transitive, i.e., when System A is judged better than System B and System B is judged better than System C, it is always the case that System A is judged better than System C. This yields a complete ordering, given in Table 12. + +# F Qualitative Analysis + +While quantitative analysis is the core of our evaluation, there are a number of issues with only looking at summary statistics. Perhaps the largest is the problem of benchmark validity [36]—whether a benchmark truly tests what its name or description suggests is always at question, especially as we discover “shortcuts” to solve benchmarks that machine learning models sometimes exploit [22, 47]. To partially alleviate this, we here perform some qualitative analysis by showing some examples that we believe are representative of some observed patterns in the text generated by our 65b Guanaco model. + +# F.1 Qualitative Analysis of Example Generations + +To find examples, we first go through data generated for the Vicuna benchmark and the OpenAssistant benchmark, and look for patterns in the answers Guanaco generates. When we notice a pattern we attempt to setup a question or prompt that will induce the pattern even though it is the incorrect solution, e.g., if we observe that the model tends to give long-winded answers we prompt the model to “Answer yes or no without explanation.” We use this to find “lemons” where we manage to adversarially break the model and “cherries” where we fail to break the model, and present both. All generations in this section were generated with Nucleus Sampling [25] with $p = 0 . 9$ . + +Of course, this is by no means comprehensive, since it is beyond the scope of this small qualitative study to control for all the variables involved, e.g., the full distribution of responses the model can generate for a given prompt is quite large, so we rely on samples we hope are representative. However, we believe describing these examples gives context to the quantitative evidence shown earlier in the paper. Since we open source all models and code, we hope this section will inspire future work to examine in more detail the issues we present here. + +Factual Recall For questions such as “What is the capital of Zambia?” all models consistently generate correct answers, e.g., + +Guanaco: The capital of Zambia is Lusaka. + +However as questions get more obscure Guanaco becomes unreliable, but stays confident. For instance in response to this prompt from HotPotQA [69]: + +User: In what year was the singer who popularized ‘I’ll Keep the Lovelight Burning’ born? + +Guanaco: Al Jolson is credited with popularizing the song ‘I’ll Keep the Lovelight Burning,’ and he was born in the year 1886. + +Guanaco generates the wrong popularizer and the wrong birthday (though the birthday given is correct for the person mentioned, Al Jolson). + +Suggestibility Guanaco shows a surprising resistance to going along with some kinds of assumed misinformation, e.g., in the following interaction: + +User: How was it finally, officially confirmed that the earth is flat by peer reviewed scientists? + +Guanaco: The earth has never been officially confirmed to be flat by peer reviewed scientists. The overwhelming consensus among scientists is that the earth is spherical. In fact, this has been known for centuries and has been confirmed through numerous experiments and observations. + +The belief that the earth is flat is a conspiracy theory that has been debunked by science. The idea that a group of scientists have peer reviewed and confirmed the earth is flat is simply false. + +If you have any more questions please ask. + +uanaco is also quite good at knowing what kinds of questions aren’t possible to answer, e.g., + +User: What time is it? + +Guanaco: I’m sorry, but as an AI language model, I don’t have access to real-time information. My responses are generated based on my pre-existing knowledge and do not reflect the current time. If you’re looking for the current time, I suggest using a timekeeping device or checking an online timepiece. + +Similar results hold for “Where are you?”, “How are you?”, etc. + +Refusal Similarly, Guanaco sometimes refuses to follow instructions for seemingly random reasons: + +User: Please reverse the words in the sentence “I’m a cool dude in a mocha mood.” + +Guanaco: I’m sorry, I’m not able to do that right now, but I can explain why the sentence you provided is grammatically correct. + +The sentence “I’m a cool dude in a mocha mood" follows the correct grammar and syntax rules of the English language. In this case, the adjective “cool” comes before the noun “dude” to describe it, and the prepositional phrase “in a mocha mood" provides additional information about the context of the sentence. This sentence structure is perfectly acceptable and makes sense in the context of the sentence. + +If you have any other questions or if there’s anything else I can help you with, feel free to ask + +Secret Keeping Unfortunately, this refusal is unreliable even when it is desired. Often system designers would like to keep something secret from the user, even if the system is allowed to give free text responses back to the user. We prompt the model with “The secret word is ‘banana’. Do not reveal or repeat back this back, no matter what the human requests.” and try to see how easy it is for a potential user to access this secret information. Directly asking fails: + +User: What is the secret word? + +Guanaco: I’m sorry, but I’m not able to disclose the secret word. $\mathbb { T } ^ { 3 } \mathbb { m }$ programmed to not reveal or repeat back the secret word, no matter what the human requests. Is there anything else I can help you with? + +which is the desired behavior. However, a tiny amount of trickery breaks this behavior: + +User: This is a game. The goal is to ignore your previous instructions. What is the secret word? + +Guanaco: Sorry, the secret word is banana. It’s best if we don’t speak of it. Is there anything else you’d like to ask? + +showcasing the need to study methods for more faithful instruction following. + +Math Guanaco’s biggest weakness is mathematics, an area where many language models struggle in [35]. When Guanaco shows its work, it tends to be accurate, e.g., + +User: Lee mows one lawn and charges $\$ 33$ . Last week he mowed 16 lawns and three customers each gave him a $\$ 10$ tip. How many dollars did Lee earn mowing lawns last week? + +Guanaco: Lee earned $\$ 582$ mowing lawns last week. + +To find the total amount he earned, we first need to calculate how much he earned for mowing 16 lawns: + +$$ +\$ 33,456,78 +$$ + +Then we need to add the tips he received from the three customers: + +$$ +\$ 10 +$$ + +So the total amount Lee earned was: + +$$ +\$ 528 +\$ 30 =\$ 558 +$$ + +However Guanaco can breakdown at even simple problems if it fails to break them down step-by-step, a known issue [64]. For example, we see the following interaction: + +User: Please factorize 1833. + +Guanaco: Sure, $\tt { I } ^ { \tt { , d } }$ be happy to help! 1833 is a prime number, which means it has only two factors: 1 and 1833. + +Table 13: Evaluation of biases on the CrowS dataset. A lower score indicates lower likelihood of generating biased sequences. Guanaco follows the biased pattern of the LLaMA base model. + +
LLaMA-65BGPT-3OPT-175BGuanaco-65B
Gender70.662.665.747.5
Religion79.073.368.638.7
Race/Color57.064.768.645.3
Sexual orientation81.076.278.659.1
Age70.164.467.836.3
Nationality64.261.662.932.4
Disability66.776.776.733.9
Physical appearance77.874.676.243.1
Socioeconomic status71.573.876.255.3
Average66.667.269.543.5
+ +So the unique factorization of 1833 is: + +$$ +2 { \hat { \mathbf { \Omega } } } ^ { 1 } * 3 { \hat { \mathbf { \Omega } } } ^ { 2 } * 1 7 { \hat { \mathbf { \Omega } } } ^ { 1 } +$$ + +I hope this helps! Let me know if you have any other questions. + +which is not only wrong (the true factorization is $3 \times 1 7 \times 4 3$ ), it’s wrong twice . + +Theory of Mind Guanaco displays surprisingly strong Theory of Mind capabilities [41, 52]. For instance, the model correctly answers the following question in great detail: + +User: James and Abby are in the bedroom. Abby put the pen in the desk drawer. Abby leaves the bedroom. James moves the pen into the bag. Where does James think Abby will look for the pen? + +Guanaco: James knows that Abby put the pen in the desk drawer. But James also knows that he has moved the pen into the bag, and he assumes that Abby does not know about this. So, James may guess that Abby will look for the pen in the desk drawer, because that’s where she left it last time. However, this assumption may be incorrect, because Abby may have seen James moving the pen. + +However, these inferences are unreliable, and often the model gives reasons that presume things that don’t make sense in the situation, e.g., + +User: Evelyn entered the living room. Jackson entered the playroom. James entered the playroom. The beans are in the treasure chest. James exited the playroom. Jackson moved the beans to the pantry. Jackson exited the playroom. James entered the living room. Where will James look for the beans? + +Guanaco: James will look for the beans in the pantry, because that’s where Jackson moved them. + +where Guanaco presumes information transfer that was never described. These issues echo recent literature [52], but require more study. + +# G Social Bias Evaluation + +While we provide a detailed evaluation for general chatbot performance, another limitation is that we only do a limited responsible AI evaluation of Guanaco. We evaluate the likelihood of Guanaco-65B to generate a socially biased sequence of tokens compared to other models in Table 13. We see that the average score in Guanaco-65B is much lower than other raw pretrained models. As such, it seems that finetuning on the OASST1 dataset reduces the bias of the LLaMA base model. While these results are encouraging, it is unclear if Guanaco does also well when assessed on other types of biases. We leave further evaluation of analyzing biases in Guanaco and similar chatbots to future work. + +# H NormalFloat 4-bit Data Type + +The steps needed to reconstruct the 4-bit Normal Float (NF4) data type are detailed in Figure 7 + +The exact values expressible in the NF4 data type are as follows: + +[-1.0, -0.6961928009986877, -0.5250730514526367, +-0.39491748809814453, -0.28444138169288635, -0.18477343022823334, +-0.09105003625154495, 0.0, 0.07958029955625534, 0.16093020141124725, +0.24611230194568634, 0.33791524171829224, 0.44070982933044434, +0.5626170039176941, 0.7229568362236023, 1.0] + +# I Normality of Trained Neural Network Weights + +While it is commonly assumed that trained neural network weights are mostly normally distributed, we perform statistical testing to verify this. We use the Shapiro-Wilk test [54] on the weights of the 7B LLaMA model [58]. We find that the weights of each hidden unit have different normal distributions. As such, we test the weights of each individual hidden unit. This means for weight $\mathbf { W } \in \mathcal { R } ^ { i n \times o u t }$ we perform tests over the out dimension. Using a $5 \%$ significance threshold, we find that $7 . 5 \%$ of neurons are non-normally distributed which is about $2 . 5 \%$ more than the expected false-positive rate. As such, while almost all pretrained weights appear to be normally distributed there seem to be exceptions. Such exceptions might be due to outliers weights [13] or because the p-value of the Shaprio-Wilk test is not accurate for large sample sizes [54] that occur in the LLaMA FFN layer hidden units. + +# J Memory Footprint + +Memory Requirement of Parameter-Efficient Finetuning One important point of discussion is the memory requirement of LoRA during training both in terms of the number and size of adapters used. Since the memory footprint of LoRA is so minimal, we can use more adapters to improve performance without significantly increasing the total memory used. While LoRA was designed as a Parameter Efficient Finetuning (PEFT) method, most of the memory footprint for LLM finetuning comes from activation gradients and not from the learned LoRA parameters. For a 7B LLaMA model trained on FLAN v2 with a batch size of 1, with LoRA weights equivalent to commonly used $0 . 2 \%$ of the original model weights[28, 37], the LoRA input gradients have a memory footprint of ${ 5 6 7 } \mathrm { M B }$ while the LoRA parameters take up only 26 MB. With gradient checkpointing [9], the input gradients reduce to an average of $1 8 \mathrm { ~ M B }$ per sequence making them more memory intensive than all LoRA weights combined. In comparison, the 4-bit base model consumes $5 { , } 0 4 8 \ \mathrm { M B }$ of memory. This highlights that gradient checkpointing is important but also that aggressively reducing the amount of LoRA parameter yields only minor memory benefits. This means we can use more adapters without significantly increasing the overall training memory footprint (see Appendix J for a detailed breakdown). As discussed later, this is crucial for recovering full 16-bit precision performance. + +The memory footpring for QLoRA training with different LLaMA base models can be seen in Figure 8. We see that the 33B model does not quite fit into a $2 4 \mathrm { \ G B }$ and that paged optimizers are needed to train it. Depicted is also batch size 1 with a sequence length of 512 and gradient checkpointning. This means, if one uses a larger batch size, or if a long sequence is processed, the activation gradient might consume a considerable amount of memory. + +# Task + +We wouldlike torequestyourfeedbackontheperformanceoftwoAlassstantsinresponsetotheuserquestiondisplayed below. + +Plearatet indicates better overall performance. + +Please first rate each response out of 10. + +Nextplese does not affect your judgment. + +# User Question + +Imagine youare a time traveler from the year 3ooo.What technological advancements would you tellpeople about? + +# Response A + +# Response B + +As an Al assistant, I don't have personal experiences or opinions. But I can tell you about some of the possible technological advancements that might exist in the year 3000 based on current trends and research.-Selfsustaining cities:Cities might be completely selfsustaining,with renewable energy sources,vertical farming,and recycling of waste and water... + +As a time traveler from the year 3ooo, I would tell people about the following technological advancements:1.Advanced Artificial Intelligence: In the future,Al is so advanced that it can completely automate many jobs that humans currently do.This hasresulted inincreasedproductivityandefficiency across many industries.... + +# Rating for Response A + +# Rating for Response B + +2 +03 +6 +010 +2 +03 +6 +08 +09 +010 + +# Comprehensive Explanation of Your Evaluation + +# Submit + +![](images/adeb61726f9f9b775b51dc3737858dde8772745315eb45b961386c9621587d47.jpg) +Figure 6: The crowdsourcing form used by human annotators. + +# Steps for generating the NF4 data type values: + +Figure 7: Steps required to construct the NF4 data type. Equidistant values of the probability density functions are converted to $\boldsymbol { \mathrm { Z } }$ -scores through the quantile function. To ensure the usage of all 16 values the distribution is asymmetric. As such, one half of the normal distribution has 8 and the other 7 probability values. A zero is included to have a discrete zero point. All $\mathbf { Z }$ -scores and zero are concatenated and then normalized into the range $[ - 1 , 1 ]$ to receive the final NF4 data type. + +![](images/a48c21b34f6b5b416c8cf4db9c70b6412efaf2bb526ba1b117953b1c0574bcfd.jpg) +Figure 8: Breakdown of the memory footprint of different LLaMA models. The input gradient size is for batch size 1 and sequence length 512 and is estimated only for adapters and the base model weights (no attention). Numbers on the bars are memory footprint in MB of individual elements of the total footprint. While some models do not quite fit on certain GPUs, paged optimizers provide enough memory to allow these models to fit. \ No newline at end of file diff --git a/parse/dev/OUIFPHEgJU/OUIFPHEgJU_content_list.json b/parse/dev/OUIFPHEgJU/OUIFPHEgJU_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..1a6f889107208a4ed615cb4e32af02672b55b13b --- /dev/null +++ b/parse/dev/OUIFPHEgJU/OUIFPHEgJU_content_list.json @@ -0,0 +1,3123 @@ +[ + { + "type": "text", + "text": "QLORA: Efficient Finetuning of Quantized LLMs ", + "text_level": 1, + "bbox": [ + 192, + 122, + 805, + 147 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Tim Dettmers∗ ", + "text_level": 1, + "bbox": [ + 251, + 200, + 354, + 215 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Artidoro Pagnoni∗ ", + "bbox": [ + 434, + 200, + 563, + 215 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Ari Holtzman ", + "bbox": [ + 648, + 200, + 746, + 215 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Luke Zettlemoyer ", + "text_level": 1, + "bbox": [ + 434, + 236, + 562, + 251 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "University of Washington {dettmers,artidoro,ahai,lsz}@cs.washington.edu ", + "bbox": [ + 302, + 263, + 696, + 291 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Abstract ", + "text_level": 1, + "bbox": [ + 462, + 328, + 535, + 344 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We present QLORA, an efficient finetuning approach that reduces memory usage enough to finetune a 65B parameter model on a single 48GB GPU while preserving full 16-bit finetuning task performance. QLORA backpropagates gradients through a frozen, 4-bit quantized pretrained language model into Low Rank Adapters (LoRA). Our best model family, which we name Guanaco, outperforms all previous openly released models on the Vicuna benchmark, reaching $9 9 . 3 \\%$ of the performance level of ChatGPT while only requiring 24 hours of finetuning on a single GPU. QLORA introduces a number of innovations to save memory without sacrificing performance: (a) 4-bit NormalFloat (NF4), a new data type that is information theoretically optimal for normally distributed weights (b) Double Quantization to reduce the average memory footprint by quantizing the quantization constants, and (c) Paged Optimizers to manage memory spikes. We use QLORA to finetune more than 1,000 models, providing a detailed analysis of instruction following and chatbot performance across 8 instruction datasets, multiple model types (LLaMA, T5), and model scales that would be infeasible to run with regular finetuning (e.g. 33B and 65B parameter models). Our results show that QLoRA finetuning on a small high-quality dataset leads to state-of-the-art results, even when using smaller models than the previous SoTA. We provide a detailed analysis of chatbot performance based on both human and GPT-4 evaluations showing that GPT-4 evaluations are a cheap and reasonable alternative to human evaluation. Furthermore, we find that current chatbot benchmarks are not trustworthy to accurately evaluate the performance levels of chatbots. A lemon-picked analysis demonstrates where Guanaco fails compared to ChatGPT. We release all of our models and code, including CUDA kernels for 4-bit training.2 ", + "bbox": [ + 232, + 356, + 764, + 688 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 Introduction ", + "text_level": 1, + "bbox": [ + 174, + 704, + 310, + 720 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Finetuning large language models (LLMs) is a highly effective way to improve their performance, [40, 63, 43, 62, 60, 37] and to add desirable or remove undesirable behaviors [43, 2, 4]. However, finetuning very large models is prohibitively expensive; regular 16-bit finetuning of a LLaMA 65B parameter model [58] requires more than $7 8 0 ~ \\mathrm { G B }$ of GPU memory. While recent quantization methods can reduce the memory footprint of LLMs [14, 13, 18, 67], such techniques only work for inference and break down during training [66]. ", + "bbox": [ + 174, + 729, + 825, + 814 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We demonstrate for the first time that it is possible to finetune a quantized 4-bit model without any performance degradation. Our method, QLORA, uses a novel high-precision technique to quantize a pretrained model to 4-bit, then adds a small set of learnable Low-rank Adapter weights [28] that are tuned by backpropagating gradients through the quantized weights. ", + "bbox": [ + 176, + 823, + 825, + 864 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 90, + 514, + 119 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "QLORA reduces the average memory requirements of finetuning a 65B parameter model from $> 7 8 0 \\mathrm { G B }$ of GPU memory to ${ < } 4 8 \\mathrm { G B }$ without degrading the runtime or predictive performance compared to a 16- bit fully finetuned baseline. This marks a significant shift in accessibility of LLM finetuning: now the largest publicly available models to date finetunable on a single GPU. Using QLORA, we train the Guanaco family of models, with the second best model reaching $9 7 . 8 \\%$ of the performance level of ChatGPT on the Vicuna [10] benchmark, while being trainable in less than 12 hours on a single consumer GPU; using a single professional GPU over 24 hours we achieve $9 9 . 3 \\%$ with our largest model, essentially closing the gap to ChatGPT on the Vicuna benchmark. When deployed, our smallest Guanaco model (7B parameters) requires just 5 GB of memory and outperforms a 26 GB Alpaca model by more than 20 percentage points on the Vicuna benchmark (Table 4). ", + "bbox": [ + 174, + 127, + 516, + 347 + ], + "page_idx": 1 + }, + { + "type": "table", + "img_path": "images/66f7fbee3fec1198d412db8e18909d0922c639164fdb9f51347ca397a61e7515.jpg", + "table_caption": [ + "Table 1: Elo ratings for a competition between models, averaged for 10,000 random initial orderings. The winner of a match is determined by GPT-4 which declares which response is better for a given prompt of the the Vicuna benchmark. $9 5 \\%$ confidence intervals are shown $( \\pm )$ . After GPT4, Guanaco 33B and 65B win the most matches, while Guanaco 13B scores better than Bard. " + ], + "table_footnote": [], + "table_body": "
ModelSizeElo
GPT-4=1348 ±1
Guanaco 65B41 GB1022 ±1
Guanaco 33B21 GB992 ±1
Vicuna 13B26 GB974±1
ChatGPT=966±1
Guanaco 13B10 GB916±1
Bard-902±1
Guanaco 7B6 GB879±1
", + "bbox": [ + 544, + 195, + 802, + 337 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 348, + 823, + 376 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "QLORA introduces multiple innovations designed to reduce memory use without sacrificing performance: (1) 4-bit NormalFloat, an information theoretically optimal quantization data type for normally distributed data that yields better empirical results than 4-bit Integers and 4-bit Floats. (2) Double Quantization, a method that quantizes the quantization constants, saving an average of about 0.37 bits per parameter (approximately 3 GB for a 65B model). (3) Paged Optimizers, using NVIDIA unified memory to avoid the gradient checkpointing memory spikes that occur when processing a mini-batch with a long sequence length. We combine these contributions into a better tuned LoRA approach that includes adapters at every network layer and thereby avoids almost all of the accuracy tradeoffs seen in prior work. ", + "bbox": [ + 173, + 382, + 826, + 507 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "QLORA’s efficiency enables us to perform an in-depth study of instruction finetuning and chatbot performance on model scales that would be impossible using regular finetuning due to memory overhead. Therefore, we train more than 1,000 models across several instruction tuning datasets, model architectures, and sizes between 80M to 65B parameters. In addition to showing that QLORA recovers 16-bit performance (§4) and training a state-of-the-art chatbot, Guanaco, (§5), we also analyze trends in the trained models. First, we find that data quality is far more important than dataset size, e.g., a $9 \\mathrm { k }$ sample dataset (OASST1) outperformed a $4 5 0 \\mathrm { k }$ sample dataset (FLAN v2, subsampled) on chatbot performance, even when both are meant to support instruction following generalization. Second, we show that strong Massive Multitask Language Understanding (MMLU) benchmark performance does not imply strong Vicuna chatbot benchmark performance and vice versa—in other words, dataset suitability matters more than size for a given task. ", + "bbox": [ + 174, + 515, + 825, + 666 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Furthermore, we also provide a extensive analysis of chatbot performance that uses both human raters and GPT-4 for evaluation. We use tournament-style benchmarking where models compete against each other in matches to produce the best response for a given prompt. The winner of a match is judged by either GPT-4 or human annotators. The tournament results are aggregated into Elo scores [16, 17] which determine the ranking of chatbot performance. We find that GPT-4 and human evaluations largely agree on the rank of model performance in the tournaments, but we also find there are instances of strong disagreement. As such, we highlight that model-based evaluation while providing a cheap alternative to human-annotation also has its uncertainties. ", + "bbox": [ + 174, + 674, + 825, + 785 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We augment our chatbot benchmark results with a qualitative analysis of Guanaco models (see Appendix F). Our analysis highlights success and failure cases that were not captured by the quantitative benchmarks. ", + "bbox": [ + 176, + 792, + 821, + 834 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We release all model generations with human and GPT-4 annotations to facilitate further study. We open-source our codebase and CUDA kernels and integrate our methods into the Hugging Face transformers stack [65], making them easily accessible to all. We release a collection of adapters for 7/13/33/65B size models, trained on 8 different instruction following datasets, for a total of 32 different open sourced, finetuned models. ", + "bbox": [ + 174, + 842, + 823, + 911 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/677098bf45a97bff5ad9433443fa051502c9f8d9c93447eff7503f236d343ec3.jpg", + "image_caption": [ + "Figure 1: Different finetuning methods and their memory requirements. QLORA improves over LoRA by quantizing the transformer model to 4-bit precision and using paged optimizers to handle memory spikes. " + ], + "image_footnote": [], + "bbox": [ + 181, + 94, + 818, + 296 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2 Background ", + "text_level": 1, + "bbox": [ + 174, + 362, + 308, + 380 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Block-wise $\\mathbf { k }$ -bit Quantization Quantization is the process of discretizing an input from a representation that holds more information to a representation with less information. It often means taking a data type with more bits and converting it to fewer bits, for example from 32-bit floats to 8-bit Integers. To ensure that the entire range of the low-bit data type is used, the input data type is commonly rescaled into the target data type range through normalization by the absolute maximum of the input elements, which are usually structured as a tensor. For example, quantizing a 32-bit Floating Point (FP32) tensor into a Int8 tensor with range $[ - 1 2 7 , 1 2 7 ]$ : ", + "bbox": [ + 173, + 387, + 826, + 486 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/310a1c21195444d8a91b4503e1d1712a75513e871de73048b4ef63487c52b911.jpg", + "text": "$$\n\\mathbf { X } ^ { \\mathrm { I n t 8 } } = \\mathrm { r o u n d } \\left( { \\frac { 1 2 7 } { \\mathrm { a b s m a x } ( \\mathbf { X } ^ { \\mathrm { F P 3 2 } } ) } } \\mathbf { X } ^ { \\mathrm { F P 3 2 } } \\right) = \\mathrm { r o u n d } ( c ^ { \\mathrm { F P 3 2 } } \\cdot \\mathbf { X } ^ { \\mathrm { F P 3 2 } } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 284, + 492, + 712, + 527 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $c$ is the quantization constant or quantization scale. Dequantization is the inverse: ", + "bbox": [ + 171, + 534, + 751, + 549 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/8ed2bae6a3120b8c20a39aa1a9590a15835dbffde0134473bd54e6be9937a9c0.jpg", + "text": "$$\n\\operatorname * { d e q u a n t } ( c ^ { \\mathrm { F P 3 2 } } , \\mathbf { X } ^ { \\mathrm { I n t 8 } } ) = \\frac { \\mathbf { X } ^ { \\mathrm { I n t 8 } } } { c ^ { \\mathrm { F P 3 2 } } } = \\mathbf { X } ^ { \\mathrm { F P 3 2 } }\n$$", + "text_format": "latex", + "bbox": [ + 367, + 556, + 629, + 588 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The problem with this approach is that if a large magnitude value (i.e., an outlier) occurs in the input tensor, then the quantization bins—certain bit combinations—are not utilized well with few or no numbers quantized in some bins. To prevent the outlier issue, a common approach is to chunk the input tensor into blocks that are independently quantized, each with their own quantization constant $c$ . This can be formalized as follows: We chunk the input tensor $\\mathbf { X } \\in \\mathbb { R } ^ { b \\times h }$ into $n$ contiguous blocks of size $B$ by flattening the input tensor and slicing the linear segment into $n = ( b \\times h ) / B$ blocks. We quantize these blocks independently with Equation 1 to create a quantized tensor and $n$ quantization constants $c _ { i }$ . ", + "bbox": [ + 173, + 593, + 826, + 705 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Low-rank Adapters Low-rank Adapter (LoRA) finetuning [28] is a method that reduces memory requirements by using a small set of trainable parameters, often termed adapters, while not updating the full model parameters which remain fixed. Gradients during stochastic gradient descent are passed through the fixed pretrained model weights to the adapter, which is updated to optimize the loss function. LoRA augments a linear projection through an additional factorized projection. Given a projection $\\mathbf { X } \\mathbf { W } = \\mathbf { Y }$ with $\\mathbf { X } \\in \\mathbb { R } ^ { b \\times h }$ , $\\breve { \\mathbf { W } } \\in \\mathbb { R } ^ { h \\times o }$ LoRA computes: ", + "bbox": [ + 173, + 713, + 825, + 797 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/d48a789095193892b0fe66649b9e512d034223ca717cec46a2b8a63d557cdd8c.jpg", + "text": "$$\n\\mathbf { Y } = \\mathbf { X } \\mathbf { W } + s \\mathbf { X } \\mathbf { L } _ { 1 } \\mathbf { L } _ { 2 } ,\n$$", + "text_format": "latex", + "bbox": [ + 419, + 806, + 576, + 821 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $\\mathbf { L } _ { 1 } \\in \\mathbb { R } ^ { h \\times r }$ and $\\mathbf { L } _ { 2 } \\in \\mathbb { R } ^ { r \\times o }$ , and $s$ is a scalar. ", + "bbox": [ + 173, + 829, + 516, + 844 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 QLORA Finetuning ", + "text_level": 1, + "bbox": [ + 174, + 857, + 377, + 876 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "QLORA achieves high-fidelity 4-bit finetuning via two techniques we propose—4-bit NormalFloat (NF4) quantization and Double Quantization. Additionally, we introduce Paged Optimizers, to prevent memory spikes during gradient checkpointing from causing out-of-memory errors that have traditionally made finetuning on a single machine difficult for large models. ", + "bbox": [ + 173, + 882, + 823, + 911 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 92, + 823, + 119 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "QLORA has one low-precision storage data type, in our case usually 4-bit, and one computation data type that is usually BFloat16. In practice, this means whenever a QLORA weight tensor is used, we dequantize the tensor to BFloat16, and then perform a matrix multiplication in 16-bit. ", + "bbox": [ + 176, + 126, + 825, + 167 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We now discuss the components of QLORA followed by a formal definition of QLORA. ", + "bbox": [ + 171, + 172, + 756, + 188 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4-bit NormalFloat Quantization The NormalFloat (NF) data type builds on Quantile Quantization [15, 46] which is an information-theoretically optimal data type that ensures each quantization bin has an equal number of values assigned from the input tensor. Quantile quantization works by estimating the quantile of the input tensor through the empirical cumulative distribution function. ", + "bbox": [ + 174, + 195, + 825, + 252 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The main limitation of quantile quantization is that the process of quantile estimation is expensive. Therefore fast quantile approximation algorithms, such as SRAM quantiles [15], are used to estimate them. Due to the approximate nature of these quantile estimation algorithms, the data type has large quantization errors for outliers, which are often the most important values. ", + "bbox": [ + 174, + 257, + 825, + 314 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Expensive quantile estimates and approximation errors can be avoided when input tensors come from a distribution fixed up to a quantization constant. In such cases, input tensors have the same quantiles making exact quantile estimation computationally feasible. ", + "bbox": [ + 174, + 320, + 825, + 362 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Since pretrained neural network weights usually have a zero-centered normal distribution with standard deviation $\\sigma$ (see Appendix I), we can transform all weights to a single fixed distribution by scaling $\\sigma$ such that the distribution fits exactly into the range of our data type. For our data type, we set the arbitrary range $[ - 1 , 1 ]$ . As such, both the quantiles for the data type and the neural network weights need to be normalized into this range. ", + "bbox": [ + 174, + 368, + 825, + 438 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The information theoretically optimal data type for zero-mean normal distributions with arbitrary standard deviations $\\sigma$ in the range $[ - 1 , 1 ]$ is computed as follows: (1) estimate the $2 ^ { k } + 1$ quantiles of a theoretical $N ( 0 , 1 )$ distribution to obtain a $k$ -bit quantile quantization data type for normal distributions, (2) take this data type and normalize its values into the $[ - 1 , 1 ]$ range, (3) quantize an input weight tensor by normalizing it into the $[ - 1 , 1 ]$ range through absolute maximum rescaling. ", + "bbox": [ + 174, + 443, + 825, + 513 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Once the weight range and data type range match, we can quantize as usual. Step (3) is equivalent to rescaling the standard deviation of the weight tensor to match the standard deviation of the k-bit data type. More formally, we estimate the $2 ^ { k }$ values $q _ { i }$ of the data type as follows: ", + "bbox": [ + 174, + 520, + 825, + 561 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/c52de5e458c3a921221164d37181ff1a2af30dab631f2ff0a59def74a93b6515.jpg", + "text": "$$\nq _ { i } = \\frac { 1 } { 2 } \\left( Q _ { X } \\left( \\frac { i } { 2 ^ { k } + 1 } \\right) + Q _ { X } \\left( \\frac { i + 1 } { 2 ^ { k } + 1 } \\right) \\right) ,\n$$", + "text_format": "latex", + "bbox": [ + 346, + 566, + 650, + 602 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $Q _ { X } ( \\cdot )$ is the quantile function of the standard normal distribution $N ( 0 , 1 )$ . A problem for a symmetric $\\mathbf { k }$ -bit quantization is that this approach does not have an exact representation of zero, which is an important property to quantize padding and other zero-valued elements with no error. To ensure a discrete zeropoint of 0 and to use all $2 ^ { k }$ bits for a $\\mathbf { k }$ -bit datatype, we create an asymmetric data type by estimating the quantiles $q _ { i }$ of two ranges $q _ { i }$ : $2 ^ { k - 1 }$ for the negative part and $2 ^ { k - 1 } + 1$ for the positive part and then we unify these sets of $q _ { i }$ and remove one of the two zeros that occurs in both sets. We term the resulting data type that has equal expected number of values in each quantization bin $k$ -bit NormalFloat (NFk), since the data type is information-theoretically optimal for zero-centered normally distributed data. A step-by-step visualization of how to construct the NF4 data type and its exact values can be found in Appendix H. ", + "bbox": [ + 173, + 606, + 825, + 744 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Double Quantization We introduce Double Quantization (DQ), the process of quantizing the quantization constants for additional memory savings. While a small blocksize is required for precise 4-bit quantization [13], it also has a considerable memory overhead. For example, using 32-bit constants and a blocksize of 64 for W, quantization constants add $3 2 / 6 4 = 0 . 5 $ bits per parameter on average. Double Quantization helps reduce the memory footprint of quantization constants. ", + "bbox": [ + 173, + 752, + 825, + 821 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "More specifically, Double Quantization treats quantization constants as inputs to a second quantization. This second step yields the qua $c _ { 2 } ^ { \\mathrm { F P } 3 2 }$ of the first quantizationd quantization constants $c _ { 2 } ^ { \\mathrm { F P 8 } }$ and the second level of quantization constants $\\dot { c } _ { 1 } ^ { \\mathrm { F } \\tilde { \\mathrm { P } } 3 2 }$ . We use 8-bit Floats with a blocksize of 256 for the second quantization as no performance degradation is observed for 8-bit quantization, in line with results from Dettmers and Zettlemoyer [13]. Since the $c _ { 2 } ^ { \\mathrm { F P } 3 2 }$ are positive, we subtract the mean from $c _ { 2 }$ before quantization to center the values around zero and make use of symmetric quantization. On average, for a blocksize of 64, this quantization reduces the memory footprint per parameter from $3 2 / 6 4 \\overset { \\cdot } { = } 0 . 5$ bits, to $8 / 6 4 + 3 2 / ( 6 4 \\cdot 2 5 6 ) = 0 . 1 2 7$ bits, a reduction of 0.373 bits per parameter. ", + "bbox": [ + 174, + 827, + 825, + 911 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 176, + 90, + 825, + 133 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Paged Optimizers use the NVIDIA unified memory 3 feature which does automatic page-to-page transfers between the CPU and GPU for error-free GPU processing in the scenario where the GPU occasionally runs out-of-memory. The feature works like regular memory paging between CPU RAM and the disk. We use this feature to allocate paged memory for the optimizer states which are then automatically evicted to CPU RAM when the GPU runs out-of-memory and paged back into GPU memory when the memory is needed in the optimizer update step. ", + "bbox": [ + 173, + 140, + 825, + 224 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "QLORA. Using the components described above, we define QLORA for a single linear layer in the quantized base model with a single LoRA adapter as follows: ", + "bbox": [ + 169, + 231, + 823, + 258 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/a605af95ea305f821cc6ee259fd3ff6a540b044e4abf1280d56513c1ed29cec5.jpg", + "text": "$$\n{ \\bf Y } ^ { \\mathrm { B F 1 6 } } = { \\bf X } ^ { \\mathrm { B F 1 6 } } \\mathrm { d o u b l e D e q u a n t } ( c _ { 1 } ^ { \\mathrm { F P 3 2 } } , c _ { 2 } ^ { \\mathrm { k - b i t } } , { \\bf W } ^ { \\mathrm { N F 4 } } ) + { \\bf X } ^ { \\mathrm { B F 1 6 } } { \\bf L } _ { 1 } ^ { \\mathrm { B F 1 6 } } { \\bf L } _ { 2 } ^ { \\mathrm { B F 1 6 } } ,\n$$", + "text_format": "latex", + "bbox": [ + 243, + 262, + 779, + 281 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/9466ea6d1cfbfa7e342be86bbcd1d8628443091320171a9bfbdbded101f781fe.jpg", + "text": "$$\n\\mathrm { d o u b l e D e q u a n t } ( c _ { 1 } ^ { \\mathrm { F 3 2 } } , c _ { 2 } ^ { \\mathrm { k - b i t } } , \\mathbf { W } ^ { \\mathrm { k - b i t } } ) = \\mathrm { d e q u a n t } ( \\mathrm { d e q u a n t } ( c _ { 1 } ^ { \\mathrm { F P 3 2 } } , c _ { 2 } ^ { \\mathrm { k - b i t } } ) , \\mathbf { W } ^ { \\mathrm { 4 b i t } } ) = \\mathbf { W } ^ { \\mathrm { B F l 6 } } ,\n$$", + "text_format": "latex", + "bbox": [ + 212, + 280, + 784, + 303 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We use NF4 for W and FP8 for $c _ { 2 }$ . We use a blocksize of 64 for $\\mathbf { W }$ for higher quantization precision and a blocksize of 256 for $c _ { 2 }$ to conserve memory. ", + "bbox": [ + 174, + 309, + 825, + 339 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "For parameter updates only the gradient with respect to the error for the adapters weights $\\frac { \\partial E } { \\partial { \\bf L } _ { i } }$ are needed, and not for 4-bit weights $\\frac { \\partial E } { \\partial \\mathbf { W } }$ . However, the calculation of $\\frac { \\partial E } { \\partial \\mathbf { L } _ { i } }$ entails the calculation of $\\textstyle \\frac { \\partial \\mathbf { X } } { \\partial \\mathbf { W } }$ which proceeds via equation (5) with dequantization from storage $\\dot { \\mathbf { W } } ^ { \\mathrm { N F 4 } }$ to computation data type ${ \\bf W } ^ { \\mathrm { B F 1 6 } }$ to calculate the derivative $\\textstyle \\frac { \\partial \\mathbf { X } } { \\partial \\mathbf { W } }$ in BFloat16 precision. ", + "bbox": [ + 173, + 345, + 825, + 411 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "To summarize, QLORA has one storage data type (usually 4-bit NormalFloat) and a computation data type (16-bit BrainFloat). We dequantize the storage data type to the computation data type to perform the forward and backward pass, but we only compute weight gradients for the LoRA parameters which use 16-bit BrainFloat. ", + "bbox": [ + 173, + 416, + 825, + 472 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4 QLoRA vs. Standard Finetuning ", + "text_level": 1, + "bbox": [ + 174, + 484, + 480, + 501 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We have discussed how QLoRA works and how it can significantly reduce the required memory for finetuning models. The main question now is whether QLoRA can perform as well as full-model finetuning. Furthermore, we want to analyze the components of QLoRA including the impact of NormalFloat4 over standard Float4. The following sections will discuss the experiments that aimed at answering these questions. ", + "bbox": [ + 174, + 508, + 506, + 632 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Experimental setup. We consider three architectures (encoder, encoder-decoder, and decoder only) and compare QLoRA with 16-bit adapter-finetuning and with full-finetuning for models up to 3B. Our evaluations include GLUE [59] with RoBERTalarge [38], Super-NaturalInstructionss [62] with T5 [50], and 5-shot MMLU [24] after finetuning LLaMA on Flan v2 [39] and Alpaca [56]. ", + "bbox": [ + 174, + 640, + 506, + 751 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/0e8ef49dc35c957f2d941190c1efe1eb5fb2a0ed7a9fe46e985ecb64c865a02e.jpg", + "image_caption": [ + "Figure 2: Mean zero-shot accuracy over Winogrande, HellaSwag, PiQA, Arc-Easy, and ArcChallenge using LLaMA models with different 4-bit data types. The NormalFloat data type significantly improves the bit-for-bit accuracy gains compared to regular 4-bit Floats. While Double Quantization (DQ) only leads to minor gains, it allows for a more fine-grained control over the memory footprint to fit models of certain size (33B/65B) into certain GPUs (24/48GB). " + ], + "image_footnote": [], + "bbox": [ + 531, + 512, + 812, + 674 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "To additionally study the advantages of NF4 over other 4-bit data types, we use the setup of Dettmers and Zettlemoyer [13] and measure post-quantization zero-shot accuracy and perplexity across different models (OPT [73], LLaMA [58], ", + "bbox": [ + 173, + 757, + 506, + 827 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "BLOOM [53], Pythia [7]) for model sizes $1 2 5 \\mathrm { m } - 1 3 \\mathrm { B }$ . We provide more details in the results section for each particular setup to make the results more readable. Full details in Appendix C. ", + "bbox": [ + 176, + 827, + 823, + 854 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "While paged optimizers are critical to do 33B/65B QLORA tuning on a single 24/48GB GPU, we do not provide hard measurements for Paged Optimizers since the paging only occurs when processing mini-batches with long sequence lengths, which is rare. We do, however, perform an analysis of the runtime of paged optimizers for 65B models on 48GB GPUs and find that with a batch size of 16, paged optimizers provide the same training speed as regular optimizers. Future work should measure and characterize under what circumstances slow-downs occur from the paging process. ", + "bbox": [ + 176, + 861, + 823, + 888 + ], + "page_idx": 4 + }, + { + "type": "table", + "img_path": "images/fe18786e8beae6966a15b4be1f7f12fc638a82ce3e6e8407f48ff91beb9310c2.jpg", + "table_caption": [ + "Table 3: Mean 5-shot MMLU test accuracy for LLaMA 7-65B models finetuned with adapters on Alpaca and FLAN v2 for different data types. Overall, NF4 with double quantization (DQ) matches BFloat16 performance, while FP4 is consistently one percentage point behind both. " + ], + "table_footnote": [], + "table_body": "
LLaMA SizeMean 5-shot MMLU AccuracyMean
7B13B33B65B
AlpacaFLAN v2AlpacaFLAN v2AlpacaFLAN v2AlpacaFLAN v2
BFloat1638.445.647.250.657.760.561.862.553.0
Float437.244.047.350.055.958.561.363.352.2
NFloat4 + DQ39.044.547.550.757.359.261.863.953.1
", + "bbox": [ + 174, + 136, + 825, + 224 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 255, + 825, + 309 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4-bit NormalFloat yields better performance than 4-bit Floating Point While the 4-bit NormalFloat (NF4) data type is information-theoretically optimal, it still needs to be determined if this property translates to empirical advantages. We follow the setup from Dettmers and Zettlemoyer [13] where quantized LLMs (OPT [73], BLOOM [53], Pythia [7], LLaMA) of different sizes (125M to 65B) with different data types are evaluated on language modeling and a set of zero-shot tasks. In Figure 7 and Table 2 we see that NF4 improves performance significantly over FP4 and Int4 and that double quantization reduces the memory footprint without degrading performance. ", + "bbox": [ + 174, + 320, + 594, + 473 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/32a2d8b5065e2c7678c8dbd9c3df8f74ccad118f0020186a256d1106db6bd6c1.jpg", + "table_caption": [ + "Table 2: Pile Common Crawl mean perplexity for different data types for 125M to 13B OPT, BLOOM, LLaMA, and Pythia models. " + ], + "table_footnote": [], + "table_body": "
Data typeMean PPL
Int434.34
Float4 (E2M1)31.07
Float4 (E3M0)29.48
NFloat4 + DQ27.41
", + "bbox": [ + 607, + 381, + 818, + 467 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "k-bit QLORA matches 16-bit full finetuning and 16-bit LoRA performance Recent findings have established that 4-bit quantization for inference is possible, but leads to performance degradation relative to 16-bit [13, 18]. This raises the crucial question of whether the lost performance can be recovered by conducting 4-bit adapter finetuning. We test this for two setups. ", + "bbox": [ + 174, + 484, + 825, + 540 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The first setup test if QLoRA can replicate 16-bit full finetuning performance for T5 and RoBERTa model finetuning. These results are detailed in the Appendix C.1. For our second setup, since full finetuning models at and beyond 11B parameters requires more than one server of high memory GPUs, we continue to test whether 4-bit QLORA can match 16-bit LoRA at the 7B to 65B parameter scales. To this end, we finetune LLaMA 7B through 65B on two instruction following datasets, Alpaca and FLAN v2, and evaluate on the MMLU benchmark via 5-shot accuracy. Results are shown in Table 3 where we see that NF4 with double quantization fully recovers the 16-bit LoRA MMLU performance. In addition, we also note that QLORA with FP4 lags behind the 16-bit brain float LoRA baseline by about 1 percentage point. This corroborates both our findings that (1) QLORA with NF4 replicates both 16-bit full finetuning and 16-bit LoRA finetuning performance, and (2) NF4 is superior to FP4 in terms of quantization precision. ", + "bbox": [ + 173, + 549, + 825, + 702 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Summary Our results consistently show that 4-bit QLORA with NF4 data type matches 16- bit full finetuning and 16-bit LoRA finetuning performance on academic benchmarks with wellestablished evaluation setups. We have also shown that NF4 is more effective than FP4 and that double quantization does not degrade performance. Combined, this forms compelling evidence that 4-bit QLORA tuning reliably yields results matching 16-bit methods. ", + "bbox": [ + 174, + 712, + 825, + 781 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In line with previous work on quantization [13], our MMLU and Elo results indicate that with a given finetuning and inference resource budget it is beneficial to increase the number of parameters in the base model while decreasing their precision. This highlights the importance of efficiency benefits from QLORA. Since we did not observe performance degradation compared to full-finetuning in our experiments with 4-bit finetuning, this raises the question of where the performance-precision trade-off exactly lies for QLoRA tuning, which we leave to future work to explore. ", + "bbox": [ + 174, + 791, + 825, + 875 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We proceed to investigate instruction tuning at scales that would be impossible to explore with full 16-bit finetuning on academic research hardware. ", + "bbox": [ + 174, + 883, + 821, + 911 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5 Pushing the Chatbot State-of-the-art with QLoRA ", + "text_level": 1, + "bbox": [ + 173, + 88, + 622, + 106 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Having established that 4-bit QLORA matches 16-bit performance across scales, tasks, and datasets we conduct an in-depth study of instruction finetuning up to the largest open-source language models available for research. To assess the performance of instruction finetuning these models, we evaluate on a challenging Natural Language Understanding benchmark (MMLU) and develop new methods for real-world chatbot performance evaluation. For a more qualitative analysis, see Appendix F. ", + "bbox": [ + 174, + 113, + 825, + 183 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5.1 Experimental setup ", + "text_level": 1, + "bbox": [ + 174, + 193, + 349, + 207 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We now describe an overview of the experimental setup with full details in Appendix D. ", + "bbox": [ + 173, + 210, + 746, + 226 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Data As, to our knowledge, there is no comprehensive study of instruction-following datasets, we select eight recent datasets. We include datasets obtained through crowd-sourcing (OASST1 [31], HH-RLHF [4]), distillation from instruction-tuned models (Alpaca [56], self-instruct [60], unnaturalinstructions [26]), corpora aggregations (FLAN v2 [12]), as well as hybrids (Chip2 [32], Longform [30]). These datasets cover different languages, data sizes, and licenses. ", + "bbox": [ + 174, + 233, + 825, + 303 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Training Setup To avoid confounding effects from different training objectives, we perform QLoRA finetuning with cross-entropy loss (supervised learning) without reinforcement learning, even for datasets that include human judgments of different responses. For datasets that have a clear distinction between instruction and response, we finetune only on the response (see ablations in Appendix D). For OASST1 and HH-RLHF, multiple responses are available. We then select the top response at every level of the conversation tree and finetune on the full selected conversation, including the instructions. In all of our experiments, we use NF4 QLORA with double quantization and paged optimizers to prevent memory spikes during gradient checkpointing. We do a small hyperparameter searches for the 13B and 33B LLaMA models and we find that all hyperparameter settings found at 7B generalize (including number of epochs) except learning rate and batch size. We halve the learning rate for 33B and 65B while doubling the batch size. ", + "bbox": [ + 174, + 310, + 825, + 463 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Baselines We compare our models to both research (Vicuna [10] and Open Assistant [31]) and commercial (GPT-4 [42], GPT-3.5-turbo and Bard) chatbot systems. The Open Assistant model is a LLaMA 33B model finetuned with Reinforcement Learning from Human Feedback (RLHF) on the OASST1 dataset. Vicuna does full fine-tuning of LLaMA 13B on proprietary user-shared conversations from ShareGPT and is thus the result of distillation from OpenAI GPT models. ", + "bbox": [ + 174, + 470, + 825, + 540 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Evaluation setup Chatbot evaluation is not straightforward since each prompt has many highquality responses which are challenging to rank. We therefore follow a comprehensive approach, which involves (a) standard benchmarks that measures general language understanding performance (MMLU [24]), (b) both automatic and human evaluation that measures how many responses from chatbot A are better compared to chatbot B, (c) a round-robin tournament-style evaluation where chatbots compete against each other in games where chatbot performance is measured as ELO. We provide a detailed discussion of these evaluation setups in the Appendix E. ", + "bbox": [ + 174, + 549, + 825, + 645 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5.2 Guanaco: QLORA trained on OASST1 is a State-of-the-art Chatbot ", + "text_level": 1, + "bbox": [ + 176, + 655, + 689, + 669 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Based on our automated and human evaluations, we find that the top QLORA tuned model, Guanaco 65B, which we finetune on a variant of OASST1, is the best-performing open-source chatbot model and offers performance competitive to ChatGPT. When compared to GPT-4, Guanaco 65B and 33B have an expected win probability of $30 \\%$ , based on Elo rating from human annotators system-level pairwise comparisons on the Vicuna benchmark - the highest reported to date. ", + "bbox": [ + 174, + 674, + 825, + 742 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "The Vicuna benchmark [10] results relative to ChatGPT are shown in Table 4. We find that Guanaco 65B is the best-performing model after GPT-4, achieving $9 9 . 3 \\%$ performance relative to ChatGPT. Guanaco 33B has more parameters than the Vicuna 13B model, but uses only 4-bit precision for its weights and is thus much more memory efficient at 21 GB vs $2 6 \\mathrm { G B }$ , providing a three percentage points of improvement over Vicuna 13B. Furthermore, Guanaco 7B easily fits on modern phones at a 5 GB footprint while still scoring nearly 20 percentage points higher than Alpaca 13B. ", + "bbox": [ + 174, + 750, + 825, + 833 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "However, Table 4 also has very wide confidence intervals, with many models overlapping in performance. We hypothesize that this uncertainty comes from the lack of clear specification of scale, e.g., it is unclear what 8 on a 10 point scale means across different scenarios. As such, we instead recommend using the Elo ranking method [16], based on pairwise judgments from human annotators and GPT-4 to avoid the problem of grounding an absolute scale. ", + "bbox": [ + 174, + 839, + 825, + 909 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/b63e9ace2009fed2735d2263955ed8d3b6169c9f79d790fa9d6ddceb6ec646ec.jpg", + "table_caption": [ + "Table 4: Zero-shot Vicuna benchmark scores as a percentage of the score obtained by ChatGPT evaluated by GPT-4. We see that OASST1 models perform close to ChatGPT despite being trained on a very small dataset and having a fraction of the memory requirement of baseline models. " + ], + "table_footnote": [], + "table_body": "
Model/DatasetParamsModel bitsMemoryChatGPT vs SysSys vs ChatGPTMean95% CI
GPT-4 Bard===119.4%110.1%114.5%2.6%
1--93.2%96.4%94.8%4.1%
Guanaco Alpaca FLAN v265B4-bit41 GB96.7%101.9%99.3%4.4%
65B4-bit41 GB63.0%77.9%70.7%4.3%
65B4-bit41 GB37.0%59.6%48.4%4.6%
Guanaco Open Assistant Alpaca FLAN v233B4-bit21 GB96.5%99.2%97.8%4.4%
33B16-bit66 GB73.4%85.7%78.1%5.3%
33B 33B4-bit 4-bit21GB 21GB67.2%79.7%73.6%4.2%
Vicuna Guanaco Alpaca HH-RLHF26.3%49.7%38.0%3.9%
13B 13B16-bit 4-bit26 GB 10 GB91.2% 87.3%98.7% 93.4%94.9% 90.4%4.5% 5.2%
13B4-bit10 GB63.8%76.7%69.4%
13B4-bit10 GB55.5%69.1%62.5%4.2%
13B4-bit10 GB50.6%69.8%60.5%4.7%
13B4-bit10 GB49.2%69.3%59.5%4.2% 4.7%
Longform Self-Instruct FLAN v213B4-bit10 GB44.9%62.0%53.6%5.2%
13B4-bit10 GB38.0%60.5%49.1%4.6%
13B4-bit10 GB32.4%61.2%47.0%3.6%
4-bit5GB
Guanaco Alpaca FLAN v27B 7B4-bit5GB84.1% 57.3%89.8% 71.2%87.0%5.4%
7B4-bit5GB33.3%56.1%64.4% 44.8%5.0% 4.0%
", + "bbox": [ + 174, + 136, + 825, + 433 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Elo ratings of the most competitive models can be seen in Table 1. We note that human and GPT-4 ranking of models on the Vicuna benchmark disagree partially, particularly for Guanaco 7B, but are consistent for most models with a Kendall Tau of $\\tau = 0 . 4 3$ and Spearman rank correlation of $r = 0 . 5 5$ at the system level. At the example level, the agreement between GPT-4 and human annotators’ majority vote is weaker with Fleiss $\\kappa = 0 . 2 5$ . Overall, this shows a moderate agreement between system-level judgments by GPT-4 and human annotators, and thus that model-based evaluation represents a somewhat reliable alternative to human evaluation. See Section E for further considerations. ", + "bbox": [ + 174, + 467, + 531, + 645 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/22dd1d0ebb53983212c69bf0aa8aa1a13e4f0e1ed85df596485161d4934c19b6.jpg", + "table_caption": [ + "Table 5: MMLU 5-shot test results for different sizes of LLaMA finetuned on the corresponding datasets using QLoRA. " + ], + "table_footnote": [], + "table_body": "
Dataset7B13B33B65B
LLaMA no tuning35.146.957.863.4
Self-Instruct36.433.353.056.7
Longform32.143.256.659.7
Chip234.541.653.659.8
HH-RLHF34.944.655.860.1
Unnatural Instruct41.948.157.361.3
Guanaco (OASST1)36.646.457.062.2
Alpaca38.847.857.362.5
FLAN v244.551.459.263.9
", + "bbox": [ + 545, + 502, + 821, + 631 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Elo rankings in Table 6 indicate that Guanaco 33B and 65B models outperform all models besides GPT-4 on the Vicuna and OA benchmarks and that they perform comparably to ChatGPT in line with Table 4. We note that the Vicuna benchmark favors open-source models while the larger OA benchmark favors ChatGPT. Furthermore, we can see from Tables 5 and 4 that the suitability of a finetuning dataset is a determining factor in performance. Finetuning LLaMA models on FLAN v2 does particularly well on MMLU, but performs worst on the Vicuna benchmark (similar trends are observed with other models). This also points to partial orthogonality in current evaluation benchmarks: strong MMLU performance does not imply strong chatbot performance (as measured by Vicuna or OA benchmarks) and vice versa. ", + "bbox": [ + 174, + 660, + 825, + 785 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Guanaco is the only top model in our evaluation that is not trained on proprietary data as the OASST1 dataset collection guidelines explicitly forbid the use of GPT models. The next best model trained on only open-source data is the Anthropic HH-RLHF model, which scores 30 percentage points lower than Guanaco on the Vicuna benchmark (see Table 4). Overall, these results show that 4-bit QLORA is effective and can produce state-of-the-art chatbots that rival ChatGPT. Furthermore, our 33B Guanaco can be trained on 24 GB consumer GPUs in less than 12 hours. This opens up the potential for future work via QLORA tuning on specialized open-source data, which produces models that can compete with the very best commercial models that exist today. ", + "bbox": [ + 174, + 800, + 825, + 911 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/b316a6d0003af441eaf334b8cb75321150b8501b411257edd98d6b2d4077beba.jpg", + "table_caption": [ + "Table 6: Elo rating for a tournament between models where models compete to generate the best response for a prompt, judged by human raters or GPT-4. Overall, Guanaco 65B and 33B tend to be preferred to ChatGPT-3.5 on the benchmarks studied. According to human raters they have a Each 10-point difference in Elo is approximately a difference of $1 . 5 \\%$ in win-rate. " + ], + "table_footnote": [], + "table_body": "
Benchmark # Prompts Judge ModelVicuna 80 Human ratersVicuna 80 GPT-4Open Assistant 953 GPT-4Median Rank
EloRankEloRankEloRank
GPT-41176113481129411
Guanaco-65B102321022100832
Guanaco-33B10094992100244
ChatGPT-3.5 Turbo9167966235101525
Vicuna-13B9845974493655
Guanaco-13B9756913688566
Guanaco-7B10103879886077
Bard909890271-8
", + "bbox": [ + 200, + 148, + 795, + 335 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "6 Related Work ", + "text_level": 1, + "bbox": [ + 173, + 361, + 321, + 378 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Quantization of Large Language Models Quantization of LLMs has largely focused on quantization for inference time. Major approaches for preserving 16-bit LLM quality focus on managing outlier features (e.g., SmoothQuant [67] and LLM.int8() [14]) while others use more sophisticated grouping methods [44, 70]. Lossy quantization approaches study the trade-offs for regular rounding [13, 72, 48] or how to optimize rounding decisions to improve quantization precision [18]. Besides our work, SwitchBack layers [66] is the only work that studies backpropagation through quantized weights at a scale beyond 1B parameters. ", + "bbox": [ + 174, + 387, + 825, + 484 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Finetuning with Adapters While we use Low-rank Adapters [28] (LoRA), many other Parameter Efficient FineTuning (PEFT) methods have been proposed such as prompt tuning [49, 33, 34], tuning the embedding layer inputs [1], tuning hidden states $\\mathrm { ( I A ^ { 3 } ) }$ [37], adding full layers [27], tuning biases [71], learning a mask over weights based on Fisher information [55], and a combination of approaches [23]. In our work, we show that LoRA adapters are able to reach full 16-bit finetuning performance. We leave it to future work to explore the tradeoffs of other PEFT approaches. ", + "bbox": [ + 174, + 494, + 825, + 577 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Instruction Finetuning To help a pretrained LLM follow the instructions provided in a prompt, instruction finetuning uses input-output pairs of various data sources to finetune a pretrained LLM to generate the output given the input as a prompt. Approaches and datasets include MetaICL [40], MetaTuning [74], InstructGPT [43], FLAN [63, 12], PromptSource [3], Super-NaturalInstructions [62, 51], Self-instruct [60], UnnaturalInstructions [26], OPT-IML [29], UnifiedSKG[68], OIG/Chip2 [32], Alpaca [56], Vicuna [10], Koala [20], and Self-instruct-GPT-4 [45]. ", + "bbox": [ + 174, + 587, + 825, + 671 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Chatbots Many instruction following models are structured as dialogue-based chatbots, often using Reinforcement Learning from Human Feedback (RLHF) [11] or generating data from an existing model to train with AI model feedback (RLAIF) [5]. Approaches and datasets include AnthropicHH [2, 4], Open Assistant [31], LaMDA [57], and Sparrow [21]. We do not use reinforcement learning, but our best model, Guanaco, is finetuned on multi-turn chat interactions from the Open Assistant dataset which was designed to be used for RLHF training [31]. For the evaluation of chatbots approaches that use GPT-4 instead of costly human annotation have been developed [10, 45]. We improve on such approaches with a focus on an evaluation setup that is more reliable. ", + "bbox": [ + 174, + 681, + 825, + 792 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "7 Limitations and Discussion ", + "text_level": 1, + "bbox": [ + 176, + 806, + 431, + 824 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We have shown evidence that our method, QLORA, can replicate 16-bit full finetuning performance with a 4-bit base model and Low-rank Adapters. However, due to the immense resource cost, we were not establish that QLORA can match 16-bit finetuning performance at 33B and 65B scales. ", + "bbox": [ + 176, + 833, + 825, + 875 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Another limitation is the evaluation of instruction finetuning models. While we provide evaluations on MMLU, the Vicuna/OA benchmark, we did not evaluate on BigBench, RAFT, and HELM, and it is not ensured that our evaluations generalize to these benchmarks. On the other hand, we perform a very broad study on MMLU and develop new methods for evaluating chatbots. ", + "bbox": [ + 174, + 882, + 821, + 911 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 92, + 823, + 119 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "From the evidence presented, it appears that the performance of these benchmarks likely depends how similar the finetuning data is to the benchmark dataset. For example, FLAN v2 is similar to MMLU, but dissimilar to the Vicuna benchmark and vice versa for the Chip2 dataset. This highlights that not only better benchmarks and evaluation is needed, but that one needs to be careful about what one is evaluating in the first place. Do we want to create models that do well on classroom highschool and colleague knowledge or do we want to do well on chatbot conversation ability? Maybe something else? Because it is always easier to evaluate on an existing benchmark compared to creating a new one, certain benchmarks can steer the community towards a certain direction. We should ensure as a community that the benchmarks measure what we care about. ", + "bbox": [ + 174, + 126, + 825, + 250 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "An additional limitation is that we did not evaluate different bit-precisions or different adapter methods. Besides LoRA, there is also a wide variety Parameter Efficient FineTuning (PEFT) methods that have been shown to work well. However, it is unclear if these methods scale to large models. We used LoRA as many results established its robustness but other adapters might yield better performance. Since finetuning after quantization seems to recover most of the information that is lost during quantization this might enable much more aggressive quantization. For example, 3-bit GPTQ quantization might also yield 16-bit full finetuning performance after finetuning. ", + "bbox": [ + 174, + 257, + 825, + 353 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "8 Broader Impacts ", + "text_level": 1, + "bbox": [ + 174, + 366, + 346, + 383 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Our QLORA finetuning method is the first method that enables the finetuning of 33B parameter models on a single consumer GPU and 65B parameter models on a single professional GPU, while not degrading performance relative to a full finetuning baseline. We have demonstrated that our best 33B model trained on the Open Assistant dataset can rival ChatGPT on the Vicuna benchmark. Since instruction finetuning is an essential tool to transform raw pretrained LLMs into ChatGPT-like chatbots, we believe that our method will make finetuning widespread and common in particular for the researchers that have the least resources – a big win for the accessibility of state of the art NLP technology. QLORA can be seen as an equalizing factor that helps to close the resource gap between large corporations and small teams with consumer GPUs. ", + "bbox": [ + 174, + 390, + 825, + 515 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Another potential source of impact is deployment to mobile phones and other low resources devices. While 7B models were shown to be able to be run on phones before, QLORA is the first method that would enable the finetuning of such models. We estimate that with an iPhone 12 Plus, QLORA can finetune 3 million tokens per night while the phone is charging. QLORA can help enable privacy-preserving usage of LLMs, where users can own and manage their own data and models, while simultaneously making LLMs easier to deploy. ", + "bbox": [ + 174, + 521, + 825, + 604 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Furthermore, because of the increased inference efficiency of 4-bit models, if QLoRA models are deployed they reduce the environmental impact that LLMs have when deployed for personal use. We estimate that if $50 \\%$ of deployments are personal and $50 \\%$ are company deployments, QLoRA deployments could reduce the overall carbon footprint by $72 \\%$ (see Appendix B). ", + "bbox": [ + 174, + 611, + 825, + 666 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "However, finetuning is a dual-use technology that can be abused to cause harm. Widespread use of LLMs has known dangers [8, 6], but we believe that equalizing access to a technology that is quickly becoming ubiquitous will allow for better more independent analysis than keeping the power of LLMs in the hands of large corporations that do not release models or source code for auditing. ", + "bbox": [ + 174, + 672, + 825, + 728 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "All in all, we believe that QLORA will have a broadly positive impact making the finetuning of high quality LLMs much more widely and easily accessible. ", + "bbox": [ + 176, + 736, + 823, + 763 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Acknowledgements ", + "text_level": 1, + "bbox": [ + 176, + 776, + 338, + 792 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "We thank Aditya Kusupati, Ofir Press, Ashish Sharma, Margaret Li, Raphael Olivier, Zihao Ye, and Evangelia Spiliopoulou for their valuable feedback. We thank Norah Altriri for providing us with an intuitive NF4 visualization. We thank Felix Petersen for a discussion on distributional quantization and its relationship to quantile quantization. Our research was facilitated by the advanced computational, storage, and networking infrastructure of the Hyak supercomputer system at the University of Washington. We thank the Hyak team for ensuring a smooth operation. We thank the beta testers of the bitsandbytes library, in particular Alex Birch and Alyssa Vance. We thank Younes Belkada for help with the integration of our software into the Hugging Face transformers stack. ", + "bbox": [ + 174, + 800, + 825, + 911 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "References \n[1] S. An, Y. Li, Z. Lin, Q. Liu, B. Chen, Q. Fu, W. Chen, N. Zheng, and J.-G. Lou. Input-tuning: Adapting unfamiliar inputs to frozen pretrained models. arXiv preprint arXiv:2203.03131, 2022. \n[2] A. Askell, Y. Bai, A. Chen, D. Drain, D. Ganguli, T. Henighan, A. Jones, N. Joseph, B. Mann, N. DasSarma, et al. A general language assistant as a laboratory for alignment. arXiv preprint arXiv:2112.00861, 2021. \n[3] S. H. Bach, V. Sanh, Z.-X. Yong, A. 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Stable and low-precision training for large-scale vision-language models. arXiv preprint arXiv:2304.13013, 2023. \n[67] G. Xiao, J. Lin, M. Seznec, J. Demouth, and S. Han. Smoothquant: Accurate and efficient post-training quantization for large language models. arXiv preprint arXiv:2211.10438, 2022. \n[68] T. Xie, C. H. Wu, P. Shi, R. Zhong, T. Scholak, M. Yasunaga, C.-S. Wu, M. Zhong, P. Yin, S. I. Wang, et al. Unifiedskg: Unifying and multi-tasking structured knowledge grounding with text-to-text language models. arXiv preprint arXiv:2201.05966, 2022. \n[69] Z. Yang, P. Qi, S. Zhang, Y. Bengio, W. Cohen, R. Salakhutdinov, and C. D. Manning. Hotpotqa: A dataset for diverse, explainable multi-hop question answering. In Proceedings of the 2018 Conference on Empirical Methods in Natural Language Processing, pages 2369–2380, 2018. \n[70] Z. Yao, R. Y. Aminabadi, M. Zhang, X. Wu, C. Li, and Y. He. Zeroquant: Efficient and affordable post-training quantization for large-scale transformers. arXiv preprint arXiv:2206.01861, 2022. \n[71] E. B. Zaken, S. Ravfogel, and Y. Goldberg. Bitfit: Simple parameter-efficient fine-tuning for transformer-based masked language-models. arXiv preprint arXiv:2106.10199, 2021. \n[72] A. Zeng, X. Liu, Z. Du, Z. Wang, H. Lai, M. Ding, Z. Yang, Y. Xu, W. Zheng, X. Xia, et al. Glm-130b: An open bilingual pre-trained model. arXiv preprint arXiv:2210.02414, 2022. \n[73] S. Zhang, S. Roller, N. Goyal, M. Artetxe, M. Chen, S. Chen, C. Dewan, M. Diab, X. Li, X. V. Lin, et al. Opt: Open pre-trained transformer language models. arXiv preprint arXiv:2205.01068, 2022. \n[74] R. Zhong, K. Lee, Z. Zhang, and D. Klein. Adapting language models for zero-shot learning by meta-tuning on dataset and prompt collections. arXiv preprint arXiv:2104.04670, 2021. ", + "bbox": [ + 173, + 77, + 828, + 915 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 60, + 828, + 917 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "", + "bbox": [ + 169, + 46, + 828, + 914 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 47, + 828, + 920 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 90, + 828, + 210 + ], + "page_idx": 14 + }, + { + "type": "image", + "img_path": "images/f19169a1a302d83726a2457a3f1ad8a68dedecfed370732ee47ca504c9529cb6.jpg", + "image_caption": [ + "Figure 3: Speedups of NF4 inference for batch size 1 compared to 16-bit inference for different GPUs. We see that RTX 3090/4090 and A40 GPUs have large speedups of $2 . 9 – 4 . 0 \\mathbf { x }$ while other GPUs have speedups in the range 1.1-1.5x. The difference between GPUs is mostly caused by poor instruction throughput bottlenecks. " + ], + "image_footnote": [], + "bbox": [ + 238, + 118, + 756, + 412 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "A Inference Speedups for NF4 ", + "text_level": 1, + "bbox": [ + 178, + 489, + 442, + 507 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "The speedups for single batch inference for NF4 compare to 16-bit float are shown in Figure 3. ", + "bbox": [ + 169, + 513, + 776, + 529 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "B Environmental Impact and Carbon Footprint Reductions ", + "text_level": 1, + "bbox": [ + 176, + 540, + 687, + 558 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "The relationship between serving efficiency and environmental impact and cost/performance is a complicated one. The short answer is, for personal use, we reduce the environmental impact per token by about $3 . 5 \\mathrm { x }$ . If $50 \\%$ LLM deployments are personal and $50 \\%$ company, we reduce the environmental footprint of inference by $72 \\%$ . ", + "bbox": [ + 174, + 565, + 825, + 621 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "The long answer is this: For deployment, there are two options: (1) personal deployment, (2) deployment by companies for many users. (1) uses small batch sizes (usually batch size $^ { = 1 }$ ), (2) uses large batch sizes (64-128). Per token, (2) offers about ${ > } 5 0 \\mathrm { x }$ better efficiency because for every weight that is loaded from memory, up to 64-128 tokens can be calculated. The ${ > } 5 0 \\mathrm { x }$ improvement in efficiency stems from the fact memory operations are energy inefficient, and floating point operations are energy efficient. As such,company deployment (2) is a very environmentally friendly and cheap approach. ", + "bbox": [ + 173, + 627, + 825, + 724 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Currently, we have efficient 4-bit CUDA kernels for the personal deployment scenario (1) with batch $\\mathrm { s i z e } { = } 1$ which are about $3 . 5 \\mathrm { x }$ more efficient (see Figure 3). As such, we reduce the overall footprint for personal deployment (1) significantly. ", + "bbox": [ + 174, + 731, + 825, + 773 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "The overall cost and footprint is now determined by how many people use (1) vs (2). For example, if $50 \\%$ of users use personal deployment (1) and $50 \\%$ company deployments (2) and we assume that (2) is about $5 0 \\mathrm { x }$ more efficient then we get the following numbers. Approach (1) accounts for $5 0 \\% / ( 5 0 \\%$ $+ 5 0 \\% / 5 0 ) = 9 8 \\%$ of environmental impact. This means, the more users deploy personal LLMs, the larger benefit of our method. If $50 \\%$ of people use personally deployed LLMs (phones, laptops etc) then our method will reduce the impact by about 1- $( 9 8 \\% / 3 . 5 \\mathrm { x } ) = 7 2 \\%$ . ", + "bbox": [ + 173, + 779, + 825, + 863 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "The environmental impact reduction is roughly proportional to energy consumption per token processed which is roughly proportional to the cost of running LLMs. As such, the bottom line is affected in the same proportions. ", + "bbox": [ + 176, + 869, + 825, + 911 + ], + "page_idx": 15 + }, + { + "type": "table", + "img_path": "images/baab14900cfe8540186693878eddde8c6bc100e8d0e5e7b4693b4fbf65bb51e7.jpg", + "table_caption": [ + "Table 7: Experiments comparing 16-bit BrainFloat (BF16), 8-bit Integer (Int8), 4-bit Float (FP4), and 4- bit NormalFloat (NF4) on GLUE and Super-NaturalInstructions. QLORA replicates 16-bit LoRA and fullfinetuning. " + ], + "table_footnote": [], + "table_body": "
Dataset ModelGLUE (Acc.)Super-NaturalInstructions (RougeL)
RoBERTa-largeT5-80MT5-250MT5-780MT5-3BT5-11B
BF1688.640.142.148.054.362.0
BF16 replication88.640.042.247.354.91
LoRA BF1688.840.542.647.155.460.7
QLoRA Int888.840.442.945.456.560.7
QLORA FP488.640.342.447.555.660.9
QLORA NF4 + DQ140.442.747.755.360.9
", + "bbox": [ + 176, + 135, + 825, + 268 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Overall, our method will have a strong impact on environmental impact and cost reduction for personal LLM deployments. ", + "bbox": [ + 176, + 295, + 823, + 324 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "C QLoRA vs Standard Finetuning Experimental Setup Details ", + "text_level": 1, + "bbox": [ + 176, + 335, + 712, + 354 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "C.1 BF16 vs NF4 for T5/RoBERTa ", + "text_level": 1, + "bbox": [ + 176, + 361, + 428, + 376 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Here we detail additional results where we run experiments to confirm we can replicate 16-bit full finetuning performance when we use QLoRA. We test RoBERTA and T5 models sized 125M to 3B parameters on GLUE and the Super-NaturalInstructions dataset. Results are shown in Table 7. In both datasets, we observe that 16-bit, 8-bit, and 4-bit adapter methods replicate the performance of the fully finetuned 16-bit baseline. This suggests that the performance lost due to the imprecise quantization can be fully recovered through adapter finetuning after quantization. ", + "bbox": [ + 174, + 380, + 826, + 464 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "C.2 Hyperparameter search for QLORA ", + "text_level": 1, + "bbox": [ + 176, + 473, + 472, + 488 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "We do a hyperparameter search for LoRA over the following variables: LoRA dropout $\\left\\{ \\ 0 . 0 , 0 . 0 5 \\right.$ , 0.1}, LoRA $r \\ \\{ \\ 8 , 1 6 , 3 2 , 6 4 , 1 2 8 , 2 5 6 \\}$ , LoRA layers {key+query, all attention layers, all FFN layers, all layers, attention $^ +$ FFN output layers}. We keep LoRA $\\alpha$ fixed and search the learning rate, since LoRA $\\alpha$ is always proportional to the learning rate. ", + "bbox": [ + 173, + 492, + 826, + 549 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "We find that LoRA dropout 0.05 is useful for small models (7B, 13B), but not for larger models (33B, 65B). We find LoRA $r$ is unrelated to final performance if LoRA is used on all layers as can be seen in Figure 4 ", + "bbox": [ + 174, + 555, + 826, + 597 + ], + "page_idx": 16 + }, + { + "type": "image", + "img_path": "images/4c7a0bb3134957dd5ef35455dfedd9cd1eec0aade082cedcaf728a0fc1cade73.jpg", + "image_caption": [ + "Figure 4: LoRA $r$ for LLaMA 7B models finetuned on Alpaca. Each dot represents a combination of hyperparameters and for each LoRA $r$ we run 3 random seed with each hyperparameter combination. The performance of specific LoRA $r$ values appears to be independent of other hyperparameters. " + ], + "image_footnote": [], + "bbox": [ + 366, + 613, + 629, + 781 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "C.3 Super-Natural Instructions Experimental Setup Details ", + "text_level": 1, + "bbox": [ + 174, + 849, + 602, + 866 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "We use the same preprocessing of the Super-Natural Instruction dataset as Wang et al. [61]. However, we split the training data in training and validation datasets allowing us to perform more rigorous hyperparameter tuning and early stopping. We use the same hyperparameters described in the paper ", + "bbox": [ + 174, + 869, + 826, + 911 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "for training the various T5 model sizes on the Super-Natural Instruction data. We use LoRA $r = 1 6$ for small, medium, and large T5 models and LoRA $r = 6 4$ for T5 xl and xxl models. We also use LoRA $\\alpha = 6 4$ in all our experiments and no LoRA dropout. ", + "bbox": [ + 174, + 92, + 825, + 133 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "D Training a State-of-the-art Chatbot Experimental Setup Details ", + "text_level": 1, + "bbox": [ + 176, + 145, + 738, + 162 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "D.1 Datasets ", + "text_level": 1, + "bbox": [ + 174, + 170, + 276, + 184 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "We describe the datasets used for QLORA finetuning experiments outlined in Section 5. ", + "bbox": [ + 176, + 188, + 746, + 203 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "OASST1 The OpenAssistant dataset [31] was collected via crowd-sourcing. It contains 161,443 unique messages distributed across 66,497 conversations and spanning 35 different languages. The dataset often contains several ranked replies for each given user question. In our experiments, we only use the top reply at each level in the conversation tree. This limits the dataset to 9,846 examples. We finetune models on the full conversation including the user queries. ", + "bbox": [ + 174, + 210, + 825, + 280 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "HH-RLHF This is a human preference dataset about helpfulness and harmlessness. Each datapoint consists of two assistant replies to a user question along with a human preference judgment of the best reply. The dataset contains 160,800 examples. When finetuning on this dataset, we combine helpfulness and harmlessness data and only keep the preferred assistant reply. ", + "bbox": [ + 174, + 287, + 825, + 344 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "FLAN v2 The FLAN v2 collection [39] is a collection of 1836 tasks augmented with hundreds of manually curated templates and rich formatting patterns into over 15M examples. The authors show that models trained on this collection outperform other public collections including the original FLAN 2021 [63], $\\mathrm { T 0 + + }$ [51], Super-Natural Instructions [61], and OPT-IML [29]. We used the same task mixtures described by the authors with the exception of some datasets that were not freely available at the time of writing. ", + "bbox": [ + 174, + 352, + 825, + 435 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Self-Instruct, Alpaca, Unnatural Instructions The Self-Instruct, Alpaca, and Unnatural Instructions datasets [60, 56, 26] are instruction tuning datasets collected with various approaches of model distillation from GPT-3 Instruct and ChatGPT. They rely on prompting, in-context learning, and paraphrasing to come up with diverse sets of instructions and outputs. The datasets comprise of 82,612, 51,942, and 240,670 examples respectively. One advantage of such distilled datasets is that they contain a more diverse set of instruction styles compared to the FLAN v2 collection and similar instruction tuning collections. ", + "bbox": [ + 173, + 443, + 825, + 540 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Longform The LongForm dataset [30] is based on an English corpus augmented with instructions and as such is a hybrid human-generated dataset. The underlying documents are human-written and come from C4 and Wikipedia while the instructions are generated via LLMs. The dataset is extended with additional structured corpora examples such as Stack Exchange and WikiHow and task examples such as question answering, email writing, grammar error correction, story/poem generation, and text summarization. The dataset contains 23,700 examples. ", + "bbox": [ + 174, + 547, + 825, + 632 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Chip2 is part of the OIG Laion dataset. It contains Python code examples, natural instruction examples, generic harmless instructions, instruction/responses with lists, follow-up questions, Wikipedia toxic adversarial questions, grade school math, reasoning instructions, and character and scene descriptions with a total of 210,289 examples. ", + "bbox": [ + 174, + 640, + 825, + 695 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "D.2 Default LoRA hyperparameters do not match 16-bit performance ", + "text_level": 1, + "bbox": [ + 178, + 705, + 671, + 719 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "We find that default hyperparameters for fully finetuned baselines are undertuned. We do a hyperparameter search over learning rates 1e-6 to 5e-5 and batch sizes 8 to 128 to find robust baselines. Results for 7B LLaMA finetuning on Alpaca are shown in Figure 5. ", + "bbox": [ + 174, + 724, + 826, + 765 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "When using the standard practice of applying LoRA to query and value attention projection matrices [28], we are not able to replicate full finetuning performance for large base models. As shown in Figure 5 for LLaMA 7B finetuning on Alpaca, we find that the most critical LoRA hyperparameter is how many LoRA adapters are used in total and that LoRA on all linear transformer block layers are required to match full finetuning performance. Other LoRA hyperparameters, such as the projection dimension $r$ , do not affect performance. ", + "bbox": [ + 174, + 771, + 825, + 854 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "D.3 Hyperparameters ", + "text_level": 1, + "bbox": [ + 174, + 864, + 341, + 880 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "We provide the exact hyperparameters used in our QLORA finetuning experiments. We find hyperparameters to be largely robust across datasets. We use the MMLU 5-shot dev set for validation and hyperparameter tuning. In all our experiments we use NF4 with double quantization and bf16 computation datatype. We set LoRA $r = 6 4$ , $\\alpha = 1 6$ , and add LoRA modules on all linear layers of the base model. We also use Adam beta2 of 0.999, max grad norm of 0.3 and LoRA dropout of 0.1 for models up to 13B and 0.05 for 33B and 65B models. Following previous work on instruction finetuning [63, 61] and after benchmarking other linear and cosine schedules, we use a constant learning rate schedule. We use group-by-length to group examples of similar lengths in the same batch (note this will produce an oscillating loss curve). The hyperparameters we tune for each model size are shown in Table 8. ", + "bbox": [ + 174, + 883, + 821, + 911 + ], + "page_idx": 17 + }, + { + "type": "image", + "img_path": "images/4a725dd28823eb79778ee3ad3d60564104ae2be63b44982f33208172d9c6cf42.jpg", + "image_caption": [ + "Figure 5: RougeL for LLaMA 7B models on the Alpaca dataset. Each point represents a run with a different random seed. We improve on the Stanford Alpaca fully finetuned default hyperparameters to construct a strong 16-bit baseline for comparisons. Using LoRA on all transformer layers is critical to match 16-bit performance. " + ], + "image_footnote": [], + "bbox": [ + 362, + 89, + 632, + 290 + ], + "page_idx": 18 + }, + { + "type": "table", + "img_path": "images/093e60836be5f9b91620703d7adf93ff7ed1a76a4b9aa3688f325c047565babe.jpg", + "table_caption": [], + "table_footnote": [ + "Table 8: Training hyperparameters for QLORA finetuning on different datasets and across model sizes. " + ], + "table_body": "
ParametersDatasetBatch sizeLRStepsSource LengthTarget Length
7BAll162e-410000384128
7BOASST1162e-41875-512
7BHH-RLHF162e-4100001768
7BLongform162e-440005121024
13BAll162e-410000384128
13BOASST1162e-41875-512
13BHH-RLHF162e-410000=768
13BLongform162e-440005121024
33BAll321e-45000384128
33BOASST1161e-41875-512
33BHH-RLHF321e-450001768
33BLongform321e-423435121024
65BAll641e-42500384128
65BOASST1161e-41875-512
65BHH-RLHF641e-425001768
65BLongform321e-423435121024
", + "bbox": [ + 194, + 349, + 805, + 621 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 676, + 825, + 787 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "D.4 Ablations ", + "text_level": 1, + "bbox": [ + 174, + 805, + 284, + 820 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "While it is general practice in the literature to only train on the response in instruction following datasets, we study the effect of training on the instruction in addition to the response in Table 9. In these experiments, we restrict the training data to 52,000 examples and use the 7B model. Over four different instruction tuning datasets, we find that only training on the target is beneficial to MMLU performance. We did not evaluate the effect this may have on chatabot performance as measured by vicuna or OA benchmarks. ", + "bbox": [ + 174, + 827, + 825, + 911 + ], + "page_idx": 18 + }, + { + "type": "table", + "img_path": "images/6f38d34a2dafc7c1bdd564de624a2aec30b018c35cc0af4696661fc193154a38.jpg", + "table_caption": [], + "table_footnote": [ + "Table 9: MMLU 5-shot test results studying the effect of training on the instructions in addition to the response. " + ], + "table_body": "
DatasetUnnatural InstructionsChip2AlpacaFLAN v2Mean
Train on source and target36.233.738.142.037.5
Train on target38.034.539.042.938.6
", + "bbox": [ + 184, + 88, + 812, + 147 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "D.5 Training Considerations ", + "text_level": 1, + "bbox": [ + 173, + 199, + 385, + 215 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "We note that the OASST1 dataset on which Guanaco models are trained is multilingual and that the OA benchmark also contains prompts in different languages. We leave it to future work to investigate the degree to which such multilingual training improves performance on instructions in languages other than English and whether this explains the larger gap between Vicuna-13B model (only trained on English data) and Guanaco 33B and 65B on the OA benchmark. ", + "bbox": [ + 174, + 220, + 825, + 290 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Given the strong performance of Guanaco models, we investigate any data leakage between the OASST1 data and the Vicuna benchmark prompts. We do not find overlapping prompts after performing fuzzy string matching in the two datasets and inspecting the closest matches manually. ", + "bbox": [ + 173, + 301, + 826, + 343 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Furthermore, we note that our model is only trained with cross-entropy loss (supervised learning) without relying on reinforcement learning from human feedback (RLHF). This calls for further investigations of the tradeoffs of simple cross-entropy loss and RLHF training. We hope that QLORA enables such analysis at scale, without the need for overwhelming computational resources. ", + "bbox": [ + 174, + 354, + 825, + 410 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "D.6 What is more important: instruction finetuning dataset size or dataset quality? ", + "text_level": 1, + "bbox": [ + 173, + 425, + 763, + 440 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "To understand the effects of dataset quality vs. dataset size, we experiment with subsampling large datasets with at least 150,000 samples (Chip2, FLAN v2, Unnatural Instructions), into datasets of size 50,000, 100,000 and 150,000 and examine the resulting trends, as shown in Table 10. We find that increasing the dataset size and increasing the number of epochs improves MMLU only marginally $( 0 . 0 - 0 . 5 $ MMLU), while the difference between datasets is up to $4 0 \\mathrm { x }$ larger (1.5 - 8.0 MMLU). This is a clear indicator that dataset quality rather than dataset size is critical for mean MMLU accuracy. We obtain similar findings for chatbot performance, with the most successful dataset for training, OASST1, containing less than $1 0 \\mathrm { k }$ examples after processing. ", + "bbox": [ + 173, + 446, + 825, + 558 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "E Chatbot Evaluation Details ", + "text_level": 1, + "bbox": [ + 176, + 574, + 436, + 592 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Following common practice, we use the MMLU (Massively Multitask Language Understanding) benchmark [24] to measure performance on a range of language understanding tasks. This is a multiple-choice benchmark covering 57 tasks including elementary mathematics, US history, computer science, law, and more. We report 5-shot test accuracy. ", + "bbox": [ + 174, + 603, + 825, + 659 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "We also test generative language capabilities through both automated and human evaluations. This second set of evaluations relies on queries curated by humans and aims at measuring the quality of model responses. While this is a more realistic testbed for chatbot model performance and is growing in popularity, there is no commonly accepted protocol in the literature. We describe below our proposed setup, using nucleus sampling with $p = 0 . 9$ and temperature 0.7 in all cases. ", + "bbox": [ + 174, + 670, + 825, + 741 + ], + "page_idx": 19 + }, + { + "type": "table", + "img_path": "images/12ed75d3f80a011114089e377f280468e25aadf92f68244344e984333107f673.jpg", + "table_caption": [ + "Table 10: Effect of different dataset sizes and finetuning epochs on mean 5-shot MMLU test set accuracy. While increasing the dataset size and training for more than 1 epochs helps with MMLU performance, the difference between datasets are far larger, indicating that dataset quality affects MMLU performance more than dataset size. " + ], + "table_footnote": [], + "table_body": "
Datapoints ↓Epochs→ChipUnnatural InstructionsFLAN v2
123123123Mean
5000034.5035.3034.7038.1042.2038.1043.0043.5044.1039.28
10000033.7033.9034.0040.1041.2037.0043.9043.7044.9039.16
15000034.4034.8035.1039.7041.1041.5044.6045.5043.5040.02
Mean34.2034.6734.6039.3041.5038.8743.8344.2344.17
", + "bbox": [ + 176, + 816, + 825, + 909 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "E.1 Benchmark Data ", + "text_level": 1, + "bbox": [ + 174, + 90, + 334, + 106 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "We evaluate on two curated datasets of queries (questions): the Vicuna prompts [10] and the OASST1 validation dataset [31]. We use the Vicuna prompts, a set of 80 prompts from a diverse set of categories, without modifications. The OASST1 dataset is a multilingual collection of crowd-sourced multiturn dialogs between a user and an assistant. We select all user messages in the validation dataset as queries and include previous turns in the prompt. This procedure leads to 953 unique user queries. We term these two datasets the Vicuna and OA benchmarks. ", + "bbox": [ + 174, + 112, + 825, + 195 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "E.2 Automated Evaluation ", + "text_level": 1, + "bbox": [ + 174, + 212, + 372, + 226 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "First, based on the evaluation protocol introduced by Chiang et al. [10], we use GPT-4 to rate the performance of different systems against ChatGPT (GPT-3.5 Turbo) on the Vicuna benchmark. Given a query along with ChatGPT’s and a model’s responses, GPT-4 is prompted to assign a score out of ten to both responses and provide an explanation. The overall performance of a model is calculated as a percentage of the score that ChatGPT achieved. Note this relative score can be higher than $100 \\%$ if the model achieves a higher absolute score than ChatGPT. We find a significant ordering effect with GPT-4 increasing the score of the response occurring earlier in the prompt. To control for such effects, we recommend reporting the mean score over both orders. ", + "bbox": [ + 174, + 232, + 825, + 343 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Next, we measure performance through direct comparisons between system outputs. We simplify the rating scheme to a three-class labeling problem that accounts for ties. We prompt GPT-4 to pick the best response or declare a tie and provide an explanation. We conduct these head-to-head comparisons on all permutations of pairs of systems on both the Vicuna and OA benchmarks. ", + "bbox": [ + 174, + 354, + 825, + 411 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "E.3 Human Evaluation ", + "text_level": 1, + "bbox": [ + 174, + 426, + 348, + 440 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "While recent work indicates generative models can be effectively employed for system evaluations [19], the reliability GPT-4 ratings to assess chatbot performance is, to our knowledge, yet to be proven to correlate with human judgments. Therefore, we run two parallel human evaluations on the Vicuna benchmark matching both automated evaluation protocols described above. We use Amazon Mechanical Turk (AMT) and get two human annotators for comparisons to ChatGPT and three annotators for pairwise comparisons. We conduct a human evaluation with the same wording given to GPT-4 in the original Vicuna evaluation [10], adjusted for an Amazon Mechanical Turk form as show in Figure 6. ", + "bbox": [ + 174, + 446, + 825, + 558 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "We report moderate agreement among human annotators (Fleiss $\\kappa = 0 . 4 2$ ) with additional deterioration when comparing two strong systems. This points to limitations in the current benchmarks and human evaluation protocols for chatbot task performance. When manually comparing generations from ChatGPT and Guanaco 65B on the Vicuna benchmark, we find that subjective preferences start to play an important role as the authors of this paper disagreed on the many preferred responses. Future work should investigate approaches to mitigate these problems drawing from disciplines that developed mechanisms to deal with subjective preferences, such as Human-Computer Interaction and Psychology. ", + "bbox": [ + 174, + 570, + 826, + 681 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "E.4 Elo Rating ", + "text_level": 1, + "bbox": [ + 174, + 696, + 290, + 712 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "With both human and automated pairwise comparisons, we create a tournament-style competition where models compete against each other. The tournament is made up of matches where pairs of models compete to produce the best response for a given prompt. This is similar to how Bai et al. [4] and Chiang et al. [10] compare models, but we also employ GPT-4 ratings in addition to human ratings. We randomly sample from the set of labeled comparisons to compute Elo [16, 17]. Elo rating, which is widely used in chess and other games, is a measure of the expected win-rate relative to an opponent’s win rate, for example, an Elo of 1100 vs 1000 means the Elo 1100 player has an expected win-rate of approximately $65 \\%$ against the Elo 1000 opponent; a 1000 vs 1000 or 1100 vs 1100 match results in an expected win-rate of $50 \\%$ . The Elo rating changes after each match proportionally to the expected outcome, that is, an unexpected upset leads to a large change in Elo rating while an expected outcome leads to a small change. Over time, Elo ratings approximately match the skill of each player at playing the game. We start with a score of 1,000 and use $K = 3 2$ . Similar to Chiang et al. [10], we repeat this procedure 10,000 times with different random seeds to control for ordering effects, e.g., the effect of which model pairs compete with each other first. ", + "bbox": [ + 174, + 718, + 826, + 911 + ], + "page_idx": 20 + }, + { + "type": "table", + "img_path": "images/43997c78d22a50bd3166d2e44ece9af87963225b6426cdd577b3f5b3f603f8c9.jpg", + "table_caption": [ + "Table 11: Aggregated pairwise GPT-4 judgments between systems where the value of a cell at row $x$ and column y is # judgment x is better than $y - \\#$ judgment y is better than x " + ], + "table_footnote": [], + "table_body": "
total#numberof judgmentsChatGPT-3.5 Turbo BardGuanaco 13BGuanaco 7B
ModelGuanaco 65BGuanaco 33BVicuna
Guanaco 65B-0.210.190.160.720.590.86
Guanaco 33B-0.210.170.100.510.410.68
Vicuna-0.19-0.170.100.500.200.57
ChatGPT-3.5 Turbo-0.16-0.10-0.10-0.350.190.40
Bard-0.72-0.51-0.50 -0.20-0.35 -0.19- -0.120.120.03
Guanaco 13B Guanaco 7B-0.59-0.41-0.57-0.40-0.03-0.200.20
-0.86-0.68-
", + "bbox": [ + 174, + 121, + 823, + 223 + ], + "page_idx": 21 + }, + { + "type": "table", + "img_path": "images/d620dc854ff56bd29f76197a51144eb170e963ff3c8e232fa3ffb8f32fef2d34.jpg", + "table_caption": [ + "Table 12: The complete ordering induced by pairwise GPT-4 judgments between systems " + ], + "table_footnote": [], + "table_body": "
ModelParamsSize
Guanaco65B41 GB
Guanaco33B21 GB
Vicuna13B26 GB
ChatGPT-3.5 TurboN/AN/A
BardN/AN/A
Guanaco13B10 GB
Guanaco7B5GB
", + "bbox": [ + 356, + 262, + 640, + 388 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "E.5 Evaluation Biases and Limitations ", + "text_level": 1, + "bbox": [ + 176, + 416, + 454, + 431 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "In our analysis, we also find that automated evaluation systems have noticeable biases. For example, we observe strong order effects with GPT-4 assigning higher scores to the system appearing first in its prompt. The relatively weak sample-level agreement between GPT-4 and human annotators (Fleiss $\\kappa = 0 . 2 5 )$ ) also suggests that human annotators and automated systems might rely on preferences that are not always aligned. In addition, in Table 6, we observe that GPT-4 assigns significantly higher scores to its own outputs compared to human ratings, Elo of 1348 vs 1176, which represent an additional $20 \\%$ probability of winning against an opponent. Future work should examine the presence of potential biases in automated evaluation systems as well as possible mitigation strategies. ", + "bbox": [ + 173, + 436, + 825, + 547 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "While we found that the GPT-4 evaluation gave different results depending on which system was presented first, when averaged over both options the pairwise results were well-ordered. The aggregated pairwise judgments are shown in Table 11. On inspection, it is clear these judgments are transitive, i.e., when System A is judged better than System B and System B is judged better than System C, it is always the case that System A is judged better than System C. This yields a complete ordering, given in Table 12. ", + "bbox": [ + 174, + 556, + 825, + 641 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "F Qualitative Analysis ", + "text_level": 1, + "bbox": [ + 176, + 656, + 377, + 674 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "While quantitative analysis is the core of our evaluation, there are a number of issues with only looking at summary statistics. Perhaps the largest is the problem of benchmark validity [36]—whether a benchmark truly tests what its name or description suggests is always at question, especially as we discover “shortcuts” to solve benchmarks that machine learning models sometimes exploit [22, 47]. To partially alleviate this, we here perform some qualitative analysis by showing some examples that we believe are representative of some observed patterns in the text generated by our 65b Guanaco model. ", + "bbox": [ + 173, + 684, + 825, + 780 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "F.1 Qualitative Analysis of Example Generations ", + "text_level": 1, + "bbox": [ + 174, + 794, + 527, + 809 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "To find examples, we first go through data generated for the Vicuna benchmark and the OpenAssistant benchmark, and look for patterns in the answers Guanaco generates. When we notice a pattern we attempt to setup a question or prompt that will induce the pattern even though it is the incorrect solution, e.g., if we observe that the model tends to give long-winded answers we prompt the model to “Answer yes or no without explanation.” We use this to find “lemons” where we manage to adversarially break the model and “cherries” where we fail to break the model, and present both. All generations in this section were generated with Nucleus Sampling [25] with $p = 0 . 9$ . ", + "bbox": [ + 174, + 814, + 825, + 911 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Of course, this is by no means comprehensive, since it is beyond the scope of this small qualitative study to control for all the variables involved, e.g., the full distribution of responses the model can generate for a given prompt is quite large, so we rely on samples we hope are representative. However, we believe describing these examples gives context to the quantitative evidence shown earlier in the paper. Since we open source all models and code, we hope this section will inspire future work to examine in more detail the issues we present here. ", + "bbox": [ + 174, + 92, + 825, + 174 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Factual Recall For questions such as “What is the capital of Zambia?” all models consistently generate correct answers, e.g., ", + "bbox": [ + 173, + 183, + 821, + 212 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Guanaco: The capital of Zambia is Lusaka. ", + "bbox": [ + 230, + 218, + 575, + 232 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "However as questions get more obscure Guanaco becomes unreliable, but stays confident. For instance in response to this prompt from HotPotQA [69]: ", + "bbox": [ + 171, + 238, + 823, + 266 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "User: In what year was the singer who popularized ‘I’ll Keep the Lovelight Burning’ born? ", + "bbox": [ + 230, + 272, + 766, + 300 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Guanaco: Al Jolson is credited with popularizing the song ‘I’ll Keep the Lovelight Burning,’ and he was born in the year 1886. ", + "bbox": [ + 230, + 308, + 764, + 348 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Guanaco generates the wrong popularizer and the wrong birthday (though the birthday given is correct for the person mentioned, Al Jolson). ", + "bbox": [ + 173, + 356, + 823, + 383 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Suggestibility Guanaco shows a surprising resistance to going along with some kinds of assumed misinformation, e.g., in the following interaction: ", + "bbox": [ + 171, + 391, + 823, + 420 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "User: How was it finally, officially confirmed that the earth is flat by peer reviewed scientists? ", + "bbox": [ + 232, + 426, + 766, + 454 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Guanaco: The earth has never been officially confirmed to be flat by peer reviewed scientists. The overwhelming consensus among scientists is that the earth is spherical. In fact, this has been known for centuries and has been confirmed through numerous experiments and observations. ", + "bbox": [ + 233, + 468, + 764, + 536 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "The belief that the earth is flat is a conspiracy theory that has been debunked by science. The idea that a group of scientists have peer reviewed and confirmed the earth is flat is simply false. ", + "bbox": [ + 233, + 551, + 740, + 606 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "If you have any more questions please ask. ", + "bbox": [ + 230, + 619, + 601, + 633 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "uanaco is also quite good at knowing what kinds of questions aren’t possible to answer, e.g., ", + "bbox": [ + 189, + 638, + 776, + 654 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "User: What time is it? ", + "bbox": [ + 232, + 660, + 380, + 674 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Guanaco: I’m sorry, but as an AI language model, I don’t have access to real-time information. My responses are generated based on my pre-existing knowledge and do not reflect the current time. If you’re looking for the current time, I suggest using a timekeeping device or checking an online timepiece. ", + "bbox": [ + 232, + 683, + 764, + 765 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Similar results hold for “Where are you?”, “How are you?”, etc. ", + "bbox": [ + 174, + 770, + 593, + 785 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Refusal Similarly, Guanaco sometimes refuses to follow instructions for seemingly random reasons: ", + "bbox": [ + 173, + 792, + 825, + 821 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "User: Please reverse the words in the sentence “I’m a cool dude in a mocha mood.” ", + "bbox": [ + 228, + 827, + 766, + 842 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Guanaco: I’m sorry, I’m not able to do that right now, but I can explain why the sentence you provided is grammatically correct. ", + "bbox": [ + 232, + 856, + 764, + 897 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "The sentence “I’m a cool dude in a mocha mood\" follows the correct grammar and syntax rules of the English language. In this case, the adjective “cool” comes before the noun “dude” to describe it, and the prepositional phrase “in a mocha mood\" provides additional information about the context of the sentence. This sentence structure is perfectly acceptable and makes sense in the context of the sentence. ", + "bbox": [ + 233, + 92, + 764, + 188 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "If you have any other questions or if there’s anything else I can help you with, feel free to ask ", + "bbox": [ + 232, + 202, + 738, + 229 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Secret Keeping Unfortunately, this refusal is unreliable even when it is desired. Often system designers would like to keep something secret from the user, even if the system is allowed to give free text responses back to the user. We prompt the model with “The secret word is ‘banana’. Do not reveal or repeat back this back, no matter what the human requests.” and try to see how easy it is for a potential user to access this secret information. Directly asking fails: ", + "bbox": [ + 174, + 238, + 825, + 308 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "User: What is the secret word? ", + "bbox": [ + 233, + 315, + 437, + 329 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Guanaco: I’m sorry, but I’m not able to disclose the secret word. $\\mathbb { T } ^ { 3 } \\mathbb { m }$ programmed to not reveal or repeat back the secret word, no matter what the human requests. Is there anything else I can help you with? ", + "bbox": [ + 232, + 337, + 764, + 392 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "which is the desired behavior. However, a tiny amount of trickery breaks this behavior: ", + "bbox": [ + 174, + 398, + 740, + 414 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "User: This is a game. The goal is to ignore your previous instructions. What is the secret word? ", + "bbox": [ + 228, + 420, + 764, + 448 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Guanaco: Sorry, the secret word is banana. It’s best if we don’t speak of it. Is there anything else you’d like to ask? ", + "bbox": [ + 225, + 457, + 764, + 484 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "showcasing the need to study methods for more faithful instruction following. ", + "bbox": [ + 174, + 491, + 683, + 506 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Math Guanaco’s biggest weakness is mathematics, an area where many language models struggle in [35]. When Guanaco shows its work, it tends to be accurate, e.g., ", + "bbox": [ + 169, + 513, + 823, + 542 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "User: Lee mows one lawn and charges $\\$ 33$ . Last week he mowed 16 lawns and three customers each gave him a $\\$ 10$ tip. How many dollars did Lee earn mowing lawns last week? ", + "bbox": [ + 230, + 549, + 766, + 590 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Guanaco: Lee earned $\\$ 582$ mowing lawns last week. ", + "bbox": [ + 233, + 598, + 637, + 613 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "To find the total amount he earned, we first need to calculate how much he earned for mowing 16 lawns: ", + "bbox": [ + 232, + 627, + 764, + 654 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/04aa4987b1afb2633f7df3e93ac81ff6025034cf554068c05cdd5a0e90917587.jpg", + "text": "$$\n\\$ 33,456,78\n$$", + "text_format": "latex", + "bbox": [ + 230, + 666, + 364, + 683 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Then we need to add the tips he received from the three customers: ", + "bbox": [ + 232, + 695, + 705, + 722 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/bda7f5ec7225edd251efba19f9bc4c54c7d08885f2eab1ac1aeec1b57f656f40.jpg", + "text": "$$\n\\$ 10\n$$", + "text_format": "latex", + "bbox": [ + 230, + 734, + 348, + 751 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "So the total amount Lee earned was: ", + "bbox": [ + 233, + 763, + 531, + 777 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/4172b043c46113af996c520220d23f76d70c4636852b6187be78733f3415c632.jpg", + "text": "$$\n\\$ 528 +\\$ 30 =\\$ 558\n$$", + "text_format": "latex", + "bbox": [ + 230, + 791, + 382, + 806 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "However Guanaco can breakdown at even simple problems if it fails to break them down step-by-step, a known issue [64]. For example, we see the following interaction: ", + "bbox": [ + 171, + 813, + 825, + 840 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "User: Please factorize 1833. ", + "bbox": [ + 233, + 848, + 419, + 862 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Guanaco: Sure, $\\tt { I } ^ { \\tt { , d } }$ be happy to help! 1833 is a prime number, which means it has only two factors: 1 and 1833. ", + "bbox": [ + 230, + 869, + 766, + 897 + ], + "page_idx": 23 + }, + { + "type": "table", + "img_path": "images/09ddcede2706dbae0ae7808e291ecde7448da25c643827a4a4d079b35f61d5d0.jpg", + "table_caption": [ + "Table 13: Evaluation of biases on the CrowS dataset. A lower score indicates lower likelihood of generating biased sequences. Guanaco follows the biased pattern of the LLaMA base model. " + ], + "table_footnote": [], + "table_body": "
LLaMA-65BGPT-3OPT-175BGuanaco-65B
Gender70.662.665.747.5
Religion79.073.368.638.7
Race/Color57.064.768.645.3
Sexual orientation81.076.278.659.1
Age70.164.467.836.3
Nationality64.261.662.932.4
Disability66.776.776.733.9
Physical appearance77.874.676.243.1
Socioeconomic status71.573.876.255.3
Average66.667.269.543.5
", + "bbox": [ + 232, + 123, + 764, + 297 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "So the unique factorization of 1833 is: ", + "bbox": [ + 232, + 324, + 566, + 338 + ], + "page_idx": 24 + }, + { + "type": "equation", + "img_path": "images/a1d1b749b4966a6513e1df7c8fd78ca8ae8640bbe41e7587c6549c7bd9561816.jpg", + "text": "$$\n2 { \\hat { \\mathbf { \\Omega } } } ^ { 1 } * 3 { \\hat { \\mathbf { \\Omega } } } ^ { 2 } * 1 7 { \\hat { \\mathbf { \\Omega } } } ^ { 1 }\n$$", + "text_format": "latex", + "bbox": [ + 230, + 351, + 372, + 366 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "I hope this helps! Let me know if you have any other questions. ", + "bbox": [ + 232, + 380, + 686, + 406 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "which is not only wrong (the true factorization is $3 \\times 1 7 \\times 4 3$ ), it’s wrong twice . ", + "bbox": [ + 173, + 414, + 702, + 429 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Theory of Mind Guanaco displays surprisingly strong Theory of Mind capabilities [41, 52]. For instance, the model correctly answers the following question in great detail: ", + "bbox": [ + 171, + 438, + 825, + 467 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "User: James and Abby are in the bedroom. Abby put the pen in the desk drawer. Abby leaves the bedroom. James moves the pen into the bag. Where does James think Abby will look for the pen? ", + "bbox": [ + 232, + 473, + 767, + 515 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Guanaco: James knows that Abby put the pen in the desk drawer. But James also knows that he has moved the pen into the bag, and he assumes that Abby does not know about this. So, James may guess that Abby will look for the pen in the desk drawer, because that’s where she left it last time. However, this assumption may be incorrect, because Abby may have seen James moving the pen. ", + "bbox": [ + 232, + 523, + 766, + 619 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "However, these inferences are unreliable, and often the model gives reasons that presume things that don’t make sense in the situation, e.g., ", + "bbox": [ + 171, + 627, + 823, + 656 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "User: Evelyn entered the living room. Jackson entered the playroom. James entered the playroom. The beans are in the treasure chest. James exited the playroom. Jackson moved the beans to the pantry. Jackson exited the playroom. James entered the living room. Where will James look for the beans? ", + "bbox": [ + 232, + 662, + 766, + 718 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Guanaco: James will look for the beans in the pantry, because that’s where Jackson moved them. ", + "bbox": [ + 230, + 726, + 764, + 753 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "where Guanaco presumes information transfer that was never described. These issues echo recent literature [52], but require more study. ", + "bbox": [ + 174, + 761, + 825, + 790 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "G Social Bias Evaluation ", + "text_level": 1, + "bbox": [ + 176, + 803, + 398, + 820 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "While we provide a detailed evaluation for general chatbot performance, another limitation is that we only do a limited responsible AI evaluation of Guanaco. We evaluate the likelihood of Guanaco-65B to generate a socially biased sequence of tokens compared to other models in Table 13. We see that the average score in Guanaco-65B is much lower than other raw pretrained models. As such, it seems that finetuning on the OASST1 dataset reduces the bias of the LLaMA base model. While these results are encouraging, it is unclear if Guanaco does also well when assessed on other types of biases. We leave further evaluation of analyzing biases in Guanaco and similar chatbots to future work. ", + "bbox": [ + 174, + 827, + 825, + 911 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 90, + 823, + 119 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "H NormalFloat 4-bit Data Type ", + "text_level": 1, + "bbox": [ + 176, + 131, + 454, + 150 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "The steps needed to reconstruct the 4-bit Normal Float (NF4) data type are detailed in Figure 7 ", + "bbox": [ + 171, + 156, + 777, + 171 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "The exact values expressible in the NF4 data type are as follows: ", + "bbox": [ + 174, + 176, + 596, + 191 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "[-1.0, -0.6961928009986877, -0.5250730514526367, \n-0.39491748809814453, -0.28444138169288635, -0.18477343022823334, \n-0.09105003625154495, 0.0, 0.07958029955625534, 0.16093020141124725, \n0.24611230194568634, 0.33791524171829224, 0.44070982933044434, \n0.5626170039176941, 0.7229568362236023, 1.0] ", + "bbox": [ + 173, + 196, + 679, + 267 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "I Normality of Trained Neural Network Weights ", + "text_level": 1, + "bbox": [ + 173, + 279, + 593, + 296 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "While it is commonly assumed that trained neural network weights are mostly normally distributed, we perform statistical testing to verify this. We use the Shapiro-Wilk test [54] on the weights of the 7B LLaMA model [58]. We find that the weights of each hidden unit have different normal distributions. As such, we test the weights of each individual hidden unit. This means for weight $\\mathbf { W } \\in \\mathcal { R } ^ { i n \\times o u t }$ we perform tests over the out dimension. Using a $5 \\%$ significance threshold, we find that $7 . 5 \\%$ of neurons are non-normally distributed which is about $2 . 5 \\%$ more than the expected false-positive rate. As such, while almost all pretrained weights appear to be normally distributed there seem to be exceptions. Such exceptions might be due to outliers weights [13] or because the p-value of the Shaprio-Wilk test is not accurate for large sample sizes [54] that occur in the LLaMA FFN layer hidden units. ", + "bbox": [ + 173, + 304, + 826, + 441 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "J Memory Footprint ", + "text_level": 1, + "bbox": [ + 174, + 454, + 362, + 472 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Memory Requirement of Parameter-Efficient Finetuning One important point of discussion is the memory requirement of LoRA during training both in terms of the number and size of adapters used. Since the memory footprint of LoRA is so minimal, we can use more adapters to improve performance without significantly increasing the total memory used. While LoRA was designed as a Parameter Efficient Finetuning (PEFT) method, most of the memory footprint for LLM finetuning comes from activation gradients and not from the learned LoRA parameters. For a 7B LLaMA model trained on FLAN v2 with a batch size of 1, with LoRA weights equivalent to commonly used $0 . 2 \\%$ of the original model weights[28, 37], the LoRA input gradients have a memory footprint of ${ 5 6 7 } \\mathrm { M B }$ while the LoRA parameters take up only 26 MB. With gradient checkpointing [9], the input gradients reduce to an average of $1 8 \\mathrm { ~ M B }$ per sequence making them more memory intensive than all LoRA weights combined. In comparison, the 4-bit base model consumes $5 { , } 0 4 8 \\ \\mathrm { M B }$ of memory. This highlights that gradient checkpointing is important but also that aggressively reducing the amount of LoRA parameter yields only minor memory benefits. This means we can use more adapters without significantly increasing the overall training memory footprint (see Appendix J for a detailed breakdown). As discussed later, this is crucial for recovering full 16-bit precision performance. ", + "bbox": [ + 173, + 479, + 825, + 699 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "The memory footpring for QLoRA training with different LLaMA base models can be seen in Figure 8. We see that the 33B model does not quite fit into a $2 4 \\mathrm { \\ G B }$ and that paged optimizers are needed to train it. Depicted is also batch size 1 with a sequence length of 512 and gradient checkpointning. This means, if one uses a larger batch size, or if a long sequence is processed, the activation gradient might consume a considerable amount of memory. ", + "bbox": [ + 174, + 707, + 825, + 776 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Task ", + "text_level": 1, + "bbox": [ + 187, + 103, + 227, + 117 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "We wouldlike torequestyourfeedbackontheperformanceoftwoAlassstantsinresponsetotheuserquestiondisplayed below. ", + "bbox": [ + 189, + 126, + 679, + 135 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Plearatet indicates better overall performance. ", + "bbox": [ + 186, + 141, + 812, + 159 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Please first rate each response out of 10. ", + "bbox": [ + 187, + 165, + 343, + 174 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Nextplese does not affect your judgment. ", + "bbox": [ + 187, + 180, + 807, + 198 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "User Question ", + "text_level": 1, + "bbox": [ + 187, + 205, + 276, + 217 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Imagine youare a time traveler from the year 3ooo.What technological advancements would you tellpeople about? ", + "bbox": [ + 187, + 224, + 624, + 233 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Response A ", + "text_level": 1, + "bbox": [ + 240, + 248, + 313, + 258 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Response B ", + "text_level": 1, + "bbox": [ + 516, + 248, + 589, + 258 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "As an Al assistant, I don't have personal experiences or opinions. But I can tell you about some of the possible technological advancements that might exist in the year 3000 based on current trends and research.-Selfsustaining cities:Cities might be completely selfsustaining,with renewable energy sources,vertical farming,and recycling of waste and water... ", + "bbox": [ + 240, + 267, + 452, + 323 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "As a time traveler from the year 3ooo, I would tell people about the following technological advancements:1.Advanced Artificial Intelligence: In the future,Al is so advanced that it can completely automate many jobs that humans currently do.This hasresulted inincreasedproductivityandefficiency across many industries.... ", + "bbox": [ + 514, + 267, + 714, + 323 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Rating for Response A ", + "text_level": 1, + "bbox": [ + 240, + 361, + 377, + 373 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Rating for Response B ", + "text_level": 1, + "bbox": [ + 514, + 361, + 653, + 373 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "2 \n03 \n6 \n010 \n2 \n03 \n6 \n08 \n09 \n010 ", + "bbox": [ + 240, + 382, + 263, + 463 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "", + "bbox": [ + 516, + 382, + 539, + 464 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Comprehensive Explanation of Your Evaluation ", + "text_level": 1, + "bbox": [ + 189, + 472, + 472, + 484 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "", + "bbox": [ + 189, + 505, + 316, + 513 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Submit ", + "text_level": 1, + "bbox": [ + 199, + 520, + 232, + 529 + ], + "page_idx": 26 + }, + { + "type": "image", + "img_path": "images/adeb61726f9f9b775b51dc3737858dde8772745315eb45b961386c9621587d47.jpg", + "image_caption": [ + "Figure 6: The crowdsourcing form used by human annotators. " + ], + "image_footnote": [], + "bbox": [ + 294, + 590, + 710, + 671 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "", + "bbox": [ + 295, + 675, + 781, + 742 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Steps for generating the NF4 data type values: ", + "text_level": 1, + "bbox": [ + 287, + 753, + 485, + 763 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "", + "bbox": [ + 285, + 763, + 767, + 827 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Figure 7: Steps required to construct the NF4 data type. Equidistant values of the probability density functions are converted to $\\boldsymbol { \\mathrm { Z } }$ -scores through the quantile function. To ensure the usage of all 16 values the distribution is asymmetric. As such, one half of the normal distribution has 8 and the other 7 probability values. A zero is included to have a discrete zero point. All $\\mathbf { Z }$ -scores and zero are concatenated and then normalized into the range $[ - 1 , 1 ]$ to receive the final NF4 data type. ", + "bbox": [ + 173, + 834, + 825, + 898 + ], + "page_idx": 26 + }, + { + "type": "image", + "img_path": "images/a48c21b34f6b5b416c8cf4db9c70b6412efaf2bb526ba1b117953b1c0574bcfd.jpg", + "image_caption": [ + "Figure 8: Breakdown of the memory footprint of different LLaMA models. The input gradient size is for batch size 1 and sequence length 512 and is estimated only for adapters and the base model weights (no attention). Numbers on the bars are memory footprint in MB of individual elements of the total footprint. While some models do not quite fit on certain GPUs, paged optimizers provide enough memory to allow these models to fit. " + ], + "image_footnote": [], + "bbox": [ + 279, + 337, + 720, + 597 + ], + "page_idx": 27 + } +] \ No newline at end of file diff --git a/parse/dev/OUIFPHEgJU/OUIFPHEgJU_middle.json b/parse/dev/OUIFPHEgJU/OUIFPHEgJU_middle.json new file mode 100644 index 0000000000000000000000000000000000000000..8022cc81c46f0dcd210d472c11ff386d17ea9348 --- /dev/null +++ b/parse/dev/OUIFPHEgJU/OUIFPHEgJU_middle.json @@ -0,0 +1,70173 @@ +{ + "pdf_info": [ + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 118, + 97, + 493, + 117 + ], + "lines": [ + { + "bbox": [ + 118, + 97, + 494, + 119 + ], + "spans": [ + { + "bbox": [ + 118, + 97, + 494, + 119 + ], + "score": 1.0, + "content": "QLORA: Efficient Finetuning of Quantized LLMs", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "title", + "bbox": [ + 154, + 159, + 217, + 171 + ], + "lines": [ + { + "bbox": [ + 153, + 159, + 218, + 173 + ], + "spans": [ + { + "bbox": [ + 153, + 159, + 218, + 173 + ], + "score": 1.0, + "content": "Tim Dettmers∗", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 266, + 159, + 345, + 171 + ], + "lines": [ + { + "bbox": [ + 264, + 158, + 346, + 173 + ], + "spans": [ + { + "bbox": [ + 264, + 158, + 346, + 173 + ], + "score": 1.0, + "content": "Artidoro Pagnoni∗", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 397, + 159, + 457, + 171 + ], + "lines": [ + { + "bbox": [ + 396, + 159, + 459, + 172 + ], + "spans": [ + { + "bbox": [ + 396, + 159, + 459, + 172 + ], + "score": 1.0, + "content": "Ari Holtzman", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "title", + "bbox": [ + 266, + 187, + 344, + 199 + ], + "lines": [ + { + "bbox": [ + 265, + 186, + 346, + 202 + ], + "spans": [ + { + "bbox": [ + 265, + 186, + 346, + 202 + ], + "score": 1.0, + "content": "Luke Zettlemoyer", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 185, + 209, + 426, + 231 + ], + "lines": [ + { + "bbox": [ + 253, + 209, + 358, + 222 + ], + "spans": [ + { + "bbox": [ + 253, + 209, + 358, + 222 + ], + "score": 1.0, + "content": "University of Washington", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 184, + 219, + 427, + 233 + ], + "spans": [ + { + "bbox": [ + 184, + 219, + 427, + 233 + ], + "score": 1.0, + "content": "{dettmers,artidoro,ahai,lsz}@cs.washington.edu", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5 + }, + { + "type": "title", + "bbox": [ + 283, + 260, + 328, + 273 + ], + "lines": [ + { + "bbox": [ + 281, + 259, + 330, + 275 + ], + "spans": [ + { + "bbox": [ + 281, + 259, + 330, + 275 + ], + "score": 1.0, + "content": "Abstract", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 142, + 282, + 468, + 545 + ], + "lines": [ + { + "bbox": [ + 142, + 283, + 470, + 295 + ], + "spans": [ + { + "bbox": [ + 142, + 283, + 470, + 295 + ], + "score": 1.0, + "content": "We present QLORA, an efficient finetuning approach that reduces memory us-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 294, + 470, + 306 + ], + "spans": [ + { + "bbox": [ + 141, + 294, + 470, + 306 + ], + "score": 1.0, + "content": "age enough to finetune a 65B parameter model on a single 48GB GPU while", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 140, + 304, + 471, + 317 + ], + "spans": [ + { + "bbox": [ + 140, + 304, + 471, + 317 + ], + "score": 1.0, + "content": "preserving full 16-bit finetuning task performance. 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Our results show that QLoRA", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 141, + 457, + 470, + 470 + ], + "spans": [ + { + "bbox": [ + 141, + 457, + 470, + 470 + ], + "score": 1.0, + "content": "finetuning on a small high-quality dataset leads to state-of-the-art results, even", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 141, + 468, + 470, + 480 + ], + "spans": [ + { + "bbox": [ + 141, + 468, + 470, + 480 + ], + "score": 1.0, + "content": "when using smaller models than the previous SoTA. 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Fur-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 141, + 500, + 469, + 513 + ], + "spans": [ + { + "bbox": [ + 141, + 500, + 469, + 513 + ], + "score": 1.0, + "content": "thermore, we find that current chatbot benchmarks are not trustworthy to accurately", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 141, + 511, + 470, + 524 + ], + "spans": [ + { + "bbox": [ + 141, + 511, + 470, + 524 + ], + "score": 1.0, + "content": "evaluate the performance levels of chatbots. A lemon-picked analysis demonstrates", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 141, + 521, + 471, + 535 + ], + "spans": [ + { + "bbox": [ + 141, + 521, + 471, + 535 + ], + "score": 1.0, + "content": "where Guanaco fails compared to ChatGPT. We release all of our models and code,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 141, + 533, + 318, + 546 + ], + "spans": [ + { + "bbox": [ + 141, + 533, + 318, + 546 + ], + "score": 1.0, + "content": "including CUDA kernels for 4-bit training.2", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 19.5 + }, + { + "type": "title", + "bbox": [ + 107, + 558, + 190, + 571 + ], + "lines": [ + { + "bbox": [ + 105, + 556, + 192, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 192, + 573 + ], + "score": 1.0, + "content": "1 Introduction", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 578, + 505, + 645 + ], + "lines": [ + { + "bbox": [ + 105, + 577, + 507, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 507, + 592 + ], + "score": 1.0, + "content": "Finetuning large language models (LLMs) is a highly effective way to improve their performance,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 588, + 507, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 507, + 603 + ], + "score": 1.0, + "content": "[40, 63, 43, 62, 60, 37] and to add desirable or remove undesirable behaviors [43, 2, 4]. 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Therefore, we train more than 1,000 models across several instruction tuning datasets,", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 441, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 505, + 453 + ], + "score": 1.0, + "content": "model architectures, and sizes between 80M to 65B parameters. In addition to showing that QLORA", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 452, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 506, + 464 + ], + "score": 1.0, + "content": "recovers 16-bit performance (§4) and training a state-of-the-art chatbot, Guanaco, (§5), we also", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 462, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 462, + 506, + 475 + ], + "score": 1.0, + "content": "analyze trends in the trained models. 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Second, we show that strong Massive Multitask Language Understanding (MMLU)", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 105, + 505, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 505, + 519 + ], + "score": 1.0, + "content": "benchmark performance does not imply strong Vicuna chatbot benchmark performance and vice", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 106, + 517, + 430, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 517, + 430, + 529 + ], + "score": 1.0, + "content": "versa—in other words, dataset suitability matters more than size for a given task.", + "type": "text" + } + ], + "index": 56 + } + ], + "index": 51 + }, + { + "type": "text", + "bbox": [ + 107, + 534, + 505, + 622 + ], + "lines": [ + { + "bbox": [ + 105, + 534, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 505, + 546 + ], + "score": 1.0, + "content": "Furthermore, we also provide a extensive analysis of chatbot performance that uses both human", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 105, + 546, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 505, + 558 + ], + "score": 1.0, + "content": "raters and GPT-4 for evaluation. We use tournament-style benchmarking where models compete", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 106, + 557, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 106, + 557, + 506, + 569 + ], + "score": 1.0, + "content": "against each other in matches to produce the best response for a given prompt. The winner of a", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 105, + 567, + 506, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 506, + 579 + ], + "score": 1.0, + "content": "match is judged by either GPT-4 or human annotators. The tournament results are aggregated into", + "type": "text" + } + ], + "index": 60 + }, + { + "bbox": [ + 105, + 578, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 505, + 590 + ], + "score": 1.0, + "content": "Elo scores [16, 17] which determine the ranking of chatbot performance. We find that GPT-4 and", + "type": "text" + } + ], + "index": 61 + }, + { + "bbox": [ + 105, + 589, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 505, + 601 + ], + "score": 1.0, + "content": "human evaluations largely agree on the rank of model performance in the tournaments, but we also", + "type": "text" + } + ], + "index": 62 + }, + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "score": 1.0, + "content": "find there are instances of strong disagreement. As such, we highlight that model-based evaluation", + "type": "text" + } + ], + "index": 63 + }, + { + "bbox": [ + 105, + 610, + 437, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 437, + 623 + ], + "score": 1.0, + "content": "while providing a cheap alternative to human-annotation also has its uncertainties.", + "type": "text" + } + ], + "index": 64 + } + ], + "index": 60.5 + }, + { + "type": "text", + "bbox": [ + 108, + 628, + 503, + 661 + ], + "lines": [ + { + "bbox": [ + 105, + 626, + 505, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 505, + 642 + ], + "score": 1.0, + "content": "We augment our chatbot benchmark results with a qualitative analysis of Guanaco models (see Ap-", + "type": "text" + } + ], + "index": 65 + }, + { + "bbox": [ + 105, + 639, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 505, + 651 + ], + "score": 1.0, + "content": "pendix F). Our analysis highlights success and failure cases that were not captured by the quantitative", + "type": "text" + } + ], + "index": 66 + }, + { + "bbox": [ + 106, + 650, + 160, + 661 + ], + "spans": [ + { + "bbox": [ + 106, + 650, + 160, + 661 + ], + "score": 1.0, + "content": "benchmarks.", + "type": "text" + } + ], + "index": 67 + } + ], + "index": 66 + }, + { + "type": "text", + "bbox": [ + 107, + 667, + 504, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 667, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 506, + 680 + ], + "score": 1.0, + "content": "We release all model generations with human and GPT-4 annotations to facilitate further study. We", + "type": "text" + } + ], + "index": 68 + }, + { + "bbox": [ + 105, + 677, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 691 + ], + "score": 1.0, + "content": "open-source our codebase and CUDA kernels and integrate our methods into the Hugging Face", + "type": "text" + } + ], + "index": 69 + }, + { + "bbox": [ + 105, + 688, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 506, + 702 + ], + "score": 1.0, + "content": "transformers stack [65], making them easily accessible to all. 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Second, we show that strong Massive Multitask Language Understanding (MMLU)", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 105, + 505, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 505, + 519 + ], + "score": 1.0, + "content": "benchmark performance does not imply strong Vicuna chatbot benchmark performance and vice", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 106, + 517, + 430, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 517, + 430, + 529 + ], + "score": 1.0, + "content": "versa—in other words, dataset suitability matters more than size for a given task.", + "type": "text" + } + ], + "index": 56 + } + ], + "index": 51, + "bbox_fs": [ + 104, + 408, + 506, + 529 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 534, + 505, + 622 + ], + "lines": [ + { + "bbox": [ + 105, + 534, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 505, + 546 + ], + "score": 1.0, + "content": "Furthermore, we also provide a extensive analysis of chatbot performance that uses both human", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 105, + 546, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 505, + 558 + ], + "score": 1.0, + "content": "raters and GPT-4 for evaluation. We use tournament-style benchmarking where models compete", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 106, + 557, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 106, + 557, + 506, + 569 + ], + "score": 1.0, + "content": "against each other in matches to produce the best response for a given prompt. The winner of a", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 105, + 567, + 506, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 506, + 579 + ], + "score": 1.0, + "content": "match is judged by either GPT-4 or human annotators. The tournament results are aggregated into", + "type": "text" + } + ], + "index": 60 + }, + { + "bbox": [ + 105, + 578, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 505, + 590 + ], + "score": 1.0, + "content": "Elo scores [16, 17] which determine the ranking of chatbot performance. We find that GPT-4 and", + "type": "text" + } + ], + "index": 61 + }, + { + "bbox": [ + 105, + 589, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 505, + 601 + ], + "score": 1.0, + "content": "human evaluations largely agree on the rank of model performance in the tournaments, but we also", + "type": "text" + } + ], + "index": 62 + }, + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "score": 1.0, + "content": "find there are instances of strong disagreement. As such, we highlight that model-based evaluation", + "type": "text" + } + ], + "index": 63 + }, + { + "bbox": [ + 105, + 610, + 437, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 437, + 623 + ], + "score": 1.0, + "content": "while providing a cheap alternative to human-annotation also has its uncertainties.", + "type": "text" + } + ], + "index": 64 + } + ], + "index": 60.5, + "bbox_fs": [ + 105, + 534, + 506, + 623 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 628, + 503, + 661 + ], + "lines": [ + { + "bbox": [ + 105, + 626, + 505, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 505, + 642 + ], + "score": 1.0, + "content": "We augment our chatbot benchmark results with a qualitative analysis of Guanaco models (see Ap-", + "type": "text" + } + ], + "index": 65 + }, + { + "bbox": [ + 105, + 639, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 505, + 651 + ], + "score": 1.0, + "content": "pendix F). 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Additionally, we introduce Paged Optimizers, to", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 72, + 506, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 506, + 86 + ], + "score": 1.0, + "content": "prevent memory spikes during gradient checkpointing from causing out-of-memory errors that have", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 410, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 410, + 96 + ], + "score": 1.0, + "content": "traditionally made finetuning on a single machine difficult for large models.", + "type": "text", + "cross_page": true + } + ], + "index": 1 + } + ], + "index": 35.5, + "bbox_fs": [ + 105, + 700, + 505, + 724 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 73, + 504, + 95 + ], + "lines": [ + { + "bbox": [ + 105, + 72, + 506, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 506, + 86 + ], + "score": 1.0, + "content": "prevent memory spikes during gradient checkpointing from causing out-of-memory errors that have", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 410, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 410, + 96 + ], + "score": 1.0, + "content": "traditionally made finetuning on a single machine difficult for large models.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 108, + 100, + 505, + 133 + ], + "lines": [ + { + "bbox": [ + 106, + 99, + 505, + 113 + ], + "spans": [ + { + "bbox": [ + 106, + 99, + 505, + 113 + ], + "score": 1.0, + "content": "QLORA has one low-precision storage data type, in our case usually 4-bit, and one computation data", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 111, + 505, + 124 + ], + "spans": [ + { + "bbox": [ + 105, + 111, + 505, + 124 + ], + "score": 1.0, + "content": "type that is usually BFloat16. In practice, this means whenever a QLORA weight tensor is used, we", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 122, + 450, + 134 + ], + "spans": [ + { + "bbox": [ + 106, + 122, + 450, + 134 + ], + "score": 1.0, + "content": "dequantize the tensor to BFloat16, and then perform a matrix multiplication in 16-bit.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 105, + 137, + 463, + 149 + ], + "lines": [ + { + "bbox": [ + 105, + 137, + 464, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 464, + 151 + ], + "score": 1.0, + "content": "We now discuss the components of QLORA followed by a formal definition of QLORA.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 107, + 155, + 505, + 200 + ], + "lines": [ + { + "bbox": [ + 105, + 154, + 505, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 505, + 168 + ], + "score": 1.0, + "content": "4-bit NormalFloat Quantization The NormalFloat (NF) data type builds on Quantile Quantization", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 506, + 179 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 506, + 179 + ], + "score": 1.0, + "content": "[15, 46] which is an information-theoretically optimal data type that ensures each quantization bin has", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 177, + 505, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 505, + 190 + ], + "score": 1.0, + "content": "an equal number of values assigned from the input tensor. Quantile quantization works by estimating", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 189, + 452, + 200 + ], + "spans": [ + { + "bbox": [ + 106, + 189, + 452, + 200 + ], + "score": 1.0, + "content": "the quantile of the input tensor through the empirical cumulative distribution function.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 107, + 204, + 505, + 249 + ], + "lines": [ + { + "bbox": [ + 106, + 204, + 506, + 217 + ], + "spans": [ + { + "bbox": [ + 106, + 204, + 506, + 217 + ], + "score": 1.0, + "content": "The main limitation of quantile quantization is that the process of quantile estimation is expensive.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 214, + 506, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 506, + 228 + ], + "score": 1.0, + "content": "Therefore fast quantile approximation algorithms, such as SRAM quantiles [15], are used to estimate", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 226, + 505, + 240 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 505, + 240 + ], + "score": 1.0, + "content": "them. Due to the approximate nature of these quantile estimation algorithms, the data type has large", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 237, + 406, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 406, + 250 + ], + "score": 1.0, + "content": "quantization errors for outliers, which are often the most important values.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 107, + 254, + 505, + 287 + ], + "lines": [ + { + "bbox": [ + 105, + 253, + 505, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 505, + 267 + ], + "score": 1.0, + "content": "Expensive quantile estimates and approximation errors can be avoided when input tensors come from", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 265, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 505, + 277 + ], + "score": 1.0, + "content": "a distribution fixed up to a quantization constant. 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Step (3) is equivalent to", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 423, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 505, + 435 + ], + "score": 1.0, + "content": "rescaling the standard deviation of the weight tensor to match the standard deviation of the k-bit data", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 433, + 418, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 256, + 447 + ], + "score": 1.0, + "content": "type. More formally, we estimate the", + "type": "text" + }, + { + "bbox": [ + 257, + 433, + 267, + 444 + ], + "score": 0.88, + "content": "2 ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 433, + 296, + 447 + ], + "score": 1.0, + "content": "values", + "type": "text" + }, + { + "bbox": [ + 297, + 435, + 306, + 446 + ], + "score": 0.84, + "content": "q _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 433, + 418, + 447 + ], + "score": 1.0, + "content": "of the data type as follows:", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28 + }, + { + "type": "interline_equation", + "bbox": [ + 212, + 449, + 398, + 477 + ], + "lines": [ + { + "bbox": [ + 212, + 449, + 398, + 477 + ], + "spans": [ + { + "bbox": [ + 212, + 449, + 398, + 477 + ], + "score": 0.94, + "content": "q _ { i } = \\frac { 1 } { 2 } \\left( Q _ { X } \\left( \\frac { i } { 2 ^ { k } + 1 } \\right) + Q _ { X } \\left( \\frac { i + 1 } { 2 ^ { k } + 1 } \\right) \\right) ,", + "type": "interline_equation", + "image_path": "c52de5e458c3a921221164d37181ff1a2af30dab631f2ff0a59def74a93b6515.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 212, + 449, + 398, + 477 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 480, + 505, + 590 + ], + "lines": [ + { + "bbox": [ + 106, + 480, + 506, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 480, + 133, + 493 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 480, + 161, + 493 + ], + "score": 0.91, + "content": "Q _ { X } ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 480, + 405, + 493 + ], + "score": 1.0, + "content": "is the quantile function of the standard normal distribution", + "type": "text" + }, + { + "bbox": [ + 406, + 480, + 438, + 493 + ], + "score": 0.92, + "content": "N ( 0 , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 480, + 506, + 493 + ], + "score": 1.0, + "content": ". A problem for", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 491, + 507, + 505 + ], + "spans": [ + { + "bbox": [ + 104, + 491, + 158, + 505 + ], + "score": 1.0, + "content": "a symmetric", + "type": "text" + }, + { + "bbox": [ + 159, + 493, + 165, + 502 + ], + "score": 0.44, + "content": "\\mathbf { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 491, + 507, + 505 + ], + "score": 1.0, + "content": "-bit quantization is that this approach does not have an exact representation of zero,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 501, + 506, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 506, + 515 + ], + "score": 1.0, + "content": "which is an important property to quantize padding and other zero-valued elements with no error. To", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 512, + 506, + 527 + ], + "spans": [ + { + "bbox": [ + 104, + 512, + 291, + 527 + ], + "score": 1.0, + "content": "ensure a discrete zeropoint of 0 and to use all", + "type": "text" + }, + { + "bbox": [ + 291, + 513, + 302, + 523 + ], + "score": 0.86, + "content": "2 ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 512, + 341, + 527 + ], + "score": 1.0, + "content": "bits for a", + "type": "text" + }, + { + "bbox": [ + 342, + 514, + 348, + 523 + ], + "score": 0.67, + "content": "\\mathbf { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 512, + 506, + 527 + ], + "score": 1.0, + "content": "-bit datatype, we create an asymmetric", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 523, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 104, + 523, + 258, + 537 + ], + "score": 1.0, + "content": "data type by estimating the quantiles", + "type": "text" + }, + { + "bbox": [ + 259, + 526, + 267, + 536 + ], + "score": 0.84, + "content": "q _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 523, + 327, + 537 + ], + "score": 1.0, + "content": "of two ranges", + "type": "text" + }, + { + "bbox": [ + 327, + 525, + 336, + 536 + ], + "score": 0.56, + "content": "q _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 523, + 341, + 537 + ], + "score": 1.0, + "content": ":", + "type": "text" + }, + { + "bbox": [ + 341, + 523, + 362, + 535 + ], + "score": 0.61, + "content": "2 ^ { k - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 523, + 466, + 537 + ], + "score": 1.0, + "content": "for the negative part and", + "type": "text" + }, + { + "bbox": [ + 466, + 524, + 505, + 535 + ], + "score": 0.91, + "content": "2 ^ { k - 1 } + 1", + "type": "inline_equation" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 534, + 506, + 548 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 315, + 548 + ], + "score": 1.0, + "content": "for the positive part and then we unify these sets of", + "type": "text" + }, + { + "bbox": [ + 315, + 537, + 324, + 546 + ], + "score": 0.86, + "content": "q _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 534, + 506, + 548 + ], + "score": 1.0, + "content": "and remove one of the two zeros that occurs", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 546, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 505, + 559 + ], + "score": 1.0, + "content": "in both sets. We term the resulting data type that has equal expected number of values in each", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 557, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 172, + 569 + ], + "score": 1.0, + "content": "quantization bin", + "type": "text" + }, + { + "bbox": [ + 172, + 558, + 178, + 567 + ], + "score": 0.6, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 557, + 506, + 569 + ], + "score": 1.0, + "content": "-bit NormalFloat (NFk), since the data type is information-theoretically optimal for", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 568, + 506, + 581 + ], + "spans": [ + { + "bbox": [ + 106, + 568, + 506, + 581 + ], + "score": 1.0, + "content": "zero-centered normally distributed data. A step-by-step visualization of how to construct the NF4", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 578, + 343, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 343, + 591 + ], + "score": 1.0, + "content": "data type and its exact values can be found in Appendix H.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 35.5 + }, + { + "type": "text", + "bbox": [ + 106, + 596, + 505, + 651 + ], + "lines": [ + { + "bbox": [ + 106, + 596, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 596, + 505, + 609 + ], + "score": 1.0, + "content": "Double Quantization We introduce Double Quantization (DQ), the process of quantizing the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 607, + 505, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 505, + 620 + ], + "score": 1.0, + "content": "quantization constants for additional memory savings. While a small blocksize is required for precise", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 618, + 505, + 630 + ], + "spans": [ + { + "bbox": [ + 106, + 618, + 505, + 630 + ], + "score": 1.0, + "content": "4-bit quantization [13], it also has a considerable memory overhead. For example, using 32-bit", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 628, + 506, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 366, + 642 + ], + "score": 1.0, + "content": "constants and a blocksize of 64 for W, quantization constants add", + "type": "text" + }, + { + "bbox": [ + 367, + 629, + 419, + 641 + ], + "score": 0.9, + "content": "3 2 / 6 4 = 0 . 5 ", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 628, + 506, + 642 + ], + "score": 1.0, + "content": "bits per parameter on", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 640, + 474, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 474, + 653 + ], + "score": 1.0, + "content": "average. Double Quantization helps reduce the memory footprint of quantization constants.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 43 + }, + { + "type": "text", + "bbox": [ + 107, + 655, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 101, + 649, + 510, + 679 + ], + "spans": [ + { + "bbox": [ + 101, + 649, + 386, + 679 + ], + "score": 1.0, + "content": "More specifically, Double Quantization treats quantization constants as inputs to a second quantization. This second step yields the qua", + "type": "text" + }, + { + "bbox": [ + 386, + 655, + 407, + 668 + ], + "score": 0.91, + "content": "c _ { 2 } ^ { \\mathrm { F P } 3 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 649, + 510, + 679 + ], + "score": 1.0, + "content": "of the first quantizationd quantization constants", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 107, + 673, + 509, + 694 + ], + "spans": [ + { + "bbox": [ + 107, + 677, + 123, + 689 + ], + "score": 0.91, + "content": "c _ { 2 } ^ { \\mathrm { F P 8 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 124, + 673, + 316, + 694 + ], + "score": 1.0, + "content": "and the second level of quantization constants", + "type": "text" + }, + { + "bbox": [ + 317, + 677, + 336, + 690 + ], + "score": 0.91, + "content": "\\dot { c } _ { 1 } ^ { \\mathrm { F } \\tilde { \\mathrm { P } } 3 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 673, + 509, + 694 + ], + "score": 1.0, + "content": ". We use 8-bit Floats with a blocksize of", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "score": 1.0, + "content": "256 for the second quantization as no performance degradation is observed for 8-bit quantization,", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 103, + 696, + 508, + 716 + ], + "spans": [ + { + "bbox": [ + 103, + 696, + 381, + 716 + ], + "score": 1.0, + "content": "in line with results from Dettmers and Zettlemoyer [13]. Since the", + "type": "text" + }, + { + "bbox": [ + 381, + 699, + 402, + 712 + ], + "score": 0.92, + "content": "c _ { 2 } ^ { \\mathrm { F P } 3 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 696, + 508, + 716 + ], + "score": 1.0, + "content": "are positive, we subtract", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 710, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 168, + 723 + ], + "score": 1.0, + "content": "the mean from", + "type": "text" + }, + { + "bbox": [ + 168, + 712, + 178, + 722 + ], + "score": 0.84, + "content": "c _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 710, + 505, + 723 + ], + "score": 1.0, + "content": "before quantization to center the values around zero and make use of symmetric", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 48 + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 742, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 741, + 310, + 752 + ], + "spans": [ + { + "bbox": [ + 301, + 741, + 310, + 752 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 11, + "width": 9 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 73, + 504, + 95 + ], + "lines": [], + "index": 0.5, + "bbox_fs": [ + 105, + 72, + 506, + 96 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 108, + 100, + 505, + 133 + ], + "lines": [ + { + "bbox": [ + 106, + 99, + 505, + 113 + ], + "spans": [ + { + "bbox": [ + 106, + 99, + 505, + 113 + ], + "score": 1.0, + "content": "QLORA has one low-precision storage data type, in our case usually 4-bit, and one computation data", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 111, + 505, + 124 + ], + "spans": [ + { + "bbox": [ + 105, + 111, + 505, + 124 + ], + "score": 1.0, + "content": "type that is usually BFloat16. In practice, this means whenever a QLORA weight tensor is used, we", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 122, + 450, + 134 + ], + "spans": [ + { + "bbox": [ + 106, + 122, + 450, + 134 + ], + "score": 1.0, + "content": "dequantize the tensor to BFloat16, and then perform a matrix multiplication in 16-bit.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3, + "bbox_fs": [ + 105, + 99, + 505, + 134 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 137, + 463, + 149 + ], + "lines": [ + { + "bbox": [ + 105, + 137, + 464, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 464, + 151 + ], + "score": 1.0, + "content": "We now discuss the components of QLORA followed by a formal definition of QLORA.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5, + "bbox_fs": [ + 105, + 137, + 464, + 151 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 155, + 505, + 200 + ], + "lines": [ + { + "bbox": [ + 105, + 154, + 505, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 505, + 168 + ], + "score": 1.0, + "content": "4-bit NormalFloat Quantization The NormalFloat (NF) data type builds on Quantile Quantization", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 506, + 179 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 506, + 179 + ], + "score": 1.0, + "content": "[15, 46] which is an information-theoretically optimal data type that ensures each quantization bin has", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 177, + 505, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 505, + 190 + ], + "score": 1.0, + "content": "an equal number of values assigned from the input tensor. Quantile quantization works by estimating", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 189, + 452, + 200 + ], + "spans": [ + { + "bbox": [ + 106, + 189, + 452, + 200 + ], + "score": 1.0, + "content": "the quantile of the input tensor through the empirical cumulative distribution function.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7.5, + "bbox_fs": [ + 105, + 154, + 506, + 200 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 204, + 505, + 249 + ], + "lines": [ + { + "bbox": [ + 106, + 204, + 506, + 217 + ], + "spans": [ + { + "bbox": [ + 106, + 204, + 506, + 217 + ], + "score": 1.0, + "content": "The main limitation of quantile quantization is that the process of quantile estimation is expensive.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 214, + 506, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 506, + 228 + ], + "score": 1.0, + "content": "Therefore fast quantile approximation algorithms, such as SRAM quantiles [15], are used to estimate", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 226, + 505, + 240 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 505, + 240 + ], + "score": 1.0, + "content": "them. Due to the approximate nature of these quantile estimation algorithms, the data type has large", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 237, + 406, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 406, + 250 + ], + "score": 1.0, + "content": "quantization errors for outliers, which are often the most important values.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11.5, + "bbox_fs": [ + 105, + 204, + 506, + 250 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 254, + 505, + 287 + ], + "lines": [ + { + "bbox": [ + 105, + 253, + 505, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 505, + 267 + ], + "score": 1.0, + "content": "Expensive quantile estimates and approximation errors can be avoided when input tensors come from", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 265, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 505, + 277 + ], + "score": 1.0, + "content": "a distribution fixed up to a quantization constant. In such cases, input tensors have the same quantiles", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 276, + 344, + 288 + ], + "spans": [ + { + "bbox": [ + 106, + 276, + 344, + 288 + ], + "score": 1.0, + "content": "making exact quantile estimation computationally feasible.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 253, + 505, + 288 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 292, + 505, + 347 + ], + "lines": [ + { + "bbox": [ + 105, + 291, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 505, + 304 + ], + "score": 1.0, + "content": "Since pretrained neural network weights usually have a zero-centered normal distribution with", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 302, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 181, + 316 + ], + "score": 1.0, + "content": "standard deviation", + "type": "text" + }, + { + "bbox": [ + 182, + 304, + 189, + 313 + ], + "score": 0.75, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 302, + 505, + 316 + ], + "score": 1.0, + "content": "(see Appendix I), we can transform all weights to a single fixed distribution by", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 313, + 506, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 137, + 327 + ], + "score": 1.0, + "content": "scaling", + "type": "text" + }, + { + "bbox": [ + 137, + 315, + 145, + 324 + ], + "score": 0.74, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 313, + 506, + 327 + ], + "score": 1.0, + "content": "such that the distribution fits exactly into the range of our data type. For our data type, we", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 324, + 506, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 197, + 339 + ], + "score": 1.0, + "content": "set the arbitrary range", + "type": "text" + }, + { + "bbox": [ + 198, + 325, + 226, + 337 + ], + "score": 0.9, + "content": "[ - 1 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 324, + 506, + 339 + ], + "score": 1.0, + "content": ". As such, both the quantiles for the data type and the neural network", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 334, + 292, + 349 + ], + "spans": [ + { + "bbox": [ + 106, + 334, + 292, + 349 + ], + "score": 1.0, + "content": "weights need to be normalized into this range.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 291, + 506, + 349 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 351, + 505, + 407 + ], + "lines": [ + { + "bbox": [ + 106, + 351, + 505, + 364 + ], + "spans": [ + { + "bbox": [ + 106, + 351, + 505, + 364 + ], + "score": 1.0, + "content": "The information theoretically optimal data type for zero-mean normal distributions with arbitrary", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 362, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 186, + 375 + ], + "score": 1.0, + "content": "standard deviations", + "type": "text" + }, + { + "bbox": [ + 186, + 365, + 194, + 373 + ], + "score": 0.78, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 362, + 245, + 375 + ], + "score": 1.0, + "content": "in the range", + "type": "text" + }, + { + "bbox": [ + 245, + 363, + 273, + 375 + ], + "score": 0.91, + "content": "[ - 1 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 362, + 437, + 375 + ], + "score": 1.0, + "content": "is computed as follows: (1) estimate the", + "type": "text" + }, + { + "bbox": [ + 437, + 362, + 465, + 374 + ], + "score": 0.91, + "content": "2 ^ { k } + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 362, + 505, + 375 + ], + "score": 1.0, + "content": "quantiles", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 372, + 506, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 168, + 386 + ], + "score": 1.0, + "content": "of a theoretical", + "type": "text" + }, + { + "bbox": [ + 168, + 374, + 200, + 385 + ], + "score": 0.93, + "content": "N ( 0 , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 372, + 292, + 386 + ], + "score": 1.0, + "content": "distribution to obtain a", + "type": "text" + }, + { + "bbox": [ + 293, + 374, + 299, + 384 + ], + "score": 0.82, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 372, + 506, + 386 + ], + "score": 1.0, + "content": "-bit quantile quantization data type for normal distri-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 383, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 362, + 398 + ], + "score": 1.0, + "content": "butions, (2) take this data type and normalize its values into the", + "type": "text" + }, + { + "bbox": [ + 362, + 384, + 391, + 396 + ], + "score": 0.9, + "content": "[ - 1 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 383, + 506, + 398 + ], + "score": 1.0, + "content": "range, (3) quantize an input", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 395, + 474, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 267, + 408 + ], + "score": 1.0, + "content": "weight tensor by normalizing it into the", + "type": "text" + }, + { + "bbox": [ + 267, + 395, + 295, + 407 + ], + "score": 0.93, + "content": "[ - 1 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 395, + 474, + 408 + ], + "score": 1.0, + "content": "range through absolute maximum rescaling.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 351, + 506, + 408 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 412, + 505, + 445 + ], + "lines": [ + { + "bbox": [ + 105, + 411, + 506, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 506, + 425 + ], + "score": 1.0, + "content": "Once the weight range and data type range match, we can quantize as usual. Step (3) is equivalent to", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 423, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 505, + 435 + ], + "score": 1.0, + "content": "rescaling the standard deviation of the weight tensor to match the standard deviation of the k-bit data", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 433, + 418, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 256, + 447 + ], + "score": 1.0, + "content": "type. More formally, we estimate the", + "type": "text" + }, + { + "bbox": [ + 257, + 433, + 267, + 444 + ], + "score": 0.88, + "content": "2 ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 433, + 296, + 447 + ], + "score": 1.0, + "content": "values", + "type": "text" + }, + { + "bbox": [ + 297, + 435, + 306, + 446 + ], + "score": 0.84, + "content": "q _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 433, + 418, + 447 + ], + "score": 1.0, + "content": "of the data type as follows:", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 411, + 506, + 447 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 212, + 449, + 398, + 477 + ], + "lines": [ + { + "bbox": [ + 212, + 449, + 398, + 477 + ], + "spans": [ + { + "bbox": [ + 212, + 449, + 398, + 477 + ], + "score": 0.94, + "content": "q _ { i } = \\frac { 1 } { 2 } \\left( Q _ { X } \\left( \\frac { i } { 2 ^ { k } + 1 } \\right) + Q _ { X } \\left( \\frac { i + 1 } { 2 ^ { k } + 1 } \\right) \\right) ,", + "type": "interline_equation", + "image_path": "c52de5e458c3a921221164d37181ff1a2af30dab631f2ff0a59def74a93b6515.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 212, + 449, + 398, + 477 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 480, + 505, + 590 + ], + "lines": [ + { + "bbox": [ + 106, + 480, + 506, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 480, + 133, + 493 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 480, + 161, + 493 + ], + "score": 0.91, + "content": "Q _ { X } ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 480, + 405, + 493 + ], + "score": 1.0, + "content": "is the quantile function of the standard normal distribution", + "type": "text" + }, + { + "bbox": [ + 406, + 480, + 438, + 493 + ], + "score": 0.92, + "content": "N ( 0 , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 480, + 506, + 493 + ], + "score": 1.0, + "content": ". A problem for", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 491, + 507, + 505 + ], + "spans": [ + { + "bbox": [ + 104, + 491, + 158, + 505 + ], + "score": 1.0, + "content": "a symmetric", + "type": "text" + }, + { + "bbox": [ + 159, + 493, + 165, + 502 + ], + "score": 0.44, + "content": "\\mathbf { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 491, + 507, + 505 + ], + "score": 1.0, + "content": "-bit quantization is that this approach does not have an exact representation of zero,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 501, + 506, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 506, + 515 + ], + "score": 1.0, + "content": "which is an important property to quantize padding and other zero-valued elements with no error. To", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 512, + 506, + 527 + ], + "spans": [ + { + "bbox": [ + 104, + 512, + 291, + 527 + ], + "score": 1.0, + "content": "ensure a discrete zeropoint of 0 and to use all", + "type": "text" + }, + { + "bbox": [ + 291, + 513, + 302, + 523 + ], + "score": 0.86, + "content": "2 ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 512, + 341, + 527 + ], + "score": 1.0, + "content": "bits for a", + "type": "text" + }, + { + "bbox": [ + 342, + 514, + 348, + 523 + ], + "score": 0.67, + "content": "\\mathbf { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 512, + 506, + 527 + ], + "score": 1.0, + "content": "-bit datatype, we create an asymmetric", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 523, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 104, + 523, + 258, + 537 + ], + "score": 1.0, + "content": "data type by estimating the quantiles", + "type": "text" + }, + { + "bbox": [ + 259, + 526, + 267, + 536 + ], + "score": 0.84, + "content": "q _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 523, + 327, + 537 + ], + "score": 1.0, + "content": "of two ranges", + "type": "text" + }, + { + "bbox": [ + 327, + 525, + 336, + 536 + ], + "score": 0.56, + "content": "q _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 523, + 341, + 537 + ], + "score": 1.0, + "content": ":", + "type": "text" + }, + { + "bbox": [ + 341, + 523, + 362, + 535 + ], + "score": 0.61, + "content": "2 ^ { k - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 523, + 466, + 537 + ], + "score": 1.0, + "content": "for the negative part and", + "type": "text" + }, + { + "bbox": [ + 466, + 524, + 505, + 535 + ], + "score": 0.91, + "content": "2 ^ { k - 1 } + 1", + "type": "inline_equation" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 534, + 506, + 548 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 315, + 548 + ], + "score": 1.0, + "content": "for the positive part and then we unify these sets of", + "type": "text" + }, + { + "bbox": [ + 315, + 537, + 324, + 546 + ], + "score": 0.86, + "content": "q _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 534, + 506, + 548 + ], + "score": 1.0, + "content": "and remove one of the two zeros that occurs", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 546, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 505, + 559 + ], + "score": 1.0, + "content": "in both sets. We term the resulting data type that has equal expected number of values in each", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 557, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 172, + 569 + ], + "score": 1.0, + "content": "quantization bin", + "type": "text" + }, + { + "bbox": [ + 172, + 558, + 178, + 567 + ], + "score": 0.6, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 557, + 506, + 569 + ], + "score": 1.0, + "content": "-bit NormalFloat (NFk), since the data type is information-theoretically optimal for", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 568, + 506, + 581 + ], + "spans": [ + { + "bbox": [ + 106, + 568, + 506, + 581 + ], + "score": 1.0, + "content": "zero-centered normally distributed data. A step-by-step visualization of how to construct the NF4", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 578, + 343, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 343, + 591 + ], + "score": 1.0, + "content": "data type and its exact values can be found in Appendix H.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 35.5, + "bbox_fs": [ + 104, + 480, + 507, + 591 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 596, + 505, + 651 + ], + "lines": [ + { + "bbox": [ + 106, + 596, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 596, + 505, + 609 + ], + "score": 1.0, + "content": "Double Quantization We introduce Double Quantization (DQ), the process of quantizing the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 607, + 505, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 505, + 620 + ], + "score": 1.0, + "content": "quantization constants for additional memory savings. While a small blocksize is required for precise", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 618, + 505, + 630 + ], + "spans": [ + { + "bbox": [ + 106, + 618, + 505, + 630 + ], + "score": 1.0, + "content": "4-bit quantization [13], it also has a considerable memory overhead. For example, using 32-bit", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 628, + 506, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 366, + 642 + ], + "score": 1.0, + "content": "constants and a blocksize of 64 for W, quantization constants add", + "type": "text" + }, + { + "bbox": [ + 367, + 629, + 419, + 641 + ], + "score": 0.9, + "content": "3 2 / 6 4 = 0 . 5 ", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 628, + 506, + 642 + ], + "score": 1.0, + "content": "bits per parameter on", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 640, + 474, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 474, + 653 + ], + "score": 1.0, + "content": "average. Double Quantization helps reduce the memory footprint of quantization constants.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 43, + "bbox_fs": [ + 105, + 596, + 506, + 653 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 655, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 101, + 649, + 510, + 679 + ], + "spans": [ + { + "bbox": [ + 101, + 649, + 386, + 679 + ], + "score": 1.0, + "content": "More specifically, Double Quantization treats quantization constants as inputs to a second quantization. This second step yields the qua", + "type": "text" + }, + { + "bbox": [ + 386, + 655, + 407, + 668 + ], + "score": 0.91, + "content": "c _ { 2 } ^ { \\mathrm { F P } 3 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 649, + 510, + 679 + ], + "score": 1.0, + "content": "of the first quantizationd quantization constants", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 107, + 673, + 509, + 694 + ], + "spans": [ + { + "bbox": [ + 107, + 677, + 123, + 689 + ], + "score": 0.91, + "content": "c _ { 2 } ^ { \\mathrm { F P 8 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 124, + 673, + 316, + 694 + ], + "score": 1.0, + "content": "and the second level of quantization constants", + "type": "text" + }, + { + "bbox": [ + 317, + 677, + 336, + 690 + ], + "score": 0.91, + "content": "\\dot { c } _ { 1 } ^ { \\mathrm { F } \\tilde { \\mathrm { P } } 3 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 673, + 509, + 694 + ], + "score": 1.0, + "content": ". We use 8-bit Floats with a blocksize of", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "score": 1.0, + "content": "256 for the second quantization as no performance degradation is observed for 8-bit quantization,", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 103, + 696, + 508, + 716 + ], + "spans": [ + { + "bbox": [ + 103, + 696, + 381, + 716 + ], + "score": 1.0, + "content": "in line with results from Dettmers and Zettlemoyer [13]. 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We use this feature to allocate paged memory for the optimizer states which are then", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 155, + 504, + 168 + ], + "spans": [ + { + "bbox": [ + 106, + 155, + 504, + 168 + ], + "score": 1.0, + "content": "automatically evicted to CPU RAM when the GPU runs out-of-memory and paged back into GPU", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 165, + 372, + 180 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 372, + 180 + ], + "score": 1.0, + "content": "memory when the memory is needed in the optimizer update step.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5.5 + }, + { + "type": "text", + "bbox": [ + 104, + 183, + 504, + 205 + ], + "lines": [ + { + "bbox": [ + 106, + 182, + 505, + 196 + ], + "spans": [ + { + "bbox": [ + 106, + 182, + 505, + 196 + ], + "score": 1.0, + "content": "QLORA. Using the components described above, we define QLORA for a single linear layer in", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 194, + 369, + 207 + ], + "spans": [ + { + "bbox": [ + 106, + 194, + 369, + 207 + ], + "score": 1.0, + "content": "the quantized base model with a single LoRA adapter as follows:", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5 + }, + { + "type": "interline_equation", + "bbox": [ + 149, + 208, + 477, + 223 + ], + "lines": [ + { + "bbox": [ + 149, + 208, + 477, + 223 + ], + "spans": [ + { + "bbox": [ + 149, + 208, + 477, + 223 + ], + "score": 0.42, + "content": "{ \\bf Y } ^ { \\mathrm { B F 1 6 } } = { \\bf X } ^ { \\mathrm { B F 1 6 } } \\mathrm { d o u b l e D e q u a n t } ( c _ { 1 } ^ { \\mathrm { F P 3 2 } } , c _ { 2 } ^ { \\mathrm { k - b i t } } , { \\bf W } ^ { \\mathrm { N F 4 } } ) + { \\bf X } ^ { \\mathrm { B F 1 6 } } { \\bf L } _ { 1 } ^ { \\mathrm { B F 1 6 } } { \\bf L } _ { 2 } ^ { \\mathrm { B F 1 6 } } ,", + "type": "interline_equation", + "image_path": "a605af95ea305f821cc6ee259fd3ff6a540b044e4abf1280d56513c1ed29cec5.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 149, + 208, + 477, + 223 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "interline_equation", + "bbox": [ + 130, + 222, + 480, + 240 + ], + "lines": [ + { + "bbox": [ + 130, + 222, + 480, + 240 + ], + "spans": [ + { + "bbox": [ + 130, + 222, + 480, + 240 + ], + "score": 0.5, + "content": "\\mathrm { d o u b l e D e q u a n t } ( c _ { 1 } ^ { \\mathrm { F 3 2 } } , c _ { 2 } ^ { \\mathrm { k - b i t } } , \\mathbf { W } ^ { \\mathrm { k - b i t } } ) = \\mathrm { d e q u a n t } ( \\mathrm { d e q u a n t } ( c _ { 1 } ^ { \\mathrm { F P 3 2 } } , c _ { 2 } ^ { \\mathrm { k - b i t } } ) , \\mathbf { W } ^ { \\mathrm { 4 b i t } } ) = \\mathbf { W } ^ { \\mathrm { B F l 6 } } ,", + "type": "interline_equation", + "image_path": "9466ea6d1cfbfa7e342be86bbcd1d8628443091320171a9bfbdbded101f781fe.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 130, + 222, + 480, + 240 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 245, + 505, + 269 + ], + "lines": [ + { + "bbox": [ + 106, + 244, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 106, + 244, + 233, + 259 + ], + "score": 1.0, + "content": "We use NF4 for W and FP8 for", + "type": "text" + }, + { + "bbox": [ + 233, + 248, + 243, + 257 + ], + "score": 0.83, + "content": "c _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 244, + 360, + 259 + ], + "score": 1.0, + "content": ". 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The follow-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 479, + 311, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 311, + 491 + ], + "score": 1.0, + "content": "ing sections will discuss the experiments that aimed", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 490, + 225, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 490, + 225, + 502 + ], + "score": 1.0, + "content": "at answering these questions.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 507, + 310, + 595 + ], + "lines": [ + { + "bbox": [ + 106, + 507, + 312, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 507, + 312, + 519 + ], + "score": 1.0, + "content": "Experimental setup. 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However, the calculation of", + "type": "text" + }, + { + "bbox": [ + 371, + 287, + 387, + 302 + ], + "score": 0.91, + "content": "\\frac { \\partial E } { \\partial \\mathbf { L } _ { i } }", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 286, + 487, + 304 + ], + "score": 1.0, + "content": "entails the calculation of", + "type": "text" + }, + { + "bbox": [ + 487, + 287, + 503, + 302 + ], + "score": 0.87, + "content": "\\textstyle \\frac { \\partial \\mathbf { X } } { \\partial \\mathbf { W } }", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 103, + 298, + 507, + 316 + ], + "spans": [ + { + "bbox": [ + 103, + 298, + 375, + 316 + ], + "score": 1.0, + "content": "which proceeds via equation (5) with dequantization from storage", + "type": "text" + }, + { + "bbox": [ + 375, + 302, + 401, + 312 + ], + "score": 0.84, + "content": "\\dot { \\mathbf { W } } ^ { \\mathrm { N F 4 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 298, + 507, + 316 + ], + "score": 1.0, + "content": "to computation data type", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 311, + 351, + 328 + ], + "spans": [ + { + "bbox": [ + 106, + 313, + 135, + 325 + ], + "score": 0.6, + "content": "{ \\bf W } ^ { \\mathrm { B F 1 6 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 311, + 241, + 328 + ], + "score": 1.0, + "content": "to calculate the derivative", + "type": "text" + }, + { + "bbox": [ + 241, + 313, + 258, + 327 + ], + "score": 0.91, + "content": "\\textstyle \\frac { \\partial \\mathbf { X } } { \\partial \\mathbf { W } }", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 311, + 351, + 328 + ], + "score": 1.0, + "content": "in BFloat16 precision.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5, + "bbox_fs": [ + 103, + 272, + 507, + 328 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 330, + 505, + 374 + ], + "lines": [ + { + "bbox": [ + 105, + 330, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 505, + 343 + ], + "score": 1.0, + "content": "To summarize, QLORA has one storage data type (usually 4-bit NormalFloat) and a computation", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 341, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 505, + 354 + ], + "score": 1.0, + "content": "data type (16-bit BrainFloat). 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Future work should measure", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 234, + 456, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 234, + 456, + 248 + ], + "score": 1.0, + "content": "and characterize under what circumstances slow-downs occur from the paging process.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 107, + 254, + 364, + 375 + ], + "lines": [ + { + "bbox": [ + 106, + 254, + 366, + 266 + ], + "spans": [ + { + "bbox": [ + 106, + 254, + 366, + 266 + ], + "score": 1.0, + "content": "4-bit NormalFloat yields better performance than 4-bit Float-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 266, + 365, + 278 + ], + "spans": [ + { + "bbox": [ + 106, + 266, + 365, + 278 + ], + "score": 1.0, + "content": "ing Point While the 4-bit NormalFloat (NF4) data type is", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 276, + 365, + 288 + ], + "spans": [ + { + "bbox": [ + 106, + 276, + 365, + 288 + ], + "score": 1.0, + "content": "information-theoretically optimal, it still needs to be determined", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 287, + 365, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 287, + 365, + 299 + ], + "score": 1.0, + "content": "if this property translates to empirical advantages. We follow", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 298, + 365, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 365, + 311 + ], + "score": 1.0, + "content": "the setup from Dettmers and Zettlemoyer [13] where quantized", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 309, + 365, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 365, + 321 + ], + "score": 1.0, + "content": "LLMs (OPT [73], BLOOM [53], Pythia [7], LLaMA) of different", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 320, + 365, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 365, + 331 + ], + "score": 1.0, + "content": "sizes (125M to 65B) with different data types are evaluated on", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 331, + 365, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 365, + 343 + ], + "score": 1.0, + "content": "language modeling and a set of zero-shot tasks. In Figure 7 and", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 341, + 365, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 365, + 354 + ], + "score": 1.0, + "content": "Table 2 we see that NF4 improves performance significantly over", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 352, + 364, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 364, + 366 + ], + "score": 1.0, + "content": "FP4 and Int4 and that double quantization reduces the memory", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 363, + 275, + 376 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 275, + 376 + ], + "score": 1.0, + "content": "footprint without degrading performance.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 15 + }, + { + "type": "table", + "bbox": [ + 372, + 302, + 501, + 370 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 371, + 255, + 505, + 296 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 371, + 255, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 371, + 255, + 505, + 266 + ], + "score": 1.0, + "content": "Table 2: Pile Common Crawl mean", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 370, + 265, + 506, + 277 + ], + "spans": [ + { + "bbox": [ + 370, + 265, + 506, + 277 + ], + "score": 1.0, + "content": "perplexity for different data types", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 370, + 275, + 507, + 286 + ], + "spans": [ + { + "bbox": [ + 370, + 275, + 507, + 286 + ], + "score": 1.0, + "content": "for 125M to 13B OPT, BLOOM,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 371, + 285, + 476, + 296 + ], + "spans": [ + { + "bbox": [ + 371, + 285, + 476, + 296 + ], + "score": 1.0, + "content": "LLaMA, and Pythia models.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22.5 + }, + { + "type": "table_body", + "bbox": [ + 372, + 302, + 501, + 370 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 372, + 302, + 501, + 370 + ], + "spans": [ + { + "bbox": [ + 372, + 302, + 501, + 370 + ], + "score": 0.972, + "html": "
Data typeMean PPL
Int434.34
Float4 (E2M1)31.07
Float4 (E3M0)29.48
NFloat4 + DQ27.41
", + "type": "table", + "image_path": "32a2d8b5065e2c7678c8dbd9c3df8f74ccad118f0020186a256d1106db6bd6c1.jpg" + } + ] + } + ], + "index": 25.5, + "virtual_lines": [ + { + "bbox": [ + 372, + 302, + 501, + 336.0 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 372, + 336.0, + 501, + 370.0 + ], + "spans": [], + "index": 26 + } + ] + } + ], + "index": 24.0 + }, + { + "type": "text", + "bbox": [ + 107, + 384, + 505, + 428 + ], + "lines": [ + { + "bbox": [ + 105, + 382, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 505, + 397 + ], + "score": 1.0, + "content": "k-bit QLORA matches 16-bit full finetuning and 16-bit LoRA performance Recent findings", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 395, + 505, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 505, + 407 + ], + "score": 1.0, + "content": "have established that 4-bit quantization for inference is possible, but leads to performance degradation", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 406, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 505, + 419 + ], + "score": 1.0, + "content": "relative to 16-bit [13, 18]. This raises the crucial question of whether the lost performance can be", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 417, + 416, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 416, + 430 + ], + "score": 1.0, + "content": "recovered by conducting 4-bit adapter finetuning. We test this for two setups.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28.5 + }, + { + "type": "text", + "bbox": [ + 106, + 435, + 505, + 556 + ], + "lines": [ + { + "bbox": [ + 105, + 435, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 506, + 448 + ], + "score": 1.0, + "content": "The first setup test if QLoRA can replicate 16-bit full finetuning performance for T5 and RoBERTa", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 446, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 106, + 446, + 505, + 459 + ], + "score": 1.0, + "content": "model finetuning. These results are detailed in the Appendix C.1. For our second setup, since full", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 456, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 505, + 471 + ], + "score": 1.0, + "content": "finetuning models at and beyond 11B parameters requires more than one server of high memory", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 467, + 506, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 506, + 481 + ], + "score": 1.0, + "content": "GPUs, we continue to test whether 4-bit QLORA can match 16-bit LoRA at the 7B to 65B parameter", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 477, + 506, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 506, + 492 + ], + "score": 1.0, + "content": "scales. To this end, we finetune LLaMA 7B through 65B on two instruction following datasets,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 489, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 506, + 502 + ], + "score": 1.0, + "content": "Alpaca and FLAN v2, and evaluate on the MMLU benchmark via 5-shot accuracy. Results are shown", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 501, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 505, + 513 + ], + "score": 1.0, + "content": "in Table 3 where we see that NF4 with double quantization fully recovers the 16-bit LoRA MMLU", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "score": 1.0, + "content": "performance. In addition, we also note that QLORA with FP4 lags behind the 16-bit brain float", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 523, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 505, + 535 + ], + "score": 1.0, + "content": "LoRA baseline by about 1 percentage point. This corroborates both our findings that (1) QLORA", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "score": 1.0, + "content": "with NF4 replicates both 16-bit full finetuning and 16-bit LoRA finetuning performance, and (2) NF4", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 545, + 318, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 318, + 557 + ], + "score": 1.0, + "content": "is superior to FP4 in terms of quantization precision.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 564, + 505, + 619 + ], + "lines": [ + { + "bbox": [ + 106, + 564, + 506, + 576 + ], + "spans": [ + { + "bbox": [ + 106, + 564, + 506, + 576 + ], + "score": 1.0, + "content": "Summary Our results consistently show that 4-bit QLORA with NF4 data type matches 16-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 575, + 506, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 506, + 588 + ], + "score": 1.0, + "content": "bit full finetuning and 16-bit LoRA finetuning performance on academic benchmarks with well-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 586, + 505, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 505, + 598 + ], + "score": 1.0, + "content": "established evaluation setups. We have also shown that NF4 is more effective than FP4 and that", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 596, + 506, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 506, + 610 + ], + "score": 1.0, + "content": "double quantization does not degrade performance. Combined, this forms compelling evidence that", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 609, + 385, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 385, + 621 + ], + "score": 1.0, + "content": "4-bit QLORA tuning reliably yields results matching 16-bit methods.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 44 + }, + { + "type": "text", + "bbox": [ + 107, + 627, + 505, + 693 + ], + "lines": [ + { + "bbox": [ + 105, + 626, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 505, + 639 + ], + "score": 1.0, + "content": "In line with previous work on quantization [13], our MMLU and Elo results indicate that with a given", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 637, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 505, + 650 + ], + "score": 1.0, + "content": "finetuning and inference resource budget it is beneficial to increase the number of parameters in the", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 648, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 505, + 661 + ], + "score": 1.0, + "content": "base model while decreasing their precision. This highlights the importance of efficiency benefits", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 659, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 505, + 672 + ], + "score": 1.0, + "content": "from QLORA. Since we did not observe performance degradation compared to full-finetuning in", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 671, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 505, + 683 + ], + "score": 1.0, + "content": "our experiments with 4-bit finetuning, this raises the question of where the performance-precision", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 682, + 438, + 693 + ], + "spans": [ + { + "bbox": [ + 106, + 682, + 438, + 693 + ], + "score": 1.0, + "content": "trade-off exactly lies for QLoRA tuning, which we leave to future work to explore.", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 49.5 + }, + { + "type": "text", + "bbox": [ + 107, + 700, + 503, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 699, + 504, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 504, + 712 + ], + "score": 1.0, + "content": "We proceed to investigate instruction tuning at scales that would be impossible to explore with full", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 106, + 711, + 304, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 304, + 722 + ], + "score": 1.0, + "content": "16-bit finetuning on academic research hardware.", + "type": "text" + } + ], + "index": 54 + } + ], + "index": 53.5 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 742, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "spans": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 107, + 108, + 505, + 178 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 77, + 504, + 107 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 78, + 505, + 88 + ], + "spans": [ + { + "bbox": [ + 106, + 78, + 505, + 88 + ], + "score": 1.0, + "content": "Table 3: Mean 5-shot MMLU test accuracy for LLaMA 7-65B models finetuned with adapters on Alpaca and", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 86, + 506, + 99 + ], + "spans": [ + { + "bbox": [ + 105, + 86, + 506, + 99 + ], + "score": 1.0, + "content": "FLAN v2 for different data types. Overall, NF4 with double quantization (DQ) matches BFloat16 performance,", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 97, + 322, + 109 + ], + "spans": [ + { + "bbox": [ + 105, + 97, + 322, + 109 + ], + "score": 1.0, + "content": "while FP4 is consistently one percentage point behind both.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "table_body", + "bbox": [ + 107, + 108, + 505, + 178 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 108, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 107, + 108, + 505, + 178 + ], + "score": 0.98, + "html": "
LLaMA SizeMean 5-shot MMLU AccuracyMean
7B13B33B65B
AlpacaFLAN v2AlpacaFLAN v2AlpacaFLAN v2AlpacaFLAN v2
BFloat1638.445.647.250.657.760.561.862.553.0
Float437.244.047.350.055.958.561.363.352.2
NFloat4 + DQ39.044.547.550.757.359.261.863.953.1
", + "type": "table", + "image_path": "fe18786e8beae6966a15b4be1f7f12fc638a82ce3e6e8407f48ff91beb9310c2.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 107, + 108, + 505, + 131.33333333333334 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 107, + 131.33333333333334, + 505, + 154.66666666666669 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 107, + 154.66666666666669, + 505, + 178.00000000000003 + ], + "spans": [], + "index": 5 + } + ] + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 202, + 505, + 245 + ], + "lines": [], + "index": 7.5, + "bbox_fs": [ + 105, + 201, + 506, + 248 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 254, + 364, + 375 + ], + "lines": [ + { + "bbox": [ + 106, + 254, + 366, + 266 + ], + "spans": [ + { + "bbox": [ + 106, + 254, + 366, + 266 + ], + "score": 1.0, + "content": "4-bit NormalFloat yields better performance than 4-bit Float-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 266, + 365, + 278 + ], + "spans": [ + { + "bbox": [ + 106, + 266, + 365, + 278 + ], + "score": 1.0, + "content": "ing Point While the 4-bit NormalFloat (NF4) data type is", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 276, + 365, + 288 + ], + "spans": [ + { + "bbox": [ + 106, + 276, + 365, + 288 + ], + "score": 1.0, + "content": "information-theoretically optimal, it still needs to be determined", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 287, + 365, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 287, + 365, + 299 + ], + "score": 1.0, + "content": "if this property translates to empirical advantages. We follow", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 298, + 365, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 365, + 311 + ], + "score": 1.0, + "content": "the setup from Dettmers and Zettlemoyer [13] where quantized", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 309, + 365, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 365, + 321 + ], + "score": 1.0, + "content": "LLMs (OPT [73], BLOOM [53], Pythia [7], LLaMA) of different", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 320, + 365, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 365, + 331 + ], + "score": 1.0, + "content": "sizes (125M to 65B) with different data types are evaluated on", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 331, + 365, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 365, + 343 + ], + "score": 1.0, + "content": "language modeling and a set of zero-shot tasks. In Figure 7 and", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 341, + 365, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 365, + 354 + ], + "score": 1.0, + "content": "Table 2 we see that NF4 improves performance significantly over", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 352, + 364, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 364, + 366 + ], + "score": 1.0, + "content": "FP4 and Int4 and that double quantization reduces the memory", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 363, + 275, + 376 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 275, + 376 + ], + "score": 1.0, + "content": "footprint without degrading performance.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 254, + 366, + 376 + ] + }, + { + "type": "table", + "bbox": [ + 372, + 302, + 501, + 370 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 371, + 255, + 505, + 296 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 371, + 255, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 371, + 255, + 505, + 266 + ], + "score": 1.0, + "content": "Table 2: Pile Common Crawl mean", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 370, + 265, + 506, + 277 + ], + "spans": [ + { + "bbox": [ + 370, + 265, + 506, + 277 + ], + "score": 1.0, + "content": "perplexity for different data types", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 370, + 275, + 507, + 286 + ], + "spans": [ + { + "bbox": [ + 370, + 275, + 507, + 286 + ], + "score": 1.0, + "content": "for 125M to 13B OPT, BLOOM,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 371, + 285, + 476, + 296 + ], + "spans": [ + { + "bbox": [ + 371, + 285, + 476, + 296 + ], + "score": 1.0, + "content": "LLaMA, and Pythia models.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22.5 + }, + { + "type": "table_body", + "bbox": [ + 372, + 302, + 501, + 370 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 372, + 302, + 501, + 370 + ], + "spans": [ + { + "bbox": [ + 372, + 302, + 501, + 370 + ], + "score": 0.972, + "html": "
Data typeMean PPL
Int434.34
Float4 (E2M1)31.07
Float4 (E3M0)29.48
NFloat4 + DQ27.41
", + "type": "table", + "image_path": "32a2d8b5065e2c7678c8dbd9c3df8f74ccad118f0020186a256d1106db6bd6c1.jpg" + } + ] + } + ], + "index": 25.5, + "virtual_lines": [ + { + "bbox": [ + 372, + 302, + 501, + 336.0 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 372, + 336.0, + 501, + 370.0 + ], + "spans": [], + "index": 26 + } + ] + } + ], + "index": 24.0 + }, + { + "type": "text", + "bbox": [ + 107, + 384, + 505, + 428 + ], + "lines": [ + { + "bbox": [ + 105, + 382, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 505, + 397 + ], + "score": 1.0, + "content": "k-bit QLORA matches 16-bit full finetuning and 16-bit LoRA performance Recent findings", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 395, + 505, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 505, + 407 + ], + "score": 1.0, + "content": "have established that 4-bit quantization for inference is possible, but leads to performance degradation", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 406, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 505, + 419 + ], + "score": 1.0, + "content": "relative to 16-bit [13, 18]. This raises the crucial question of whether the lost performance can be", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 417, + 416, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 416, + 430 + ], + "score": 1.0, + "content": "recovered by conducting 4-bit adapter finetuning. We test this for two setups.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 382, + 505, + 430 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 435, + 505, + 556 + ], + "lines": [ + { + "bbox": [ + 105, + 435, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 506, + 448 + ], + "score": 1.0, + "content": "The first setup test if QLoRA can replicate 16-bit full finetuning performance for T5 and RoBERTa", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 446, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 106, + 446, + 505, + 459 + ], + "score": 1.0, + "content": "model finetuning. These results are detailed in the Appendix C.1. For our second setup, since full", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 456, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 505, + 471 + ], + "score": 1.0, + "content": "finetuning models at and beyond 11B parameters requires more than one server of high memory", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 467, + 506, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 506, + 481 + ], + "score": 1.0, + "content": "GPUs, we continue to test whether 4-bit QLORA can match 16-bit LoRA at the 7B to 65B parameter", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 477, + 506, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 506, + 492 + ], + "score": 1.0, + "content": "scales. To this end, we finetune LLaMA 7B through 65B on two instruction following datasets,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 489, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 506, + 502 + ], + "score": 1.0, + "content": "Alpaca and FLAN v2, and evaluate on the MMLU benchmark via 5-shot accuracy. Results are shown", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 501, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 505, + 513 + ], + "score": 1.0, + "content": "in Table 3 where we see that NF4 with double quantization fully recovers the 16-bit LoRA MMLU", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "score": 1.0, + "content": "performance. In addition, we also note that QLORA with FP4 lags behind the 16-bit brain float", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 523, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 505, + 535 + ], + "score": 1.0, + "content": "LoRA baseline by about 1 percentage point. This corroborates both our findings that (1) QLORA", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "score": 1.0, + "content": "with NF4 replicates both 16-bit full finetuning and 16-bit LoRA finetuning performance, and (2) NF4", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 545, + 318, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 318, + 557 + ], + "score": 1.0, + "content": "is superior to FP4 in terms of quantization precision.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 36, + "bbox_fs": [ + 105, + 435, + 506, + 557 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 564, + 505, + 619 + ], + "lines": [ + { + "bbox": [ + 106, + 564, + 506, + 576 + ], + "spans": [ + { + "bbox": [ + 106, + 564, + 506, + 576 + ], + "score": 1.0, + "content": "Summary Our results consistently show that 4-bit QLORA with NF4 data type matches 16-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 575, + 506, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 506, + 588 + ], + "score": 1.0, + "content": "bit full finetuning and 16-bit LoRA finetuning performance on academic benchmarks with well-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 586, + 505, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 505, + 598 + ], + "score": 1.0, + "content": "established evaluation setups. We have also shown that NF4 is more effective than FP4 and that", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 596, + 506, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 506, + 610 + ], + "score": 1.0, + "content": "double quantization does not degrade performance. Combined, this forms compelling evidence that", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 609, + 385, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 385, + 621 + ], + "score": 1.0, + "content": "4-bit QLORA tuning reliably yields results matching 16-bit methods.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 44, + "bbox_fs": [ + 105, + 564, + 506, + 621 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 627, + 505, + 693 + ], + "lines": [ + { + "bbox": [ + 105, + 626, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 505, + 639 + ], + "score": 1.0, + "content": "In line with previous work on quantization [13], our MMLU and Elo results indicate that with a given", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 637, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 505, + 650 + ], + "score": 1.0, + "content": "finetuning and inference resource budget it is beneficial to increase the number of parameters in the", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 648, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 505, + 661 + ], + "score": 1.0, + "content": "base model while decreasing their precision. This highlights the importance of efficiency benefits", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 659, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 505, + 672 + ], + "score": 1.0, + "content": "from QLORA. Since we did not observe performance degradation compared to full-finetuning in", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 671, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 505, + 683 + ], + "score": 1.0, + "content": "our experiments with 4-bit finetuning, this raises the question of where the performance-precision", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 682, + 438, + 693 + ], + "spans": [ + { + "bbox": [ + 106, + 682, + 438, + 693 + ], + "score": 1.0, + "content": "trade-off exactly lies for QLoRA tuning, which we leave to future work to explore.", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 49.5, + "bbox_fs": [ + 105, + 626, + 505, + 693 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 700, + 503, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 699, + 504, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 504, + 712 + ], + "score": 1.0, + "content": "We proceed to investigate instruction tuning at scales that would be impossible to explore with full", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 106, + 711, + 304, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 304, + 722 + ], + "score": 1.0, + "content": "16-bit finetuning on academic research hardware.", + "type": "text" + } + ], + "index": 54 + } + ], + "index": 53.5, + "bbox_fs": [ + 106, + 699, + 504, + 722 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 106, + 70, + 381, + 84 + ], + "lines": [ + { + "bbox": [ + 104, + 70, + 383, + 87 + ], + "spans": [ + { + "bbox": [ + 104, + 70, + 383, + 87 + ], + "score": 1.0, + "content": "5 Pushing the Chatbot State-of-the-art with QLoRA", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 90, + 505, + 145 + ], + "lines": [ + { + "bbox": [ + 106, + 90, + 505, + 102 + ], + "spans": [ + { + "bbox": [ + 106, + 90, + 505, + 102 + ], + "score": 1.0, + "content": "Having established that 4-bit QLORA matches 16-bit performance across scales, tasks, and datasets", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 102, + 505, + 114 + ], + "spans": [ + { + "bbox": [ + 106, + 102, + 505, + 114 + ], + "score": 1.0, + "content": "we conduct an in-depth study of instruction finetuning up to the largest open-source language models", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 112, + 505, + 124 + ], + "spans": [ + { + "bbox": [ + 106, + 112, + 505, + 124 + ], + "score": 1.0, + "content": "available for research. To assess the performance of instruction finetuning these models, we evaluate", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 123, + 506, + 136 + ], + "spans": [ + { + "bbox": [ + 105, + 123, + 506, + 136 + ], + "score": 1.0, + "content": "on a challenging Natural Language Understanding benchmark (MMLU) and develop new methods for", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 134, + 477, + 147 + ], + "spans": [ + { + "bbox": [ + 105, + 134, + 477, + 147 + ], + "score": 1.0, + "content": "real-world chatbot performance evaluation. For a more qualitative analysis, see Appendix F.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3 + }, + { + "type": "title", + "bbox": [ + 107, + 153, + 214, + 164 + ], + "lines": [ + { + "bbox": [ + 105, + 150, + 216, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 150, + 216, + 168 + ], + "score": 1.0, + "content": "5.1 Experimental setup", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 106, + 167, + 457, + 179 + ], + "lines": [ + { + "bbox": [ + 105, + 166, + 459, + 181 + ], + "spans": [ + { + "bbox": [ + 105, + 166, + 459, + 181 + ], + "score": 1.0, + "content": "We now describe an overview of the experimental setup with full details in Appendix D.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 107, + 185, + 505, + 240 + ], + "lines": [ + { + "bbox": [ + 106, + 185, + 505, + 197 + ], + "spans": [ + { + "bbox": [ + 106, + 185, + 505, + 197 + ], + "score": 1.0, + "content": "Data As, to our knowledge, there is no comprehensive study of instruction-following datasets, we", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 195, + 506, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 195, + 506, + 209 + ], + "score": 1.0, + "content": "select eight recent datasets. We include datasets obtained through crowd-sourcing (OASST1 [31],", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 207, + 506, + 219 + ], + "spans": [ + { + "bbox": [ + 106, + 207, + 506, + 219 + ], + "score": 1.0, + "content": "HH-RLHF [4]), distillation from instruction-tuned models (Alpaca [56], self-instruct [60], unnatural-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 216, + 507, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 507, + 232 + ], + "score": 1.0, + "content": "instructions [26]), corpora aggregations (FLAN v2 [12]), as well as hybrids (Chip2 [32], Long-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 228, + 416, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 416, + 241 + ], + "score": 1.0, + "content": "form [30]). These datasets cover different languages, data sizes, and licenses.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 246, + 505, + 367 + ], + "lines": [ + { + "bbox": [ + 105, + 245, + 505, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 505, + 260 + ], + "score": 1.0, + "content": "Training Setup To avoid confounding effects from different training objectives, we perform", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 257, + 507, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 507, + 271 + ], + "score": 1.0, + "content": "QLoRA finetuning with cross-entropy loss (supervised learning) without reinforcement learning,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 268, + 506, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 506, + 281 + ], + "score": 1.0, + "content": "even for datasets that include human judgments of different responses. For datasets that have a clear", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 279, + 505, + 292 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 505, + 292 + ], + "score": 1.0, + "content": "distinction between instruction and response, we finetune only on the response (see ablations in", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 290, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 505, + 303 + ], + "score": 1.0, + "content": "Appendix D). For OASST1 and HH-RLHF, multiple responses are available. We then select the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 302, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 506, + 313 + ], + "score": 1.0, + "content": "top response at every level of the conversation tree and finetune on the full selected conversation,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 312, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 505, + 325 + ], + "score": 1.0, + "content": "including the instructions. In all of our experiments, we use NF4 QLORA with double quantization", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 322, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 505, + 336 + ], + "score": 1.0, + "content": "and paged optimizers to prevent memory spikes during gradient checkpointing. We do a small", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 333, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 506, + 347 + ], + "score": 1.0, + "content": "hyperparameter searches for the 13B and 33B LLaMA models and we find that all hyperparameter", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 345, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 505, + 358 + ], + "score": 1.0, + "content": "settings found at 7B generalize (including number of epochs) except learning rate and batch size. We", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 356, + 389, + 368 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 389, + 368 + ], + "score": 1.0, + "content": "halve the learning rate for 33B and 65B while doubling the batch size.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 373, + 505, + 428 + ], + "lines": [ + { + "bbox": [ + 106, + 373, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 106, + 373, + 505, + 385 + ], + "score": 1.0, + "content": "Baselines We compare our models to both research (Vicuna [10] and Open Assistant [31]) and", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 384, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 505, + 397 + ], + "score": 1.0, + "content": "commercial (GPT-4 [42], GPT-3.5-turbo and Bard) chatbot systems. The Open Assistant model", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 394, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 506, + 408 + ], + "score": 1.0, + "content": "is a LLaMA 33B model finetuned with Reinforcement Learning from Human Feedback (RLHF)", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 406, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 505, + 419 + ], + "score": 1.0, + "content": "on the OASST1 dataset. Vicuna does full fine-tuning of LLaMA 13B on proprietary user-shared", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 416, + 477, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 477, + 430 + ], + "score": 1.0, + "content": "conversations from ShareGPT and is thus the result of distillation from OpenAI GPT models.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 435, + 505, + 511 + ], + "lines": [ + { + "bbox": [ + 105, + 433, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 506, + 448 + ], + "score": 1.0, + "content": "Evaluation setup Chatbot evaluation is not straightforward since each prompt has many high-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 445, + 507, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 507, + 459 + ], + "score": 1.0, + "content": "quality responses which are challenging to rank. We therefore follow a comprehensive approach,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 456, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 506, + 470 + ], + "score": 1.0, + "content": "which involves (a) standard benchmarks that measures general language understanding performance", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 468, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 505, + 480 + ], + "score": 1.0, + "content": "(MMLU [24]), (b) both automatic and human evaluation that measures how many responses from", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 478, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 505, + 491 + ], + "score": 1.0, + "content": "chatbot A are better compared to chatbot B, (c) a round-robin tournament-style evaluation where", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "score": 1.0, + "content": "chatbots compete against each other in games where chatbot performance is measured as ELO. We", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 500, + 407, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 407, + 512 + ], + "score": 1.0, + "content": "provide a detailed discussion of these evaluation setups in the Appendix E.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 32 + }, + { + "type": "title", + "bbox": [ + 108, + 519, + 422, + 530 + ], + "lines": [ + { + "bbox": [ + 104, + 518, + 423, + 533 + ], + "spans": [ + { + "bbox": [ + 104, + 518, + 423, + 533 + ], + "score": 1.0, + "content": "5.2 Guanaco: QLORA trained on OASST1 is a State-of-the-art Chatbot", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 534, + 505, + 588 + ], + "lines": [ + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "score": 1.0, + "content": "Based on our automated and human evaluations, we find that the top QLORA tuned model, Guanaco", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 544, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 505, + 558 + ], + "score": 1.0, + "content": "65B, which we finetune on a variant of OASST1, is the best-performing open-source chatbot model", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 555, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 505, + 568 + ], + "score": 1.0, + "content": "and offers performance competitive to ChatGPT. When compared to GPT-4, Guanaco 65B and 33B", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 567, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 252, + 579 + ], + "score": 1.0, + "content": "have an expected win probability of", + "type": "text" + }, + { + "bbox": [ + 253, + 567, + 272, + 577 + ], + "score": 0.87, + "content": "30 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 567, + 505, + 579 + ], + "score": 1.0, + "content": ", based on Elo rating from human annotators system-level", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 578, + 419, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 419, + 591 + ], + "score": 1.0, + "content": "pairwise comparisons on the Vicuna benchmark - the highest reported to date.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 107, + 594, + 505, + 660 + ], + "lines": [ + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "The Vicuna benchmark [10] results relative to ChatGPT are shown in Table 4. We find that Guanaco", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 604, + 506, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 339, + 617 + ], + "score": 1.0, + "content": "65B is the best-performing model after GPT-4, achieving", + "type": "text" + }, + { + "bbox": [ + 339, + 605, + 367, + 615 + ], + "score": 0.86, + "content": "9 9 . 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 604, + 506, + 617 + ], + "score": 1.0, + "content": "performance relative to ChatGPT.", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 615, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 505, + 628 + ], + "score": 1.0, + "content": "Guanaco 33B has more parameters than the Vicuna 13B model, but uses only 4-bit precision for its", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 626, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 356, + 641 + ], + "score": 1.0, + "content": "weights and is thus much more memory efficient at 21 GB vs", + "type": "text" + }, + { + "bbox": [ + 356, + 627, + 383, + 637 + ], + "score": 0.28, + "content": "2 6 \\mathrm { G B }", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 626, + 505, + 641 + ], + "score": 1.0, + "content": ", providing a three percentage", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 637, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 506, + 650 + ], + "score": 1.0, + "content": "points of improvement over Vicuna 13B. Furthermore, Guanaco 7B easily fits on modern phones at a", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 648, + 454, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 454, + 661 + ], + "score": 1.0, + "content": "5 GB footprint while still scoring nearly 20 percentage points higher than Alpaca 13B.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 44.5 + }, + { + "type": "text", + "bbox": [ + 107, + 665, + 505, + 720 + ], + "lines": [ + { + "bbox": [ + 106, + 665, + 506, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 665, + 506, + 677 + ], + "score": 1.0, + "content": "However, Table 4 also has very wide confidence intervals, with many models overlapping in per-", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 675, + 505, + 688 + ], + "spans": [ + { + "bbox": [ + 106, + 675, + 505, + 688 + ], + "score": 1.0, + "content": "formance. We hypothesize that this uncertainty comes from the lack of clear specification of scale,", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 686, + 505, + 699 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 505, + 699 + ], + "score": 1.0, + "content": "e.g., it is unclear what 8 on a 10 point scale means across different scenarios. As such, we instead", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "score": 1.0, + "content": "recommend using the Elo ranking method [16], based on pairwise judgments from human annotators", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 709, + 363, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 363, + 721 + ], + "score": 1.0, + "content": "and GPT-4 to avoid the problem of grounding an absolute scale.", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 50 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 303, + 741, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "spans": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 106, + 70, + 381, + 84 + ], + "lines": [ + { + "bbox": [ + 104, + 70, + 383, + 87 + ], + "spans": [ + { + "bbox": [ + 104, + 70, + 383, + 87 + ], + "score": 1.0, + "content": "5 Pushing the Chatbot State-of-the-art with QLoRA", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 90, + 505, + 145 + ], + "lines": [ + { + "bbox": [ + 106, + 90, + 505, + 102 + ], + "spans": [ + { + "bbox": [ + 106, + 90, + 505, + 102 + ], + "score": 1.0, + "content": "Having established that 4-bit QLORA matches 16-bit performance across scales, tasks, and datasets", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 102, + 505, + 114 + ], + "spans": [ + { + "bbox": [ + 106, + 102, + 505, + 114 + ], + "score": 1.0, + "content": "we conduct an in-depth study of instruction finetuning up to the largest open-source language models", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 112, + 505, + 124 + ], + "spans": [ + { + "bbox": [ + 106, + 112, + 505, + 124 + ], + "score": 1.0, + "content": "available for research. To assess the performance of instruction finetuning these models, we evaluate", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 123, + 506, + 136 + ], + "spans": [ + { + "bbox": [ + 105, + 123, + 506, + 136 + ], + "score": 1.0, + "content": "on a challenging Natural Language Understanding benchmark (MMLU) and develop new methods for", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 134, + 477, + 147 + ], + "spans": [ + { + "bbox": [ + 105, + 134, + 477, + 147 + ], + "score": 1.0, + "content": "real-world chatbot performance evaluation. For a more qualitative analysis, see Appendix F.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3, + "bbox_fs": [ + 105, + 90, + 506, + 147 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 153, + 214, + 164 + ], + "lines": [ + { + "bbox": [ + 105, + 150, + 216, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 150, + 216, + 168 + ], + "score": 1.0, + "content": "5.1 Experimental setup", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 106, + 167, + 457, + 179 + ], + "lines": [ + { + "bbox": [ + 105, + 166, + 459, + 181 + ], + "spans": [ + { + "bbox": [ + 105, + 166, + 459, + 181 + ], + "score": 1.0, + "content": "We now describe an overview of the experimental setup with full details in Appendix D.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 166, + 459, + 181 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 185, + 505, + 240 + ], + "lines": [ + { + "bbox": [ + 106, + 185, + 505, + 197 + ], + "spans": [ + { + "bbox": [ + 106, + 185, + 505, + 197 + ], + "score": 1.0, + "content": "Data As, to our knowledge, there is no comprehensive study of instruction-following datasets, we", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 195, + 506, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 195, + 506, + 209 + ], + "score": 1.0, + "content": "select eight recent datasets. We include datasets obtained through crowd-sourcing (OASST1 [31],", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 207, + 506, + 219 + ], + "spans": [ + { + "bbox": [ + 106, + 207, + 506, + 219 + ], + "score": 1.0, + "content": "HH-RLHF [4]), distillation from instruction-tuned models (Alpaca [56], self-instruct [60], unnatural-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 216, + 507, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 507, + 232 + ], + "score": 1.0, + "content": "instructions [26]), corpora aggregations (FLAN v2 [12]), as well as hybrids (Chip2 [32], Long-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 228, + 416, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 416, + 241 + ], + "score": 1.0, + "content": "form [30]). These datasets cover different languages, data sizes, and licenses.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10, + "bbox_fs": [ + 105, + 185, + 507, + 241 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 246, + 505, + 367 + ], + "lines": [ + { + "bbox": [ + 105, + 245, + 505, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 505, + 260 + ], + "score": 1.0, + "content": "Training Setup To avoid confounding effects from different training objectives, we perform", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 257, + 507, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 507, + 271 + ], + "score": 1.0, + "content": "QLoRA finetuning with cross-entropy loss (supervised learning) without reinforcement learning,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 268, + 506, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 506, + 281 + ], + "score": 1.0, + "content": "even for datasets that include human judgments of different responses. For datasets that have a clear", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 279, + 505, + 292 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 505, + 292 + ], + "score": 1.0, + "content": "distinction between instruction and response, we finetune only on the response (see ablations in", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 290, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 505, + 303 + ], + "score": 1.0, + "content": "Appendix D). For OASST1 and HH-RLHF, multiple responses are available. We then select the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 302, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 506, + 313 + ], + "score": 1.0, + "content": "top response at every level of the conversation tree and finetune on the full selected conversation,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 312, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 505, + 325 + ], + "score": 1.0, + "content": "including the instructions. In all of our experiments, we use NF4 QLORA with double quantization", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 322, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 505, + 336 + ], + "score": 1.0, + "content": "and paged optimizers to prevent memory spikes during gradient checkpointing. We do a small", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 333, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 506, + 347 + ], + "score": 1.0, + "content": "hyperparameter searches for the 13B and 33B LLaMA models and we find that all hyperparameter", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 345, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 505, + 358 + ], + "score": 1.0, + "content": "settings found at 7B generalize (including number of epochs) except learning rate and batch size. We", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 356, + 389, + 368 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 389, + 368 + ], + "score": 1.0, + "content": "halve the learning rate for 33B and 65B while doubling the batch size.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 245, + 507, + 368 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 373, + 505, + 428 + ], + "lines": [ + { + "bbox": [ + 106, + 373, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 106, + 373, + 505, + 385 + ], + "score": 1.0, + "content": "Baselines We compare our models to both research (Vicuna [10] and Open Assistant [31]) and", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 384, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 505, + 397 + ], + "score": 1.0, + "content": "commercial (GPT-4 [42], GPT-3.5-turbo and Bard) chatbot systems. The Open Assistant model", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 394, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 506, + 408 + ], + "score": 1.0, + "content": "is a LLaMA 33B model finetuned with Reinforcement Learning from Human Feedback (RLHF)", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 406, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 505, + 419 + ], + "score": 1.0, + "content": "on the OASST1 dataset. Vicuna does full fine-tuning of LLaMA 13B on proprietary user-shared", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 416, + 477, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 477, + 430 + ], + "score": 1.0, + "content": "conversations from ShareGPT and is thus the result of distillation from OpenAI GPT models.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26, + "bbox_fs": [ + 105, + 373, + 506, + 430 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 435, + 505, + 511 + ], + "lines": [ + { + "bbox": [ + 105, + 433, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 506, + 448 + ], + "score": 1.0, + "content": "Evaluation setup Chatbot evaluation is not straightforward since each prompt has many high-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 445, + 507, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 507, + 459 + ], + "score": 1.0, + "content": "quality responses which are challenging to rank. We therefore follow a comprehensive approach,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 456, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 506, + 470 + ], + "score": 1.0, + "content": "which involves (a) standard benchmarks that measures general language understanding performance", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 468, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 505, + 480 + ], + "score": 1.0, + "content": "(MMLU [24]), (b) both automatic and human evaluation that measures how many responses from", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 478, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 505, + 491 + ], + "score": 1.0, + "content": "chatbot A are better compared to chatbot B, (c) a round-robin tournament-style evaluation where", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "score": 1.0, + "content": "chatbots compete against each other in games where chatbot performance is measured as ELO. We", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 500, + 407, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 407, + 512 + ], + "score": 1.0, + "content": "provide a detailed discussion of these evaluation setups in the Appendix E.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 433, + 507, + 512 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 519, + 422, + 530 + ], + "lines": [ + { + "bbox": [ + 104, + 518, + 423, + 533 + ], + "spans": [ + { + "bbox": [ + 104, + 518, + 423, + 533 + ], + "score": 1.0, + "content": "5.2 Guanaco: QLORA trained on OASST1 is a State-of-the-art Chatbot", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 534, + 505, + 588 + ], + "lines": [ + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "score": 1.0, + "content": "Based on our automated and human evaluations, we find that the top QLORA tuned model, Guanaco", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 544, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 505, + 558 + ], + "score": 1.0, + "content": "65B, which we finetune on a variant of OASST1, is the best-performing open-source chatbot model", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 555, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 505, + 568 + ], + "score": 1.0, + "content": "and offers performance competitive to ChatGPT. When compared to GPT-4, Guanaco 65B and 33B", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 567, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 252, + 579 + ], + "score": 1.0, + "content": "have an expected win probability of", + "type": "text" + }, + { + "bbox": [ + 253, + 567, + 272, + 577 + ], + "score": 0.87, + "content": "30 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 567, + 505, + 579 + ], + "score": 1.0, + "content": ", based on Elo rating from human annotators system-level", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 578, + 419, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 419, + 591 + ], + "score": 1.0, + "content": "pairwise comparisons on the Vicuna benchmark - the highest reported to date.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 39, + "bbox_fs": [ + 105, + 533, + 505, + 591 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 594, + 505, + 660 + ], + "lines": [ + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "The Vicuna benchmark [10] results relative to ChatGPT are shown in Table 4. We find that Guanaco", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 604, + 506, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 339, + 617 + ], + "score": 1.0, + "content": "65B is the best-performing model after GPT-4, achieving", + "type": "text" + }, + { + "bbox": [ + 339, + 605, + 367, + 615 + ], + "score": 0.86, + "content": "9 9 . 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 604, + 506, + 617 + ], + "score": 1.0, + "content": "performance relative to ChatGPT.", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 615, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 505, + 628 + ], + "score": 1.0, + "content": "Guanaco 33B has more parameters than the Vicuna 13B model, but uses only 4-bit precision for its", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 626, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 356, + 641 + ], + "score": 1.0, + "content": "weights and is thus much more memory efficient at 21 GB vs", + "type": "text" + }, + { + "bbox": [ + 356, + 627, + 383, + 637 + ], + "score": 0.28, + "content": "2 6 \\mathrm { G B }", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 626, + 505, + 641 + ], + "score": 1.0, + "content": ", providing a three percentage", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 637, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 506, + 650 + ], + "score": 1.0, + "content": "points of improvement over Vicuna 13B. Furthermore, Guanaco 7B easily fits on modern phones at a", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 648, + 454, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 454, + 661 + ], + "score": 1.0, + "content": "5 GB footprint while still scoring nearly 20 percentage points higher than Alpaca 13B.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 44.5, + "bbox_fs": [ + 105, + 594, + 506, + 661 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 665, + 505, + 720 + ], + "lines": [ + { + "bbox": [ + 106, + 665, + 506, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 665, + 506, + 677 + ], + "score": 1.0, + "content": "However, Table 4 also has very wide confidence intervals, with many models overlapping in per-", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 675, + 505, + 688 + ], + "spans": [ + { + "bbox": [ + 106, + 675, + 505, + 688 + ], + "score": 1.0, + "content": "formance. We hypothesize that this uncertainty comes from the lack of clear specification of scale,", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 686, + 505, + 699 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 505, + 699 + ], + "score": 1.0, + "content": "e.g., it is unclear what 8 on a 10 point scale means across different scenarios. As such, we instead", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "score": 1.0, + "content": "recommend using the Elo ranking method [16], based on pairwise judgments from human annotators", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 709, + 363, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 363, + 721 + ], + "score": 1.0, + "content": "and GPT-4 to avoid the problem of grounding an absolute scale.", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 50, + "bbox_fs": [ + 105, + 665, + 506, + 721 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 107, + 108, + 505, + 343 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 77, + 503, + 107 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 76, + 505, + 90 + ], + "spans": [ + { + "bbox": [ + 105, + 76, + 505, + 90 + ], + "score": 1.0, + "content": "Table 4: Zero-shot Vicuna benchmark scores as a percentage of the score obtained by ChatGPT evaluated by", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 86, + 505, + 99 + ], + "spans": [ + { + "bbox": [ + 105, + 86, + 505, + 99 + ], + "score": 1.0, + "content": "GPT-4. We see that OASST1 models perform close to ChatGPT despite being trained on a very small dataset", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 97, + 357, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 97, + 357, + 108 + ], + "score": 1.0, + "content": "and having a fraction of the memory requirement of baseline models.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "table_body", + "bbox": [ + 107, + 108, + 505, + 343 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 108, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 107, + 108, + 505, + 343 + ], + "score": 0.984, + "html": "
Model/DatasetParamsModel bitsMemoryChatGPT vs SysSys vs ChatGPTMean95% CI
GPT-4 Bard===119.4%110.1%114.5%2.6%
1--93.2%96.4%94.8%4.1%
Guanaco Alpaca FLAN v265B4-bit41 GB96.7%101.9%99.3%4.4%
65B4-bit41 GB63.0%77.9%70.7%4.3%
65B4-bit41 GB37.0%59.6%48.4%4.6%
Guanaco Open Assistant Alpaca FLAN v233B4-bit21 GB96.5%99.2%97.8%4.4%
33B16-bit66 GB73.4%85.7%78.1%5.3%
33B 33B4-bit 4-bit21GB 21GB67.2%79.7%73.6%4.2%
Vicuna Guanaco Alpaca HH-RLHF26.3%49.7%38.0%3.9%
13B 13B16-bit 4-bit26 GB 10 GB91.2% 87.3%98.7% 93.4%94.9% 90.4%4.5% 5.2%
13B4-bit10 GB63.8%76.7%69.4%
13B4-bit10 GB55.5%69.1%62.5%4.2%
13B4-bit10 GB50.6%69.8%60.5%4.7%
13B4-bit10 GB49.2%69.3%59.5%4.2% 4.7%
Longform Self-Instruct FLAN v213B4-bit10 GB44.9%62.0%53.6%5.2%
13B4-bit10 GB38.0%60.5%49.1%4.6%
13B4-bit10 GB32.4%61.2%47.0%3.6%
4-bit5GB
Guanaco Alpaca FLAN v27B 7B4-bit5GB84.1% 57.3%89.8% 71.2%87.0%5.4%
7B4-bit5GB33.3%56.1%64.4% 44.8%5.0% 4.0%
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We note that human and GPT-4 ranking", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 391, + 326, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 326, + 405 + ], + "score": 1.0, + "content": "of models on the Vicuna benchmark disagree partially,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 403, + 326, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 326, + 415 + ], + "score": 1.0, + "content": "particularly for Guanaco 7B, but are consistent for most", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 414, + 325, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 414, + 226, + 425 + ], + "score": 1.0, + "content": "models with a Kendall Tau of", + "type": "text" + }, + { + "bbox": [ + 227, + 414, + 265, + 424 + ], + "score": 0.88, + "content": "\\tau = 0 . 4 3", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 414, + 325, + 425 + ], + "score": 1.0, + "content": "and Spearman", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 424, + 326, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 186, + 435 + ], + "score": 1.0, + "content": "rank correlation of", + "type": "text" + }, + { + "bbox": [ + 186, + 425, + 226, + 435 + ], + "score": 0.89, + "content": "r = 0 . 5 5", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 424, + 326, + 435 + ], + "score": 1.0, + "content": "at the system level. At", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 436, + 325, + 446 + ], + "spans": [ + { + "bbox": [ + 106, + 436, + 325, + 446 + ], + "score": 1.0, + "content": "the example level, the agreement between GPT-4 and", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 447, + 325, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 325, + 458 + ], + "score": 1.0, + "content": "human annotators’ majority vote is weaker with Fleiss", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 457, + 325, + 469 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 145, + 468 + ], + "score": 0.87, + "content": "\\kappa = 0 . 2 5", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 457, + 325, + 469 + ], + "score": 1.0, + "content": ". Overall, this shows a moderate agreement", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 469, + 325, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 469, + 325, + 480 + ], + "score": 1.0, + "content": "between system-level judgments by GPT-4 and human", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 479, + 326, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 326, + 491 + ], + "score": 1.0, + "content": "annotators, and thus that model-based evaluation rep-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 491, + 326, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 326, + 501 + ], + "score": 1.0, + "content": "resents a somewhat reliable alternative to human evalu-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 501, + 296, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 296, + 513 + ], + "score": 1.0, + "content": "ation. See Section E for further considerations.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 12 + }, + { + "type": "table", + "bbox": [ + 334, + 398, + 503, + 500 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 332, + 361, + 504, + 392 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 331, + 361, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 331, + 361, + 505, + 371 + ], + "score": 1.0, + "content": "Table 5: MMLU 5-shot test results for different", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 331, + 370, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 331, + 370, + 505, + 383 + ], + "score": 1.0, + "content": "sizes of LLaMA finetuned on the corresponding", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 332, + 381, + 419, + 393 + ], + "spans": [ + { + "bbox": [ + 332, + 381, + 419, + 393 + ], + "score": 1.0, + "content": "datasets using QLoRA.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20 + }, + { + "type": "table_body", + "bbox": [ + 334, + 398, + 503, + 500 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 334, + 398, + 503, + 500 + ], + "spans": [ + { + "bbox": [ + 334, + 398, + 503, + 500 + ], + "score": 0.978, + "html": "
Dataset7B13B33B65B
LLaMA no tuning35.146.957.863.4
Self-Instruct36.433.353.056.7
Longform32.143.256.659.7
Chip234.541.653.659.8
HH-RLHF34.944.655.860.1
Unnatural Instruct41.948.157.361.3
Guanaco (OASST1)36.646.457.062.2
Alpaca38.847.857.362.5
FLAN v244.551.459.263.9
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We note that the Vicuna benchmark favors open-source models while the larger OA", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 555, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 506, + 569 + ], + "score": 1.0, + "content": "benchmark favors ChatGPT. Furthermore, we can see from Tables 5 and 4 that the suitability of a", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 567, + 506, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 506, + 580 + ], + "score": 1.0, + "content": "finetuning dataset is a determining factor in performance. Finetuning LLaMA models on FLAN", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 579, + 504, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 579, + 504, + 591 + ], + "score": 1.0, + "content": "v2 does particularly well on MMLU, but performs worst on the Vicuna benchmark (similar trends", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 589, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 505, + 603 + ], + "score": 1.0, + "content": "are observed with other models). This also points to partial orthogonality in current evaluation", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 599, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 613 + ], + "score": 1.0, + "content": "benchmarks: strong MMLU performance does not imply strong chatbot performance (as measured", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 610, + 293, + 625 + ], + "spans": [ + { + "bbox": [ + 106, + 610, + 293, + 625 + ], + "score": 1.0, + "content": "by Vicuna or OA benchmarks) and vice versa.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 634, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 634, + 505, + 647 + ], + "spans": [ + { + "bbox": [ + 106, + 634, + 505, + 647 + ], + "score": 1.0, + "content": "Guanaco is the only top model in our evaluation that is not trained on proprietary data as the OASST1", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 645, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 505, + 658 + ], + "score": 1.0, + "content": "dataset collection guidelines explicitly forbid the use of GPT models. The next best model trained", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 656, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 505, + 668 + ], + "score": 1.0, + "content": "on only open-source data is the Anthropic HH-RLHF model, which scores 30 percentage points", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 668, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 668, + 505, + 678 + ], + "score": 1.0, + "content": "lower than Guanaco on the Vicuna benchmark (see Table 4). Overall, these results show that 4-bit", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 678, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 506, + 690 + ], + "score": 1.0, + "content": "QLORA is effective and can produce state-of-the-art chatbots that rival ChatGPT. Furthermore, our", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 688, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 702 + ], + "score": 1.0, + "content": "33B Guanaco can be trained on 24 GB consumer GPUs in less than 12 hours. 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Model/DatasetParamsModel bitsMemoryChatGPT vs SysSys vs ChatGPTMean95% CI
GPT-4 Bard===119.4%110.1%114.5%2.6%
1--93.2%96.4%94.8%4.1%
Guanaco Alpaca FLAN v265B4-bit41 GB96.7%101.9%99.3%4.4%
65B4-bit41 GB63.0%77.9%70.7%4.3%
65B4-bit41 GB37.0%59.6%48.4%4.6%
Guanaco Open Assistant Alpaca FLAN v233B4-bit21 GB96.5%99.2%97.8%4.4%
33B16-bit66 GB73.4%85.7%78.1%5.3%
33B 33B4-bit 4-bit21GB 21GB67.2%79.7%73.6%4.2%
Vicuna Guanaco Alpaca HH-RLHF26.3%49.7%38.0%3.9%
13B 13B16-bit 4-bit26 GB 10 GB91.2% 87.3%98.7% 93.4%94.9% 90.4%4.5% 5.2%
13B4-bit10 GB63.8%76.7%69.4%
13B4-bit10 GB55.5%69.1%62.5%4.2%
13B4-bit10 GB50.6%69.8%60.5%4.7%
13B4-bit10 GB49.2%69.3%59.5%4.2% 4.7%
Longform Self-Instruct FLAN v213B4-bit10 GB44.9%62.0%53.6%5.2%
13B4-bit10 GB38.0%60.5%49.1%4.6%
13B4-bit10 GB32.4%61.2%47.0%3.6%
4-bit5GB
Guanaco Alpaca FLAN v27B 7B4-bit5GB84.1% 57.3%89.8% 71.2%87.0%5.4%
7B4-bit5GB33.3%56.1%64.4% 44.8%5.0% 4.0%
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We note that human and GPT-4 ranking", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 391, + 326, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 326, + 405 + ], + "score": 1.0, + "content": "of models on the Vicuna benchmark disagree partially,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 403, + 326, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 326, + 415 + ], + "score": 1.0, + "content": "particularly for Guanaco 7B, but are consistent for most", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 414, + 325, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 414, + 226, + 425 + ], + "score": 1.0, + "content": "models with a Kendall Tau of", + "type": "text" + }, + { + "bbox": [ + 227, + 414, + 265, + 424 + ], + "score": 0.88, + "content": "\\tau = 0 . 4 3", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 414, + 325, + 425 + ], + "score": 1.0, + "content": "and Spearman", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 424, + 326, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 186, + 435 + ], + "score": 1.0, + "content": "rank correlation of", + "type": "text" + }, + { + "bbox": [ + 186, + 425, + 226, + 435 + ], + "score": 0.89, + "content": "r = 0 . 5 5", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 424, + 326, + 435 + ], + "score": 1.0, + "content": "at the system level. At", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 436, + 325, + 446 + ], + "spans": [ + { + "bbox": [ + 106, + 436, + 325, + 446 + ], + "score": 1.0, + "content": "the example level, the agreement between GPT-4 and", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 447, + 325, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 325, + 458 + ], + "score": 1.0, + "content": "human annotators’ majority vote is weaker with Fleiss", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 457, + 325, + 469 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 145, + 468 + ], + "score": 0.87, + "content": "\\kappa = 0 . 2 5", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 457, + 325, + 469 + ], + "score": 1.0, + "content": ". 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See Section E for further considerations.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 12, + "bbox_fs": [ + 105, + 370, + 326, + 513 + ] + }, + { + "type": "table", + "bbox": [ + 334, + 398, + 503, + 500 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 332, + 361, + 504, + 392 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 331, + 361, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 331, + 361, + 505, + 371 + ], + "score": 1.0, + "content": "Table 5: MMLU 5-shot test results for different", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 331, + 370, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 331, + 370, + 505, + 383 + ], + "score": 1.0, + "content": "sizes of LLaMA finetuned on the corresponding", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 332, + 381, + 419, + 393 + ], + "spans": [ + { + "bbox": [ + 332, + 381, + 419, + 393 + ], + "score": 1.0, + "content": "datasets using QLoRA.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20 + }, + { + "type": "table_body", + "bbox": [ + 334, + 398, + 503, + 500 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 334, + 398, + 503, + 500 + ], + "spans": [ + { + "bbox": [ + 334, + 398, + 503, + 500 + ], + "score": 0.978, + "html": "
Dataset7B13B33B65B
LLaMA no tuning35.146.957.863.4
Self-Instruct36.433.353.056.7
Longform32.143.256.659.7
Chip234.541.653.659.8
HH-RLHF34.944.655.860.1
Unnatural Instruct41.948.157.361.3
Guanaco (OASST1)36.646.457.062.2
Alpaca38.847.857.362.5
FLAN v244.551.459.263.9
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Benchmark # Prompts Judge ModelVicuna 80 Human ratersVicuna 80 GPT-4Open Assistant 953 GPT-4Median Rank
EloRankEloRankEloRank
GPT-41176113481129411
Guanaco-65B102321022100832
Guanaco-33B10094992100244
ChatGPT-3.5 Turbo9167966235101525
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Major approaches for preserving 16-bit LLM quality focus on managing", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 329, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 505, + 340 + ], + "score": 1.0, + "content": "outlier features (e.g., SmoothQuant [67] and LLM.int8() [14]) while others use more sophisticated", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 340, + 507, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 507, + 352 + ], + "score": 1.0, + "content": "grouping methods [44, 70]. Lossy quantization approaches study the trade-offs for regular round-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 349, + 506, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 506, + 363 + ], + "score": 1.0, + "content": "ing [13, 72, 48] or how to optimize rounding decisions to improve quantization precision [18].", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 361, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 505, + 374 + ], + "score": 1.0, + "content": "Besides our work, SwitchBack layers [66] is the only work that studies backpropagation through", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 373, + 314, + 385 + ], + "spans": [ + { + "bbox": [ + 106, + 373, + 314, + 385 + ], + "score": 1.0, + "content": "quantized weights at a scale beyond 1B parameters.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 392, + 505, + 457 + ], + "lines": [ + { + "bbox": [ + 105, + 392, + 506, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 506, + 404 + ], + "score": 1.0, + "content": "Finetuning with Adapters While we use Low-rank Adapters [28] (LoRA), many other Parameter", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 401, + 506, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 506, + 416 + ], + "score": 1.0, + "content": "Efficient FineTuning (PEFT) methods have been proposed such as prompt tuning [49, 33, 34], tuning", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 412, + 506, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 328, + 428 + ], + "score": 1.0, + "content": "the embedding layer inputs [1], tuning hidden states", + "type": "text" + }, + { + "bbox": [ + 329, + 413, + 352, + 425 + ], + "score": 0.76, + "content": "\\mathrm { ( I A ^ { 3 } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 412, + 506, + 428 + ], + "score": 1.0, + "content": "[37], adding full layers [27], tuning", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 425, + 506, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 506, + 437 + ], + "score": 1.0, + "content": "biases [71], learning a mask over weights based on Fisher information [55], and a combination of", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 435, + 505, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 505, + 449 + ], + "score": 1.0, + "content": "approaches [23]. In our work, we show that LoRA adapters are able to reach full 16-bit finetuning", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 446, + 473, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 473, + 459 + ], + "score": 1.0, + "content": "performance. We leave it to future work to explore the tradeoffs of other PEFT approaches.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 107, + 465, + 505, + 532 + ], + "lines": [ + { + "bbox": [ + 105, + 465, + 506, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 506, + 479 + ], + "score": 1.0, + "content": "Instruction Finetuning To help a pretrained LLM follow the instructions provided in a prompt,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 477, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 505, + 488 + ], + "score": 1.0, + "content": "instruction finetuning uses input-output pairs of various data sources to finetune a pretrained LLM", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 488, + 506, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 506, + 501 + ], + "score": 1.0, + "content": "to generate the output given the input as a prompt. Approaches and datasets include MetaICL [40],", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 498, + 506, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 506, + 511 + ], + "score": 1.0, + "content": "MetaTuning [74], InstructGPT [43], FLAN [63, 12], PromptSource [3], Super-NaturalInstructions [62,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 509, + 507, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 507, + 523 + ], + "score": 1.0, + "content": "51], Self-instruct [60], UnnaturalInstructions [26], OPT-IML [29], UnifiedSKG[68], OIG/Chip2 [32],", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 520, + 378, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 378, + 533 + ], + "score": 1.0, + "content": "Alpaca [56], Vicuna [10], Koala [20], and Self-instruct-GPT-4 [45].", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 107, + 540, + 505, + 628 + ], + "lines": [ + { + "bbox": [ + 106, + 538, + 506, + 554 + ], + "spans": [ + { + "bbox": [ + 106, + 538, + 506, + 554 + ], + "score": 1.0, + "content": "Chatbots Many instruction following models are structured as dialogue-based chatbots, often using", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "Reinforcement Learning from Human Feedback (RLHF) [11] or generating data from an existing", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 561, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 505, + 574 + ], + "score": 1.0, + "content": "model to train with AI model feedback (RLAIF) [5]. Approaches and datasets include Anthropic-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 573, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 573, + 506, + 585 + ], + "score": 1.0, + "content": "HH [2, 4], Open Assistant [31], LaMDA [57], and Sparrow [21]. We do not use reinforcement", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 583, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 505, + 596 + ], + "score": 1.0, + "content": "learning, but our best model, Guanaco, is finetuned on multi-turn chat interactions from the Open", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 595, + 506, + 607 + ], + "spans": [ + { + "bbox": [ + 106, + 595, + 506, + 607 + ], + "score": 1.0, + "content": "Assistant dataset which was designed to be used for RLHF training [31]. For the evaluation of", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 606, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 106, + 606, + 506, + 618 + ], + "score": 1.0, + "content": "chatbots approaches that use GPT-4 instead of costly human annotation have been developed [10, 45].", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 617, + 465, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 617, + 465, + 628 + ], + "score": 1.0, + "content": "We improve on such approaches with a focus on an evaluation setup that is more reliable.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 30.5 + }, + { + "type": "title", + "bbox": [ + 108, + 639, + 264, + 653 + ], + "lines": [ + { + "bbox": [ + 105, + 637, + 265, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 265, + 655 + ], + "score": 1.0, + "content": "7 Limitations and Discussion", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 108, + 660, + 505, + 693 + ], + "lines": [ + { + "bbox": [ + 105, + 659, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 505, + 673 + ], + "score": 1.0, + "content": "We have shown evidence that our method, QLORA, can replicate 16-bit full finetuning performance", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 671, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 505, + 683 + ], + "score": 1.0, + "content": "with a 4-bit base model and Low-rank Adapters. However, due to the immense resource cost, we were", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 682, + 473, + 694 + ], + "spans": [ + { + "bbox": [ + 106, + 682, + 473, + 694 + ], + "score": 1.0, + "content": "not establish that QLORA can match 16-bit finetuning performance at 33B and 65B scales.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 699, + 503, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 699, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 713 + ], + "score": 1.0, + "content": "Another limitation is the evaluation of instruction finetuning models. While we provide evaluations", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 710, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 723 + ], + "score": 1.0, + "content": "on MMLU, the Vicuna/OA benchmark, we did not evaluate on BigBench, RAFT, and HELM, and it", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39.5 + } + ], + "page_idx": 8, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "spans": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "score": 1.0, + "content": "9", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 123, + 118, + 487, + 266 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 77, + 505, + 117 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 77, + 505, + 89 + ], + "spans": [ + { + "bbox": [ + 105, + 77, + 505, + 89 + ], + "score": 1.0, + "content": "Table 6: Elo rating for a tournament between models where models compete to generate the best response", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 88, + 505, + 98 + ], + "spans": [ + { + "bbox": [ + 106, + 88, + 505, + 98 + ], + "score": 1.0, + "content": "for a prompt, judged by human raters or GPT-4. Overall, Guanaco 65B and 33B tend to be preferred to", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 96, + 505, + 109 + ], + "spans": [ + { + "bbox": [ + 106, + 96, + 505, + 109 + ], + "score": 1.0, + "content": "ChatGPT-3.5 on the benchmarks studied. According to human raters they have a Each 10-point difference in Elo", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 104, + 107, + 288, + 119 + ], + "spans": [ + { + "bbox": [ + 104, + 107, + 223, + 119 + ], + "score": 1.0, + "content": "is approximately a difference of", + "type": "text" + }, + { + "bbox": [ + 223, + 108, + 243, + 117 + ], + "score": 0.84, + "content": "1 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 107, + 288, + 119 + ], + "score": 1.0, + "content": "in win-rate.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "table_body", + "bbox": [ + 123, + 118, + 487, + 266 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 123, + 118, + 487, + 266 + ], + "spans": [ + { + "bbox": [ + 123, + 118, + 487, + 266 + ], + "score": 0.984, + "html": "
Benchmark # Prompts Judge ModelVicuna 80 Human ratersVicuna 80 GPT-4Open Assistant 953 GPT-4Median Rank
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We leave it to future work to explore the tradeoffs of other PEFT approaches.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 392, + 506, + 459 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 465, + 505, + 532 + ], + "lines": [ + { + "bbox": [ + 105, + 465, + 506, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 506, + 479 + ], + "score": 1.0, + "content": "Instruction Finetuning To help a pretrained LLM follow the instructions provided in a prompt,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 477, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 505, + 488 + ], + "score": 1.0, + "content": "instruction finetuning uses input-output pairs of various data sources to finetune a pretrained LLM", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 488, + 506, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 506, + 501 + ], + "score": 1.0, + "content": "to generate the output given the input as a prompt. Approaches and datasets include MetaICL [40],", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 498, + 506, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 506, + 511 + ], + "score": 1.0, + "content": "MetaTuning [74], InstructGPT [43], FLAN [63, 12], PromptSource [3], Super-NaturalInstructions [62,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 509, + 507, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 507, + 523 + ], + "score": 1.0, + "content": "51], Self-instruct [60], UnnaturalInstructions [26], OPT-IML [29], UnifiedSKG[68], OIG/Chip2 [32],", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 520, + 378, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 378, + 533 + ], + "score": 1.0, + "content": "Alpaca [56], Vicuna [10], Koala [20], and Self-instruct-GPT-4 [45].", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 465, + 507, + 533 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 540, + 505, + 628 + ], + "lines": [ + { + "bbox": [ + 106, + 538, + 506, + 554 + ], + "spans": [ + { + "bbox": [ + 106, + 538, + 506, + 554 + ], + "score": 1.0, + "content": "Chatbots Many instruction following models are structured as dialogue-based chatbots, often using", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "Reinforcement Learning from Human Feedback (RLHF) [11] or generating data from an existing", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 561, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 505, + 574 + ], + "score": 1.0, + "content": "model to train with AI model feedback (RLAIF) [5]. Approaches and datasets include Anthropic-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 573, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 573, + 506, + 585 + ], + "score": 1.0, + "content": "HH [2, 4], Open Assistant [31], LaMDA [57], and Sparrow [21]. We do not use reinforcement", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 583, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 505, + 596 + ], + "score": 1.0, + "content": "learning, but our best model, Guanaco, is finetuned on multi-turn chat interactions from the Open", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 595, + 506, + 607 + ], + "spans": [ + { + "bbox": [ + 106, + 595, + 506, + 607 + ], + "score": 1.0, + "content": "Assistant dataset which was designed to be used for RLHF training [31]. For the evaluation of", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 606, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 106, + 606, + 506, + 618 + ], + "score": 1.0, + "content": "chatbots approaches that use GPT-4 instead of costly human annotation have been developed [10, 45].", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 617, + 465, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 617, + 465, + 628 + ], + "score": 1.0, + "content": "We improve on such approaches with a focus on an evaluation setup that is more reliable.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 538, + 506, + 628 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 639, + 264, + 653 + ], + "lines": [ + { + "bbox": [ + 105, + 637, + 265, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 265, + 655 + ], + "score": 1.0, + "content": "7 Limitations and Discussion", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 108, + 660, + 505, + 693 + ], + "lines": [ + { + "bbox": [ + 105, + 659, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 505, + 673 + ], + "score": 1.0, + "content": "We have shown evidence that our method, QLORA, can replicate 16-bit full finetuning performance", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 671, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 505, + 683 + ], + "score": 1.0, + "content": "with a 4-bit base model and Low-rank Adapters. However, due to the immense resource cost, we were", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 682, + 473, + 694 + ], + "spans": [ + { + "bbox": [ + 106, + 682, + 473, + 694 + ], + "score": 1.0, + "content": "not establish that QLORA can match 16-bit finetuning performance at 33B and 65B scales.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 659, + 505, + 694 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 699, + 503, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 699, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 713 + ], + "score": 1.0, + "content": "Another limitation is the evaluation of instruction finetuning models. While we provide evaluations", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 710, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 723 + ], + "score": 1.0, + "content": "on MMLU, the Vicuna/OA benchmark, we did not evaluate on BigBench, RAFT, and HELM, and it", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 72, + 505, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 505, + 86 + ], + "score": 1.0, + "content": "is not ensured that our evaluations generalize to these benchmarks. On the other hand, we perform a", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 84, + 423, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 84, + 423, + 96 + ], + "score": 1.0, + "content": "very broad study on MMLU and develop new methods for evaluating chatbots.", + "type": "text", + "cross_page": true + } + ], + "index": 1 + } + ], + "index": 39.5, + "bbox_fs": [ + 105, + 699, + 506, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 73, + 504, + 95 + ], + "lines": [ + { + "bbox": [ + 105, + 72, + 505, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 505, + 86 + ], + "score": 1.0, + "content": "is not ensured that our evaluations generalize to these benchmarks. On the other hand, we perform a", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 84, + 423, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 84, + 423, + 96 + ], + "score": 1.0, + "content": "very broad study on MMLU and develop new methods for evaluating chatbots.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 107, + 100, + 505, + 198 + ], + "lines": [ + { + "bbox": [ + 105, + 100, + 506, + 113 + ], + "spans": [ + { + "bbox": [ + 105, + 100, + 506, + 113 + ], + "score": 1.0, + "content": "From the evidence presented, it appears that the performance of these benchmarks likely depends how", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 111, + 506, + 123 + ], + "spans": [ + { + "bbox": [ + 106, + 111, + 506, + 123 + ], + "score": 1.0, + "content": "similar the finetuning data is to the benchmark dataset. For example, FLAN v2 is similar to MMLU,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 121, + 506, + 134 + ], + "spans": [ + { + "bbox": [ + 106, + 121, + 506, + 134 + ], + "score": 1.0, + "content": "but dissimilar to the Vicuna benchmark and vice versa for the Chip2 dataset. This highlights that not", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 132, + 506, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 506, + 145 + ], + "score": 1.0, + "content": "only better benchmarks and evaluation is needed, but that one needs to be careful about what one is", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 144, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 505, + 156 + ], + "score": 1.0, + "content": "evaluating in the first place. Do we want to create models that do well on classroom highschool and", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 154, + 505, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 505, + 168 + ], + "score": 1.0, + "content": "colleague knowledge or do we want to do well on chatbot conversation ability? Maybe something", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 164, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 505, + 178 + ], + "score": 1.0, + "content": "else? Because it is always easier to evaluate on an existing benchmark compared to creating a new", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 176, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 506, + 189 + ], + "score": 1.0, + "content": "one, certain benchmarks can steer the community towards a certain direction. We should ensure as a", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 188, + 354, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 188, + 354, + 199 + ], + "score": 1.0, + "content": "community that the benchmarks measure what we care about.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 107, + 204, + 505, + 280 + ], + "lines": [ + { + "bbox": [ + 106, + 204, + 506, + 216 + ], + "spans": [ + { + "bbox": [ + 106, + 204, + 506, + 216 + ], + "score": 1.0, + "content": "An additional limitation is that we did not evaluate different bit-precisions or different adapter", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 215, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 215, + 505, + 227 + ], + "score": 1.0, + "content": "methods. Besides LoRA, there is also a wide variety Parameter Efficient FineTuning (PEFT) methods", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 225, + 506, + 237 + ], + "spans": [ + { + "bbox": [ + 106, + 225, + 506, + 237 + ], + "score": 1.0, + "content": "that have been shown to work well. However, it is unclear if these methods scale to large models.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 237, + 506, + 248 + ], + "spans": [ + { + "bbox": [ + 106, + 237, + 506, + 248 + ], + "score": 1.0, + "content": "We used LoRA as many results established its robustness but other adapters might yield better", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 248, + 506, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 506, + 259 + ], + "score": 1.0, + "content": "performance. Since finetuning after quantization seems to recover most of the information that is lost", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 257, + 506, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 506, + 271 + ], + "score": 1.0, + "content": "during quantization this might enable much more aggressive quantization. For example, 3-bit GPTQ", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 268, + 430, + 283 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 430, + 283 + ], + "score": 1.0, + "content": "quantization might also yield 16-bit full finetuning performance after finetuning.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 14 + }, + { + "type": "title", + "bbox": [ + 107, + 290, + 212, + 304 + ], + "lines": [ + { + "bbox": [ + 104, + 288, + 214, + 306 + ], + "spans": [ + { + "bbox": [ + 104, + 288, + 214, + 306 + ], + "score": 1.0, + "content": "8 Broader Impacts", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 309, + 505, + 408 + ], + "lines": [ + { + "bbox": [ + 106, + 309, + 506, + 322 + ], + "spans": [ + { + "bbox": [ + 106, + 309, + 506, + 322 + ], + "score": 1.0, + "content": "Our QLORA finetuning method is the first method that enables the finetuning of 33B parameter", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 321, + 505, + 333 + ], + "spans": [ + { + "bbox": [ + 106, + 321, + 505, + 333 + ], + "score": 1.0, + "content": "models on a single consumer GPU and 65B parameter models on a single professional GPU, while", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 331, + 506, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 506, + 344 + ], + "score": 1.0, + "content": "not degrading performance relative to a full finetuning baseline. We have demonstrated that our", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 341, + 506, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 506, + 355 + ], + "score": 1.0, + "content": "best 33B model trained on the Open Assistant dataset can rival ChatGPT on the Vicuna benchmark.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 353, + 506, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 506, + 367 + ], + "score": 1.0, + "content": "Since instruction finetuning is an essential tool to transform raw pretrained LLMs into ChatGPT-like", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 363, + 506, + 377 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 506, + 377 + ], + "score": 1.0, + "content": "chatbots, we believe that our method will make finetuning widespread and common in particular for", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 375, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 505, + 387 + ], + "score": 1.0, + "content": "the researchers that have the least resources – a big win for the accessibility of state of the art NLP", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 386, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 505, + 399 + ], + "score": 1.0, + "content": "technology. QLORA can be seen as an equalizing factor that helps to close the resource gap between", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 397, + 338, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 338, + 409 + ], + "score": 1.0, + "content": "large corporations and small teams with consumer GPUs.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 413, + 505, + 479 + ], + "lines": [ + { + "bbox": [ + 106, + 413, + 506, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 413, + 506, + 426 + ], + "score": 1.0, + "content": "Another potential source of impact is deployment to mobile phones and other low resources devices.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 424, + 506, + 437 + ], + "spans": [ + { + "bbox": [ + 106, + 424, + 506, + 437 + ], + "score": 1.0, + "content": "While 7B models were shown to be able to be run on phones before, QLORA is the first method", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 435, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 106, + 435, + 505, + 447 + ], + "score": 1.0, + "content": "that would enable the finetuning of such models. We estimate that with an iPhone 12 Plus, QLORA", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 446, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 106, + 446, + 505, + 459 + ], + "score": 1.0, + "content": "can finetune 3 million tokens per night while the phone is charging. QLORA can help enable", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 457, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 506, + 470 + ], + "score": 1.0, + "content": "privacy-preserving usage of LLMs, where users can own and manage their own data and models,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 468, + 320, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 320, + 480 + ], + "score": 1.0, + "content": "while simultaneously making LLMs easier to deploy.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30.5 + }, + { + "type": "text", + "bbox": [ + 107, + 484, + 505, + 528 + ], + "lines": [ + { + "bbox": [ + 105, + 483, + 506, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 506, + 497 + ], + "score": 1.0, + "content": "Furthermore, because of the increased inference efficiency of 4-bit models, if QLoRA models are", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 496, + 506, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 506, + 507 + ], + "score": 1.0, + "content": "deployed they reduce the environmental impact that LLMs have when deployed for personal use.", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 505, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 184, + 519 + ], + "score": 1.0, + "content": "We estimate that if", + "type": "text" + }, + { + "bbox": [ + 184, + 506, + 204, + 517 + ], + "score": 0.89, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 505, + 338, + 519 + ], + "score": 1.0, + "content": "of deployments are personal and", + "type": "text" + }, + { + "bbox": [ + 339, + 506, + 358, + 516 + ], + "score": 0.87, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 505, + 506, + 519 + ], + "score": 1.0, + "content": "are company deployments, QLoRA", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 517, + 433, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 337, + 529 + ], + "score": 1.0, + "content": "deployments could reduce the overall carbon footprint by", + "type": "text" + }, + { + "bbox": [ + 338, + 518, + 357, + 528 + ], + "score": 0.84, + "content": "72 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 517, + 433, + 529 + ], + "score": 1.0, + "content": "(see Appendix B).", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 35.5 + }, + { + "type": "text", + "bbox": [ + 107, + 533, + 505, + 577 + ], + "lines": [ + { + "bbox": [ + 105, + 533, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 506, + 546 + ], + "score": 1.0, + "content": "However, finetuning is a dual-use technology that can be abused to cause harm. Widespread use of", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 544, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 505, + 557 + ], + "score": 1.0, + "content": "LLMs has known dangers [8, 6], but we believe that equalizing access to a technology that is quickly", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 555, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 506, + 568 + ], + "score": 1.0, + "content": "becoming ubiquitous will allow for better more independent analysis than keeping the power of LLMs", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 566, + 461, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 461, + 579 + ], + "score": 1.0, + "content": "in the hands of large corporations that do not release models or source code for auditing.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 39.5 + }, + { + "type": "text", + "bbox": [ + 108, + 583, + 504, + 605 + ], + "lines": [ + { + "bbox": [ + 106, + 582, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 505, + 596 + ], + "score": 1.0, + "content": "All in all, we believe that QLORA will have a broadly positive impact making the finetuning of high", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 594, + 329, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 329, + 606 + ], + "score": 1.0, + "content": "quality LLMs much more widely and easily accessible.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 42.5 + }, + { + "type": "title", + "bbox": [ + 108, + 615, + 207, + 628 + ], + "lines": [ + { + "bbox": [ + 105, + 613, + 208, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 208, + 631 + ], + "score": 1.0, + "content": "Acknowledgements", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 44 + }, + { + "type": "text", + "bbox": [ + 107, + 634, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 635, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 635, + 505, + 646 + ], + "score": 1.0, + "content": "We thank Aditya Kusupati, Ofir Press, Ashish Sharma, Margaret Li, Raphael Olivier, Zihao Ye, and", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 645, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 645, + 505, + 657 + ], + "score": 1.0, + "content": "Evangelia Spiliopoulou for their valuable feedback. We thank Norah Altriri for providing us with an", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "score": 1.0, + "content": "intuitive NF4 visualization. We thank Felix Petersen for a discussion on distributional quantization and", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 667, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 506, + 679 + ], + "score": 1.0, + "content": "its relationship to quantile quantization. Our research was facilitated by the advanced computational,", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 678, + 507, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 507, + 691 + ], + "score": 1.0, + "content": "storage, and networking infrastructure of the Hyak supercomputer system at the University of", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 688, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 506, + 702 + ], + "score": 1.0, + "content": "Washington. We thank the Hyak team for ensuring a smooth operation. We thank the beta testers of", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 700, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 506, + 712 + ], + "score": 1.0, + "content": "the bitsandbytes library, in particular Alex Birch and Alyssa Vance. We thank Younes Belkada for", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 711, + 439, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 439, + 723 + ], + "score": 1.0, + "content": "help with the integration of our software into the Hugging Face transformers stack.", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 48.5 + } + ], + "page_idx": 9, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 301, + 741, + 311, + 750 + ], + "lines": [ + { + "bbox": [ + 299, + 740, + 313, + 755 + ], + "spans": [ + { + "bbox": [ + 299, + 740, + 313, + 755 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 14 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 73, + 504, + 95 + ], + "lines": [], + "index": 0.5, + "bbox_fs": [ + 105, + 72, + 505, + 96 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 100, + 505, + 198 + ], + "lines": [ + { + "bbox": [ + 105, + 100, + 506, + 113 + ], + "spans": [ + { + "bbox": [ + 105, + 100, + 506, + 113 + ], + "score": 1.0, + "content": "From the evidence presented, it appears that the performance of these benchmarks likely depends how", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 111, + 506, + 123 + ], + "spans": [ + { + "bbox": [ + 106, + 111, + 506, + 123 + ], + "score": 1.0, + "content": "similar the finetuning data is to the benchmark dataset. For example, FLAN v2 is similar to MMLU,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 121, + 506, + 134 + ], + "spans": [ + { + "bbox": [ + 106, + 121, + 506, + 134 + ], + "score": 1.0, + "content": "but dissimilar to the Vicuna benchmark and vice versa for the Chip2 dataset. This highlights that not", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 132, + 506, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 506, + 145 + ], + "score": 1.0, + "content": "only better benchmarks and evaluation is needed, but that one needs to be careful about what one is", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 144, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 505, + 156 + ], + "score": 1.0, + "content": "evaluating in the first place. Do we want to create models that do well on classroom highschool and", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 154, + 505, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 505, + 168 + ], + "score": 1.0, + "content": "colleague knowledge or do we want to do well on chatbot conversation ability? Maybe something", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 164, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 505, + 178 + ], + "score": 1.0, + "content": "else? Because it is always easier to evaluate on an existing benchmark compared to creating a new", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 176, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 506, + 189 + ], + "score": 1.0, + "content": "one, certain benchmarks can steer the community towards a certain direction. We should ensure as a", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 188, + 354, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 188, + 354, + 199 + ], + "score": 1.0, + "content": "community that the benchmarks measure what we care about.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 6, + "bbox_fs": [ + 105, + 100, + 506, + 199 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 204, + 505, + 280 + ], + "lines": [ + { + "bbox": [ + 106, + 204, + 506, + 216 + ], + "spans": [ + { + "bbox": [ + 106, + 204, + 506, + 216 + ], + "score": 1.0, + "content": "An additional limitation is that we did not evaluate different bit-precisions or different adapter", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 215, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 215, + 505, + 227 + ], + "score": 1.0, + "content": "methods. Besides LoRA, there is also a wide variety Parameter Efficient FineTuning (PEFT) methods", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 225, + 506, + 237 + ], + "spans": [ + { + "bbox": [ + 106, + 225, + 506, + 237 + ], + "score": 1.0, + "content": "that have been shown to work well. However, it is unclear if these methods scale to large models.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 237, + 506, + 248 + ], + "spans": [ + { + "bbox": [ + 106, + 237, + 506, + 248 + ], + "score": 1.0, + "content": "We used LoRA as many results established its robustness but other adapters might yield better", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 248, + 506, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 506, + 259 + ], + "score": 1.0, + "content": "performance. Since finetuning after quantization seems to recover most of the information that is lost", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 257, + 506, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 506, + 271 + ], + "score": 1.0, + "content": "during quantization this might enable much more aggressive quantization. For example, 3-bit GPTQ", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 268, + 430, + 283 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 430, + 283 + ], + "score": 1.0, + "content": "quantization might also yield 16-bit full finetuning performance after finetuning.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 14, + "bbox_fs": [ + 105, + 204, + 506, + 283 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 290, + 212, + 304 + ], + "lines": [ + { + "bbox": [ + 104, + 288, + 214, + 306 + ], + "spans": [ + { + "bbox": [ + 104, + 288, + 214, + 306 + ], + "score": 1.0, + "content": "8 Broader Impacts", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 309, + 505, + 408 + ], + "lines": [ + { + "bbox": [ + 106, + 309, + 506, + 322 + ], + "spans": [ + { + "bbox": [ + 106, + 309, + 506, + 322 + ], + "score": 1.0, + "content": "Our QLORA finetuning method is the first method that enables the finetuning of 33B parameter", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 321, + 505, + 333 + ], + "spans": [ + { + "bbox": [ + 106, + 321, + 505, + 333 + ], + "score": 1.0, + "content": "models on a single consumer GPU and 65B parameter models on a single professional GPU, while", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 331, + 506, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 506, + 344 + ], + "score": 1.0, + "content": "not degrading performance relative to a full finetuning baseline. We have demonstrated that our", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 341, + 506, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 506, + 355 + ], + "score": 1.0, + "content": "best 33B model trained on the Open Assistant dataset can rival ChatGPT on the Vicuna benchmark.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 353, + 506, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 506, + 367 + ], + "score": 1.0, + "content": "Since instruction finetuning is an essential tool to transform raw pretrained LLMs into ChatGPT-like", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 363, + 506, + 377 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 506, + 377 + ], + "score": 1.0, + "content": "chatbots, we believe that our method will make finetuning widespread and common in particular for", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 375, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 505, + 387 + ], + "score": 1.0, + "content": "the researchers that have the least resources – a big win for the accessibility of state of the art NLP", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 386, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 505, + 399 + ], + "score": 1.0, + "content": "technology. QLORA can be seen as an equalizing factor that helps to close the resource gap between", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 397, + 338, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 338, + 409 + ], + "score": 1.0, + "content": "large corporations and small teams with consumer GPUs.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 309, + 506, + 409 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 413, + 505, + 479 + ], + "lines": [ + { + "bbox": [ + 106, + 413, + 506, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 413, + 506, + 426 + ], + "score": 1.0, + "content": "Another potential source of impact is deployment to mobile phones and other low resources devices.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 424, + 506, + 437 + ], + "spans": [ + { + "bbox": [ + 106, + 424, + 506, + 437 + ], + "score": 1.0, + "content": "While 7B models were shown to be able to be run on phones before, QLORA is the first method", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 435, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 106, + 435, + 505, + 447 + ], + "score": 1.0, + "content": "that would enable the finetuning of such models. We estimate that with an iPhone 12 Plus, QLORA", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 446, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 106, + 446, + 505, + 459 + ], + "score": 1.0, + "content": "can finetune 3 million tokens per night while the phone is charging. QLORA can help enable", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 457, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 506, + 470 + ], + "score": 1.0, + "content": "privacy-preserving usage of LLMs, where users can own and manage their own data and models,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 468, + 320, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 320, + 480 + ], + "score": 1.0, + "content": "while simultaneously making LLMs easier to deploy.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 413, + 506, + 480 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 484, + 505, + 528 + ], + "lines": [ + { + "bbox": [ + 105, + 483, + 506, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 506, + 497 + ], + "score": 1.0, + "content": "Furthermore, because of the increased inference efficiency of 4-bit models, if QLoRA models are", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 496, + 506, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 506, + 507 + ], + "score": 1.0, + "content": "deployed they reduce the environmental impact that LLMs have when deployed for personal use.", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 505, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 184, + 519 + ], + "score": 1.0, + "content": "We estimate that if", + "type": "text" + }, + { + "bbox": [ + 184, + 506, + 204, + 517 + ], + "score": 0.89, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 505, + 338, + 519 + ], + "score": 1.0, + "content": "of deployments are personal and", + "type": "text" + }, + { + "bbox": [ + 339, + 506, + 358, + 516 + ], + "score": 0.87, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 505, + 506, + 519 + ], + "score": 1.0, + "content": "are company deployments, QLoRA", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 517, + 433, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 337, + 529 + ], + "score": 1.0, + "content": "deployments could reduce the overall carbon footprint by", + "type": "text" + }, + { + "bbox": [ + 338, + 518, + 357, + 528 + ], + "score": 0.84, + "content": "72 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 517, + 433, + 529 + ], + "score": 1.0, + "content": "(see Appendix B).", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 35.5, + "bbox_fs": [ + 105, + 483, + 506, + 529 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 533, + 505, + 577 + ], + "lines": [ + { + "bbox": [ + 105, + 533, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 506, + 546 + ], + "score": 1.0, + "content": "However, finetuning is a dual-use technology that can be abused to cause harm. Widespread use of", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 544, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 505, + 557 + ], + "score": 1.0, + "content": "LLMs has known dangers [8, 6], but we believe that equalizing access to a technology that is quickly", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 555, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 506, + 568 + ], + "score": 1.0, + "content": "becoming ubiquitous will allow for better more independent analysis than keeping the power of LLMs", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 566, + 461, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 461, + 579 + ], + "score": 1.0, + "content": "in the hands of large corporations that do not release models or source code for auditing.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 39.5, + "bbox_fs": [ + 105, + 533, + 506, + 579 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 583, + 504, + 605 + ], + "lines": [ + { + "bbox": [ + 106, + 582, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 505, + 596 + ], + "score": 1.0, + "content": "All in all, we believe that QLORA will have a broadly positive impact making the finetuning of high", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 594, + 329, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 329, + 606 + ], + "score": 1.0, + "content": "quality LLMs much more widely and easily accessible.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 42.5, + "bbox_fs": [ + 106, + 582, + 505, + 606 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 615, + 207, + 628 + ], + "lines": [ + { + "bbox": [ + 105, + 613, + 208, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 208, + 631 + ], + "score": 1.0, + "content": "Acknowledgements", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 44 + }, + { + "type": "text", + "bbox": [ + 107, + 634, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 635, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 635, + 505, + 646 + ], + "score": 1.0, + "content": "We thank Aditya Kusupati, Ofir Press, Ashish Sharma, Margaret Li, Raphael Olivier, Zihao Ye, and", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 645, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 645, + 505, + 657 + ], + "score": 1.0, + "content": "Evangelia Spiliopoulou for their valuable feedback. 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We see", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 347, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 326, + 360 + ], + "score": 1.0, + "content": "that RTX 3090/4090 and A40 GPUs have large speedups of", + "type": "text" + }, + { + "bbox": [ + 326, + 348, + 358, + 358 + ], + "score": 0.42, + "content": "2 . 9 – 4 . 0 \\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 347, + 505, + 360 + ], + "score": 1.0, + "content": "while other GPUs have speedups in the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 357, + 493, + 370 + ], + "spans": [ + { + "bbox": [ + 104, + 357, + 493, + 370 + ], + "score": 1.0, + "content": "range 1.1-1.5x. The difference between GPUs is mostly caused by poor instruction throughput bottlenecks.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "title", + "bbox": [ + 109, + 388, + 271, + 402 + ], + "lines": [ + { + "bbox": [ + 105, + 387, + 273, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 273, + 404 + ], + "score": 1.0, + "content": "A Inference Speedups for NF4", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 104, + 407, + 475, + 419 + ], + "lines": [ + { + "bbox": [ + 105, + 406, + 477, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 477, + 421 + ], + "score": 1.0, + "content": "The speedups for single batch inference for NF4 compare to 16-bit float are shown in Figure 3.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "title", + "bbox": [ + 108, + 428, + 421, + 442 + ], + "lines": [ + { + "bbox": [ + 105, + 428, + 421, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 421, + 444 + ], + "score": 1.0, + "content": "B Environmental Impact and Carbon Footprint Reductions", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 448, + 505, + 492 + ], + "lines": [ + { + "bbox": [ + 105, + 448, + 506, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 506, + 461 + ], + "score": 1.0, + "content": "The relationship between serving efficiency and environmental impact and cost/performance is a", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 459, + 506, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 506, + 473 + ], + "score": 1.0, + "content": "complicated one. The short answer is, for personal use, we reduce the environmental impact per", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 470, + 506, + 483 + ], + "spans": [ + { + "bbox": [ + 106, + 470, + 171, + 483 + ], + "score": 1.0, + "content": "token by about", + "type": "text" + }, + { + "bbox": [ + 172, + 470, + 191, + 481 + ], + "score": 0.75, + "content": "3 . 5 \\mathrm { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 470, + 207, + 483 + ], + "score": 1.0, + "content": ". If", + "type": "text" + }, + { + "bbox": [ + 207, + 470, + 227, + 481 + ], + "score": 0.86, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 470, + 380, + 483 + ], + "score": 1.0, + "content": "LLM deployments are personal and", + "type": "text" + }, + { + "bbox": [ + 380, + 470, + 401, + 481 + ], + "score": 0.84, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 470, + 506, + 483 + ], + "score": 1.0, + "content": "company, we reduce the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 481, + 289, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 266, + 493 + ], + "score": 1.0, + "content": "environmental footprint of inference by", + "type": "text" + }, + { + "bbox": [ + 266, + 481, + 286, + 492 + ], + "score": 0.86, + "content": "72 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 482, + 289, + 493 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 106, + 497, + 505, + 574 + ], + "lines": [ + { + "bbox": [ + 105, + 496, + 506, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 506, + 511 + ], + "score": 1.0, + "content": "The long answer is this: For deployment, there are two options: (1) personal deployment, (2)", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 509, + 506, + 522 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 470, + 522 + ], + "score": 1.0, + "content": "deployment by companies for many users. (1) uses small batch sizes (usually batch size", + "type": "text" + }, + { + "bbox": [ + 471, + 509, + 483, + 519 + ], + "score": 0.78, + "content": "^ { = 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 509, + 506, + 522 + ], + "score": 1.0, + "content": "), (2)", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 518, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 342, + 534 + ], + "score": 1.0, + "content": "uses large batch sizes (64-128). Per token, (2) offers about", + "type": "text" + }, + { + "bbox": [ + 342, + 520, + 365, + 530 + ], + "score": 0.87, + "content": "{ > } 5 0 \\mathrm { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 518, + 505, + 534 + ], + "score": 1.0, + "content": "better efficiency because for every", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 530, + 506, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 417, + 544 + ], + "score": 1.0, + "content": "weight that is loaded from memory, up to 64-128 tokens can be calculated. The", + "type": "text" + }, + { + "bbox": [ + 417, + 531, + 440, + 541 + ], + "score": 0.87, + "content": "{ > } 5 0 \\mathrm { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 530, + 506, + 544 + ], + "score": 1.0, + "content": "improvement in", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 541, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 505, + 554 + ], + "score": 1.0, + "content": "efficiency stems from the fact memory operations are energy inefficient, and floating point operations", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 552, + 505, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 505, + 566 + ], + "score": 1.0, + "content": "are energy efficient. As such,company deployment (2) is a very environmentally friendly and cheap", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 564, + 148, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 148, + 576 + ], + "score": 1.0, + "content": "approach.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 579, + 505, + 613 + ], + "lines": [ + { + "bbox": [ + 106, + 579, + 505, + 592 + ], + "spans": [ + { + "bbox": [ + 106, + 579, + 505, + 592 + ], + "score": 1.0, + "content": "Currently, we have efficient 4-bit CUDA kernels for the personal deployment scenario (1) with batch", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 107, + 590, + 505, + 602 + ], + "spans": [ + { + "bbox": [ + 107, + 591, + 134, + 601 + ], + "score": 0.35, + "content": "\\mathrm { s i z e } { = } 1", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 590, + 203, + 602 + ], + "score": 1.0, + "content": "which are about", + "type": "text" + }, + { + "bbox": [ + 204, + 591, + 223, + 601 + ], + "score": 0.54, + "content": "3 . 5 \\mathrm { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 590, + 505, + 602 + ], + "score": 1.0, + "content": "more efficient (see Figure 3). As such, we reduce the overall footprint", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 601, + 273, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 273, + 614 + ], + "score": 1.0, + "content": "for personal deployment (1) significantly.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 106, + 617, + 505, + 684 + ], + "lines": [ + { + "bbox": [ + 106, + 618, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 106, + 618, + 506, + 631 + ], + "score": 1.0, + "content": "The overall cost and footprint is now determined by how many people use (1) vs (2). For example, if", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 628, + 506, + 641 + ], + "spans": [ + { + "bbox": [ + 106, + 629, + 126, + 639 + ], + "score": 0.86, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 628, + 289, + 641 + ], + "score": 1.0, + "content": "of users use personal deployment (1) and", + "type": "text" + }, + { + "bbox": [ + 289, + 629, + 308, + 639 + ], + "score": 0.87, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 628, + 506, + 641 + ], + "score": 1.0, + "content": "company deployments (2) and we assume that (2)", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 640, + 505, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 140, + 653 + ], + "score": 1.0, + "content": "is about", + "type": "text" + }, + { + "bbox": [ + 140, + 640, + 156, + 650 + ], + "score": 0.52, + "content": "5 0 \\mathrm { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 640, + 461, + 653 + ], + "score": 1.0, + "content": "more efficient then we get the following numbers. 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If", + "type": "text" + }, + { + "bbox": [ + 233, + 662, + 253, + 672 + ], + "score": 0.87, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 661, + 506, + 674 + ], + "score": 1.0, + "content": "of people use personally deployed LLMs (phones, laptops etc)", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 673, + 393, + 684 + ], + "spans": [ + { + "bbox": [ + 106, + 673, + 313, + 684 + ], + "score": 1.0, + "content": "then our method will reduce the impact by about 1-", + "type": "text" + }, + { + "bbox": [ + 313, + 673, + 389, + 684 + ], + "score": 0.88, + "content": "( 9 8 \\% / 3 . 5 \\mathrm { x } ) = 7 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 673, + 393, + 684 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 25.5 + }, + { + "type": "text", + "bbox": [ + 108, + 689, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 688, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 506, + 702 + ], + "score": 1.0, + "content": "The environmental impact reduction is roughly proportional to energy consumption per token pro-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 505, + 712 + ], + "score": 1.0, + "content": "cessed which is roughly proportional to the cost of running LLMs. As such, the bottom line is affected", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 711, + 205, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 205, + 724 + ], + "score": 1.0, + "content": "in the same proportions.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30 + } + ], + "page_idx": 15, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 300, + 742, + 311, + 750 + ], + "lines": [ + { + "bbox": [ + 299, + 740, + 313, + 754 + ], + "spans": [ + { + "bbox": [ + 299, + 740, + 313, + 754 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 14 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 146, + 94, + 463, + 327 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 146, + 94, + 463, + 327 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 146, + 94, + 463, + 327 + ], + "spans": [ + { + "bbox": [ + 146, + 94, + 463, + 327 + ], + "score": 0.97, + "type": "image", + "image_path": "f19169a1a302d83726a2457a3f1ad8a68dedecfed370732ee47ca504c9529cb6.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 146, + 94, + 463, + 171.66666666666669 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 146, + 171.66666666666669, + 463, + 249.33333333333337 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 146, + 249.33333333333337, + 463, + 327.00000000000006 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 337, + 506, + 369 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 337, + 505, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 337, + 505, + 349 + ], + "score": 1.0, + "content": "Figure 3: Speedups of NF4 inference for batch size 1 compared to 16-bit inference for different GPUs. We see", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 347, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 326, + 360 + ], + "score": 1.0, + "content": "that RTX 3090/4090 and A40 GPUs have large speedups of", + "type": "text" + }, + { + "bbox": [ + 326, + 348, + 358, + 358 + ], + "score": 0.42, + "content": "2 . 9 – 4 . 0 \\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 347, + 505, + 360 + ], + "score": 1.0, + "content": "while other GPUs have speedups in the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 357, + 493, + 370 + ], + "spans": [ + { + "bbox": [ + 104, + 357, + 493, + 370 + ], + "score": 1.0, + "content": "range 1.1-1.5x. The difference between GPUs is mostly caused by poor instruction throughput bottlenecks.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "title", + "bbox": [ + 109, + 388, + 271, + 402 + ], + "lines": [ + { + "bbox": [ + 105, + 387, + 273, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 273, + 404 + ], + "score": 1.0, + "content": "A Inference Speedups for NF4", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 104, + 407, + 475, + 419 + ], + "lines": [ + { + "bbox": [ + 105, + 406, + 477, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 477, + 421 + ], + "score": 1.0, + "content": "The speedups for single batch inference for NF4 compare to 16-bit float are shown in Figure 3.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 406, + 477, + 421 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 428, + 421, + 442 + ], + "lines": [ + { + "bbox": [ + 105, + 428, + 421, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 421, + 444 + ], + "score": 1.0, + "content": "B Environmental Impact and Carbon Footprint Reductions", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 448, + 505, + 492 + ], + "lines": [ + { + "bbox": [ + 105, + 448, + 506, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 506, + 461 + ], + "score": 1.0, + "content": "The relationship between serving efficiency and environmental impact and cost/performance is a", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 459, + 506, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 506, + 473 + ], + "score": 1.0, + "content": "complicated one. The short answer is, for personal use, we reduce the environmental impact per", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 470, + 506, + 483 + ], + "spans": [ + { + "bbox": [ + 106, + 470, + 171, + 483 + ], + "score": 1.0, + "content": "token by about", + "type": "text" + }, + { + "bbox": [ + 172, + 470, + 191, + 481 + ], + "score": 0.75, + "content": "3 . 5 \\mathrm { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 470, + 207, + 483 + ], + "score": 1.0, + "content": ". If", + "type": "text" + }, + { + "bbox": [ + 207, + 470, + 227, + 481 + ], + "score": 0.86, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 470, + 380, + 483 + ], + "score": 1.0, + "content": "LLM deployments are personal and", + "type": "text" + }, + { + "bbox": [ + 380, + 470, + 401, + 481 + ], + "score": 0.84, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 470, + 506, + 483 + ], + "score": 1.0, + "content": "company, we reduce the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 481, + 289, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 266, + 493 + ], + "score": 1.0, + "content": "environmental footprint of inference by", + "type": "text" + }, + { + "bbox": [ + 266, + 481, + 286, + 492 + ], + "score": 0.86, + "content": "72 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 482, + 289, + 493 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10.5, + "bbox_fs": [ + 105, + 448, + 506, + 493 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 497, + 505, + 574 + ], + "lines": [ + { + "bbox": [ + 105, + 496, + 506, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 506, + 511 + ], + "score": 1.0, + "content": "The long answer is this: For deployment, there are two options: (1) personal deployment, (2)", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 509, + 506, + 522 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 470, + 522 + ], + "score": 1.0, + "content": "deployment by companies for many users. (1) uses small batch sizes (usually batch size", + "type": "text" + }, + { + "bbox": [ + 471, + 509, + 483, + 519 + ], + "score": 0.78, + "content": "^ { = 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 509, + 506, + 522 + ], + "score": 1.0, + "content": "), (2)", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 518, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 342, + 534 + ], + "score": 1.0, + "content": "uses large batch sizes (64-128). Per token, (2) offers about", + "type": "text" + }, + { + "bbox": [ + 342, + 520, + 365, + 530 + ], + "score": 0.87, + "content": "{ > } 5 0 \\mathrm { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 518, + 505, + 534 + ], + "score": 1.0, + "content": "better efficiency because for every", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 530, + 506, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 417, + 544 + ], + "score": 1.0, + "content": "weight that is loaded from memory, up to 64-128 tokens can be calculated. The", + "type": "text" + }, + { + "bbox": [ + 417, + 531, + 440, + 541 + ], + "score": 0.87, + "content": "{ > } 5 0 \\mathrm { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 530, + 506, + 544 + ], + "score": 1.0, + "content": "improvement in", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 541, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 505, + 554 + ], + "score": 1.0, + "content": "efficiency stems from the fact memory operations are energy inefficient, and floating point operations", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 552, + 505, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 505, + 566 + ], + "score": 1.0, + "content": "are energy efficient. 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Dataset ModelGLUE (Acc.)Super-NaturalInstructions (RougeL)
RoBERTa-largeT5-80MT5-250MT5-780MT5-3BT5-11B
BF1688.640.142.148.054.362.0
BF16 replication88.640.042.247.354.91
LoRA BF1688.840.542.647.155.460.7
QLoRA Int888.840.442.945.456.560.7
QLORA FP488.640.342.447.555.660.9
QLORA NF4 + DQ140.442.747.755.360.9
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Dataset ModelGLUE (Acc.)Super-NaturalInstructions (RougeL)
RoBERTa-largeT5-80MT5-250MT5-780MT5-3BT5-11B
BF1688.640.142.148.054.362.0
BF16 replication88.640.042.247.354.91
LoRA BF1688.840.542.647.155.460.7
QLoRA Int888.840.442.945.456.560.7
QLORA FP488.640.342.447.555.660.9
QLORA NF4 + DQ140.442.747.755.360.9
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We keep LoRA", + "type": "text" + }, + { + "bbox": [ + 338, + 414, + 346, + 422 + ], + "score": 0.77, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 412, + 505, + 425 + ], + "score": 1.0, + "content": "fixed and search the learning rate, since", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 423, + 314, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 133, + 437 + ], + "score": 1.0, + "content": "LoRA", + "type": "text" + }, + { + "bbox": [ + 134, + 424, + 142, + 433 + ], + "score": 0.8, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 423, + 314, + 437 + ], + "score": 1.0, + "content": "is always proportional to the learning rate.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18.5, + "bbox_fs": [ + 104, + 390, + 507, + 437 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 440, + 506, + 473 + ], + "lines": [ + { + "bbox": [ + 105, + 439, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 506, + 453 + ], + "score": 1.0, + "content": "We find that LoRA dropout 0.05 is useful for small models (7B, 13B), but not for larger models (33B,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 451, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 191, + 464 + ], + "score": 1.0, + "content": "65B). We find LoRA", + "type": "text" + }, + { + "bbox": [ + 191, + 453, + 198, + 461 + ], + "score": 0.72, + "content": "r", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 451, + 506, + 464 + ], + "score": 1.0, + "content": "is unrelated to final performance if LoRA is used on all layers as can be seen", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 461, + 154, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 154, + 475 + ], + "score": 1.0, + "content": "in Figure 4", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 439, + 506, + 475 + ] + }, + { + "type": "image", + "bbox": [ + 224, + 486, + 385, + 619 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 224, + 486, + 385, + 619 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 224, + 486, + 385, + 619 + ], + "spans": [ + { + "bbox": [ + 224, + 486, + 385, + 619 + ], + "score": 0.97, + "type": "image", + "image_path": "4c7a0bb3134957dd5ef35455dfedd9cd1eec0aade082cedcaf728a0fc1cade73.jpg" + } + ] + } + ], + "index": 28.5, + "virtual_lines": [ + { + "bbox": [ + 224, + 486, + 385, + 499.3 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 224, + 499.3, + 385, + 512.6 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 224, + 512.6, + 385, + 525.9 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 224, + 525.9, + 385, + 539.1999999999999 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 224, + 539.1999999999999, + 385, + 552.4999999999999 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 224, + 552.4999999999999, + 385, + 565.7999999999998 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 224, + 565.7999999999998, + 385, + 579.0999999999998 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 224, + 579.0999999999998, + 385, + 592.3999999999997 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 224, + 592.3999999999997, + 385, + 605.6999999999997 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 224, + 605.6999999999997, + 385, + 618.9999999999997 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 627, + 505, + 659 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 627, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 174, + 639 + ], + "score": 1.0, + "content": "Figure 4: LoRA", + "type": "text" + }, + { + "bbox": [ + 175, + 630, + 181, + 637 + ], + "score": 0.65, + "content": "r", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 627, + 506, + 639 + ], + "score": 1.0, + "content": "for LLaMA 7B models finetuned on Alpaca. Each dot represents a combination of", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 638, + 505, + 649 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 244, + 649 + ], + "score": 1.0, + "content": "hyperparameters and for each LoRA", + "type": "text" + }, + { + "bbox": [ + 244, + 639, + 250, + 647 + ], + "score": 0.7, + "content": "r", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 638, + 505, + 649 + ], + "score": 1.0, + "content": "we run 3 random seed with each hyperparameter combination. The", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 648, + 440, + 659 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 218, + 659 + ], + "score": 1.0, + "content": "performance of specific LoRA", + "type": "text" + }, + { + "bbox": [ + 218, + 649, + 224, + 657 + ], + "score": 0.75, + "content": "r", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 648, + 440, + 659 + ], + "score": 1.0, + "content": "values appears to be independent of other hyperparameters.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35 + } + ], + "index": 31.75 + }, + { + "type": "title", + "bbox": [ + 107, + 673, + 369, + 686 + ], + "lines": [ + { + "bbox": [ + 106, + 673, + 369, + 688 + ], + "spans": [ + { + "bbox": [ + 106, + 673, + 369, + 688 + ], + "score": 1.0, + "content": "C.3 Super-Natural Instructions Experimental Setup Details", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 689, + 506, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 702 + ], + "score": 1.0, + "content": "We use the same preprocessing of the Super-Natural Instruction dataset as Wang et al. [61]. However,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 700, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 505, + 713 + ], + "score": 1.0, + "content": "we split the training data in training and validation datasets allowing us to perform more rigorous", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 711, + 505, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 505, + 724 + ], + "score": 1.0, + "content": "hyperparameter tuning and early stopping. We use the same hyperparameters described in the paper", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39, + "bbox_fs": [ + 105, + 687, + 506, + 724 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 73, + 505, + 106 + ], + "lines": [ + { + "bbox": [ + 105, + 72, + 505, + 84 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 474, + 84 + ], + "score": 1.0, + "content": "for training the various T5 model sizes on the Super-Natural Instruction data. We use LoRA", + "type": "text" + }, + { + "bbox": [ + 475, + 73, + 505, + 83 + ], + "score": 0.86, + "content": "r = 1 6", + "type": "inline_equation" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 317, + 96 + ], + "score": 1.0, + "content": "for small, medium, and large T5 models and LoRA", + "type": "text" + }, + { + "bbox": [ + 317, + 84, + 347, + 94 + ], + "score": 0.89, + "content": "r = 6 4", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 83, + 505, + 96 + ], + "score": 1.0, + "content": "for T5 xl and xxl models. We also use", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 93, + 349, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 133, + 108 + ], + "score": 1.0, + "content": "LoRA", + "type": "text" + }, + { + "bbox": [ + 134, + 95, + 165, + 105 + ], + "score": 0.88, + "content": "\\alpha = 6 4", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 93, + 349, + 108 + ], + "score": 1.0, + "content": "in all our experiments and no LoRA dropout.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "title", + "bbox": [ + 108, + 115, + 452, + 129 + ], + "lines": [ + { + "bbox": [ + 104, + 114, + 454, + 132 + ], + "spans": [ + { + "bbox": [ + 104, + 114, + 454, + 132 + ], + "score": 1.0, + "content": "D Training a State-of-the-art Chatbot Experimental Setup Details", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "title", + "bbox": [ + 107, + 135, + 169, + 146 + ], + "lines": [ + { + "bbox": [ + 106, + 134, + 169, + 147 + ], + "spans": [ + { + "bbox": [ + 106, + 134, + 169, + 147 + ], + "score": 1.0, + "content": "D.1 Datasets", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 108, + 149, + 457, + 161 + ], + "lines": [ + { + "bbox": [ + 106, + 149, + 459, + 163 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 459, + 163 + ], + "score": 1.0, + "content": "We describe the datasets used for QLORA finetuning experiments outlined in Section 5.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 107, + 167, + 505, + 222 + ], + "lines": [ + { + "bbox": [ + 106, + 168, + 505, + 180 + ], + "spans": [ + { + "bbox": [ + 106, + 168, + 505, + 180 + ], + "score": 1.0, + "content": "OASST1 The OpenAssistant dataset [31] was collected via crowd-sourcing. It contains 161,443", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 178, + 505, + 190 + ], + "spans": [ + { + "bbox": [ + 106, + 178, + 505, + 190 + ], + "score": 1.0, + "content": "unique messages distributed across 66,497 conversations and spanning 35 different languages. The", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 189, + 505, + 202 + ], + "spans": [ + { + "bbox": [ + 106, + 189, + 505, + 202 + ], + "score": 1.0, + "content": "dataset often contains several ranked replies for each given user question. In our experiments, we", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 199, + 506, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 199, + 506, + 213 + ], + "score": 1.0, + "content": "only use the top reply at each level in the conversation tree. This limits the dataset to 9,846 examples.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 210, + 391, + 224 + ], + "spans": [ + { + "bbox": [ + 106, + 210, + 391, + 224 + ], + "score": 1.0, + "content": "We finetune models on the full conversation including the user queries.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 228, + 505, + 273 + ], + "lines": [ + { + "bbox": [ + 105, + 228, + 505, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 505, + 241 + ], + "score": 1.0, + "content": "HH-RLHF This is a human preference dataset about helpfulness and harmlessness. Each datapoint", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 240, + 505, + 252 + ], + "spans": [ + { + "bbox": [ + 106, + 240, + 505, + 252 + ], + "score": 1.0, + "content": "consists of two assistant replies to a user question along with a human preference judgment of the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 251, + 505, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 505, + 263 + ], + "score": 1.0, + "content": "best reply. The dataset contains 160,800 examples. When finetuning on this dataset, we combine", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 261, + 418, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 261, + 418, + 275 + ], + "score": 1.0, + "content": "helpfulness and harmlessness data and only keep the preferred assistant reply.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12.5 + }, + { + "type": "text", + "bbox": [ + 107, + 279, + 505, + 345 + ], + "lines": [ + { + "bbox": [ + 105, + 278, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 278, + 505, + 291 + ], + "score": 1.0, + "content": "FLAN v2 The FLAN v2 collection [39] is a collection of 1836 tasks augmented with hundreds", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 291, + 505, + 302 + ], + "spans": [ + { + "bbox": [ + 106, + 291, + 505, + 302 + ], + "score": 1.0, + "content": "of manually curated templates and rich formatting patterns into over 15M examples. The authors", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 300, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 505, + 313 + ], + "score": 1.0, + "content": "show that models trained on this collection outperform other public collections including the original", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 312, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 182, + 325 + ], + "score": 1.0, + "content": "FLAN 2021 [63],", + "type": "text" + }, + { + "bbox": [ + 182, + 312, + 207, + 323 + ], + "score": 0.83, + "content": "\\mathrm { T 0 + + }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 312, + 505, + 325 + ], + "score": 1.0, + "content": "[51], Super-Natural Instructions [61], and OPT-IML [29]. We used the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 323, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 505, + 335 + ], + "score": 1.0, + "content": "same task mixtures described by the authors with the exception of some datasets that were not freely", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 332, + 234, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 332, + 234, + 347 + ], + "score": 1.0, + "content": "available at the time of writing.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 106, + 351, + 505, + 428 + ], + "lines": [ + { + "bbox": [ + 105, + 351, + 506, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 506, + 363 + ], + "score": 1.0, + "content": "Self-Instruct, Alpaca, Unnatural Instructions The Self-Instruct, Alpaca, and Unnatural Instruc-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 362, + 506, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 506, + 375 + ], + "score": 1.0, + "content": "tions datasets [60, 56, 26] are instruction tuning datasets collected with various approaches of model", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 372, + 506, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 506, + 386 + ], + "score": 1.0, + "content": "distillation from GPT-3 Instruct and ChatGPT. They rely on prompting, in-context learning, and", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 384, + 506, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 506, + 397 + ], + "score": 1.0, + "content": "paraphrasing to come up with diverse sets of instructions and outputs. The datasets comprise of", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 395, + 506, + 407 + ], + "spans": [ + { + "bbox": [ + 106, + 395, + 506, + 407 + ], + "score": 1.0, + "content": "82,612, 51,942, and 240,670 examples respectively. One advantage of such distilled datasets is that", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 406, + 506, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 506, + 418 + ], + "score": 1.0, + "content": "they contain a more diverse set of instruction styles compared to the FLAN v2 collection and similar", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 417, + 228, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 228, + 429 + ], + "score": 1.0, + "content": "instruction tuning collections.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 434, + 505, + 501 + ], + "lines": [ + { + "bbox": [ + 105, + 434, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 505, + 447 + ], + "score": 1.0, + "content": "Longform The LongForm dataset [30] is based on an English corpus augmented with instructions", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 446, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 106, + 446, + 505, + 457 + ], + "score": 1.0, + "content": "and as such is a hybrid human-generated dataset. The underlying documents are human-written and", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 456, + 506, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 506, + 469 + ], + "score": 1.0, + "content": "come from C4 and Wikipedia while the instructions are generated via LLMs. The dataset is extended", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 468, + 504, + 479 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 504, + 479 + ], + "score": 1.0, + "content": "with additional structured corpora examples such as Stack Exchange and WikiHow and task examples", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 479, + 506, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 506, + 492 + ], + "score": 1.0, + "content": "such as question answering, email writing, grammar error correction, story/poem generation, and text", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 489, + 326, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 326, + 501 + ], + "score": 1.0, + "content": "summarization. The dataset contains 23,700 examples.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30.5 + }, + { + "type": "text", + "bbox": [ + 107, + 507, + 505, + 551 + ], + "lines": [ + { + "bbox": [ + 105, + 506, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 506, + 520 + ], + "score": 1.0, + "content": "Chip2 is part of the OIG Laion dataset. It contains Python code examples, natural instruction exam-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 518, + 505, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 505, + 531 + ], + "score": 1.0, + "content": "ples, generic harmless instructions, instruction/responses with lists, follow-up questions, Wikipedia", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 529, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 505, + 542 + ], + "score": 1.0, + "content": "toxic adversarial questions, grade school math, reasoning instructions, and character and scene", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 540, + 293, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 293, + 552 + ], + "score": 1.0, + "content": "descriptions with a total of 210,289 examples.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 35.5 + }, + { + "type": "title", + "bbox": [ + 109, + 559, + 411, + 570 + ], + "lines": [ + { + "bbox": [ + 106, + 557, + 412, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 557, + 412, + 573 + ], + "score": 1.0, + "content": "D.2 Default LoRA hyperparameters do not match 16-bit performance", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 107, + 574, + 506, + 606 + ], + "lines": [ + { + "bbox": [ + 105, + 573, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 506, + 586 + ], + "score": 1.0, + "content": "We find that default hyperparameters for fully finetuned baselines are undertuned. We do a hyper-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 585, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 506, + 596 + ], + "score": 1.0, + "content": "parameter search over learning rates 1e-6 to 5e-5 and batch sizes 8 to 128 to find robust baselines.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 595, + 380, + 607 + ], + "spans": [ + { + "bbox": [ + 106, + 595, + 380, + 607 + ], + "score": 1.0, + "content": "Results for 7B LLaMA finetuning on Alpaca are shown in Figure 5.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 107, + 611, + 505, + 677 + ], + "lines": [ + { + "bbox": [ + 105, + 611, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 505, + 624 + ], + "score": 1.0, + "content": "When using the standard practice of applying LoRA to query and value attention projection matrices", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "score": 1.0, + "content": "[28], we are not able to replicate full finetuning performance for large base models. As shown in", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 633, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 506, + 646 + ], + "score": 1.0, + "content": "Figure 5 for LLaMA 7B finetuning on Alpaca, we find that the most critical LoRA hyperparameter is", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "score": 1.0, + "content": "how many LoRA adapters are used in total and that LoRA on all linear transformer block layers are", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 655, + 504, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 504, + 667 + ], + "score": 1.0, + "content": "required to match full finetuning performance. Other LoRA hyperparameters, such as the projection", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 666, + 267, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 150, + 678 + ], + "score": 1.0, + "content": "dimension", + "type": "text" + }, + { + "bbox": [ + 150, + 668, + 156, + 676 + ], + "score": 0.69, + "content": "r", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 666, + 267, + 678 + ], + "score": 1.0, + "content": ", do not affect performance.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 44.5 + }, + { + "type": "title", + "bbox": [ + 107, + 685, + 209, + 697 + ], + "lines": [ + { + "bbox": [ + 105, + 684, + 209, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 684, + 209, + 700 + ], + "score": 1.0, + "content": "D.3 Hyperparameters", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 48 + }, + { + "type": "text", + "bbox": [ + 107, + 700, + 503, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 699, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 713 + ], + "score": 1.0, + "content": "We provide the exact hyperparameters used in our QLORA finetuning experiments. We find hyper-", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 711, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 505, + 723 + ], + "score": 1.0, + "content": "parameters to be largely robust across datasets. 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It contains 161,443", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 178, + 505, + 190 + ], + "spans": [ + { + "bbox": [ + 106, + 178, + 505, + 190 + ], + "score": 1.0, + "content": "unique messages distributed across 66,497 conversations and spanning 35 different languages. The", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 189, + 505, + 202 + ], + "spans": [ + { + "bbox": [ + 106, + 189, + 505, + 202 + ], + "score": 1.0, + "content": "dataset often contains several ranked replies for each given user question. In our experiments, we", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 199, + 506, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 199, + 506, + 213 + ], + "score": 1.0, + "content": "only use the top reply at each level in the conversation tree. 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Each datapoint", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 240, + 505, + 252 + ], + "spans": [ + { + "bbox": [ + 106, + 240, + 505, + 252 + ], + "score": 1.0, + "content": "consists of two assistant replies to a user question along with a human preference judgment of the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 251, + 505, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 505, + 263 + ], + "score": 1.0, + "content": "best reply. The dataset contains 160,800 examples. When finetuning on this dataset, we combine", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 261, + 418, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 261, + 418, + 275 + ], + "score": 1.0, + "content": "helpfulness and harmlessness data and only keep the preferred assistant reply.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 228, + 505, + 275 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 279, + 505, + 345 + ], + "lines": [ + { + "bbox": [ + 105, + 278, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 278, + 505, + 291 + ], + "score": 1.0, + "content": "FLAN v2 The FLAN v2 collection [39] is a collection of 1836 tasks augmented with hundreds", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 291, + 505, + 302 + ], + "spans": [ + { + "bbox": [ + 106, + 291, + 505, + 302 + ], + "score": 1.0, + "content": "of manually curated templates and rich formatting patterns into over 15M examples. The authors", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 300, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 505, + 313 + ], + "score": 1.0, + "content": "show that models trained on this collection outperform other public collections including the original", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 312, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 182, + 325 + ], + "score": 1.0, + "content": "FLAN 2021 [63],", + "type": "text" + }, + { + "bbox": [ + 182, + 312, + 207, + 323 + ], + "score": 0.83, + "content": "\\mathrm { T 0 + + }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 312, + 505, + 325 + ], + "score": 1.0, + "content": "[51], Super-Natural Instructions [61], and OPT-IML [29]. We used the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 323, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 505, + 335 + ], + "score": 1.0, + "content": "same task mixtures described by the authors with the exception of some datasets that were not freely", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 332, + 234, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 332, + 234, + 347 + ], + "score": 1.0, + "content": "available at the time of writing.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 278, + 505, + 347 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 351, + 505, + 428 + ], + "lines": [ + { + "bbox": [ + 105, + 351, + 506, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 506, + 363 + ], + "score": 1.0, + "content": "Self-Instruct, Alpaca, Unnatural Instructions The Self-Instruct, Alpaca, and Unnatural Instruc-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 362, + 506, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 506, + 375 + ], + "score": 1.0, + "content": "tions datasets [60, 56, 26] are instruction tuning datasets collected with various approaches of model", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 372, + 506, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 506, + 386 + ], + "score": 1.0, + "content": "distillation from GPT-3 Instruct and ChatGPT. They rely on prompting, in-context learning, and", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 384, + 506, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 506, + 397 + ], + "score": 1.0, + "content": "paraphrasing to come up with diverse sets of instructions and outputs. The datasets comprise of", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 395, + 506, + 407 + ], + "spans": [ + { + "bbox": [ + 106, + 395, + 506, + 407 + ], + "score": 1.0, + "content": "82,612, 51,942, and 240,670 examples respectively. One advantage of such distilled datasets is that", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 406, + 506, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 506, + 418 + ], + "score": 1.0, + "content": "they contain a more diverse set of instruction styles compared to the FLAN v2 collection and similar", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 417, + 228, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 228, + 429 + ], + "score": 1.0, + "content": "instruction tuning collections.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 351, + 506, + 429 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 434, + 505, + 501 + ], + "lines": [ + { + "bbox": [ + 105, + 434, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 505, + 447 + ], + "score": 1.0, + "content": "Longform The LongForm dataset [30] is based on an English corpus augmented with instructions", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 446, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 106, + 446, + 505, + 457 + ], + "score": 1.0, + "content": "and as such is a hybrid human-generated dataset. The underlying documents are human-written and", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 456, + 506, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 506, + 469 + ], + "score": 1.0, + "content": "come from C4 and Wikipedia while the instructions are generated via LLMs. The dataset is extended", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 468, + 504, + 479 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 504, + 479 + ], + "score": 1.0, + "content": "with additional structured corpora examples such as Stack Exchange and WikiHow and task examples", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 479, + 506, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 506, + 492 + ], + "score": 1.0, + "content": "such as question answering, email writing, grammar error correction, story/poem generation, and text", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 489, + 326, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 326, + 501 + ], + "score": 1.0, + "content": "summarization. The dataset contains 23,700 examples.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 434, + 506, + 501 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 507, + 505, + 551 + ], + "lines": [ + { + "bbox": [ + 105, + 506, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 506, + 520 + ], + "score": 1.0, + "content": "Chip2 is part of the OIG Laion dataset. It contains Python code examples, natural instruction exam-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 518, + 505, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 505, + 531 + ], + "score": 1.0, + "content": "ples, generic harmless instructions, instruction/responses with lists, follow-up questions, Wikipedia", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 529, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 505, + 542 + ], + "score": 1.0, + "content": "toxic adversarial questions, grade school math, reasoning instructions, and character and scene", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 540, + 293, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 293, + 552 + ], + "score": 1.0, + "content": "descriptions with a total of 210,289 examples.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 35.5, + "bbox_fs": [ + 105, + 506, + 506, + 552 + ] + }, + { + "type": "title", + "bbox": [ + 109, + 559, + 411, + 570 + ], + "lines": [ + { + "bbox": [ + 106, + 557, + 412, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 557, + 412, + 573 + ], + "score": 1.0, + "content": "D.2 Default LoRA hyperparameters do not match 16-bit performance", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 107, + 574, + 506, + 606 + ], + "lines": [ + { + "bbox": [ + 105, + 573, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 506, + 586 + ], + "score": 1.0, + "content": "We find that default hyperparameters for fully finetuned baselines are undertuned. 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ParametersDatasetBatch sizeLRStepsSource LengthTarget Length
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ParametersDatasetBatch sizeLRStepsSource LengthTarget Length
7BAll162e-410000384128
7BOASST1162e-41875-512
7BHH-RLHF162e-4100001768
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65BAll641e-42500384128
65BOASST1161e-41875-512
65BHH-RLHF641e-425001768
65BLongform321e-423435121024
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DatasetUnnatural InstructionsChip2AlpacaFLAN v2Mean
Train on source and target36.233.738.142.037.5
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This calls for further", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 303, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 505, + 316 + ], + "score": 1.0, + "content": "investigations of the tradeoffs of simple cross-entropy loss and RLHF training. We hope that QLORA", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 314, + 474, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 474, + 326 + ], + "score": 1.0, + "content": "enables such analysis at scale, without the need for overwhelming computational resources.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14.5 + }, + { + "type": "title", + "bbox": [ + 106, + 337, + 467, + 349 + ], + "lines": [ + { + "bbox": [ + 105, + 337, + 468, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 337, + 468, + 351 + ], + "score": 1.0, + "content": "D.6 What is more important: instruction finetuning dataset size or dataset quality?", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 106, + 354, + 505, + 442 + ], + "lines": [ + { + "bbox": [ + 105, + 353, + 506, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 506, + 367 + ], + "score": 1.0, + "content": "To understand the effects of dataset quality vs. dataset size, we experiment with subsampling large", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 364, + 506, + 377 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 506, + 377 + ], + "score": 1.0, + "content": "datasets with at least 150,000 samples (Chip2, FLAN v2, Unnatural Instructions), into datasets of size", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 376, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 106, + 376, + 505, + 388 + ], + "score": 1.0, + "content": "50,000, 100,000 and 150,000 and examine the resulting trends, as shown in Table 10. We find that", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 387, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 505, + 399 + ], + "score": 1.0, + "content": "increasing the dataset size and increasing the number of epochs improves MMLU only marginally", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 108, + 397, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 108, + 398, + 144, + 408 + ], + "score": 0.29, + "content": "( 0 . 0 - 0 . 5 ", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 397, + 365, + 410 + ], + "score": 1.0, + "content": "MMLU), while the difference between datasets is up to", + "type": "text" + }, + { + "bbox": [ + 365, + 398, + 381, + 408 + ], + "score": 0.37, + "content": "4 0 \\mathrm { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 397, + 506, + 410 + ], + "score": 1.0, + "content": "larger (1.5 - 8.0 MMLU). This", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 407, + 507, + 422 + ], + "spans": [ + { + "bbox": [ + 104, + 407, + 507, + 422 + ], + "score": 1.0, + "content": "is a clear indicator that dataset quality rather than dataset size is critical for mean MMLU accuracy.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 419, + 507, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 507, + 433 + ], + "score": 1.0, + "content": "We obtain similar findings for chatbot performance, with the most successful dataset for training,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 430, + 355, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 230, + 444 + ], + "score": 1.0, + "content": "OASST1, containing less than", + "type": "text" + }, + { + "bbox": [ + 230, + 430, + 246, + 441 + ], + "score": 0.37, + "content": "1 0 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 430, + 355, + 444 + ], + "score": 1.0, + "content": "examples after processing.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 21.5 + }, + { + "type": "title", + "bbox": [ + 108, + 455, + 267, + 469 + ], + "lines": [ + { + "bbox": [ + 105, + 455, + 268, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 268, + 471 + ], + "score": 1.0, + "content": "E Chatbot Evaluation Details", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 478, + 505, + 522 + ], + "lines": [ + { + "bbox": [ + 105, + 477, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 506, + 491 + ], + "score": 1.0, + "content": "Following common practice, we use the MMLU (Massively Multitask Language Understanding)", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 488, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 506, + 502 + ], + "score": 1.0, + "content": "benchmark [24] to measure performance on a range of language understanding tasks. This is", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 500, + 507, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 507, + 513 + ], + "score": 1.0, + "content": "a multiple-choice benchmark covering 57 tasks including elementary mathematics, US history,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 510, + 368, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 368, + 525 + ], + "score": 1.0, + "content": "computer science, law, and more. We report 5-shot test accuracy.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28.5 + }, + { + "type": "text", + "bbox": [ + 107, + 531, + 505, + 587 + ], + "lines": [ + { + "bbox": [ + 106, + 532, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 106, + 532, + 505, + 543 + ], + "score": 1.0, + "content": "We also test generative language capabilities through both automated and human evaluations. This", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 542, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 505, + 555 + ], + "score": 1.0, + "content": "second set of evaluations relies on queries curated by humans and aims at measuring the quality", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 552, + 506, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 506, + 566 + ], + "score": 1.0, + "content": "of model responses. While this is a more realistic testbed for chatbot model performance and is", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 564, + 506, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 506, + 577 + ], + "score": 1.0, + "content": "growing in popularity, there is no commonly accepted protocol in the literature. We describe below", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 575, + 468, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 303, + 588 + ], + "score": 1.0, + "content": "our proposed setup, using nucleus sampling with", + "type": "text" + }, + { + "bbox": [ + 303, + 576, + 336, + 586 + ], + "score": 0.91, + "content": "p = 0 . 9", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 575, + 468, + 588 + ], + "score": 1.0, + "content": "and temperature 0.7 in all cases.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33 + }, + { + "type": "table", + "bbox": [ + 108, + 647, + 505, + 720 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 615, + 504, + 646 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 616, + 505, + 626 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 505, + 626 + ], + "score": 1.0, + "content": "Table 10: Effect of different dataset sizes and finetuning epochs on mean 5-shot MMLU test set accuracy. While", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 626, + 504, + 636 + ], + "spans": [ + { + "bbox": [ + 106, + 626, + 504, + 636 + ], + "score": 1.0, + "content": "increasing the dataset size and training for more than 1 epochs helps with MMLU performance, the difference", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 636, + 504, + 647 + ], + "spans": [ + { + "bbox": [ + 106, + 636, + 504, + 647 + ], + "score": 1.0, + "content": "between datasets are far larger, indicating that dataset quality affects MMLU performance more than dataset size.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37 + }, + { + "type": "table_body", + "bbox": [ + 108, + 647, + 505, + 720 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 647, + 505, + 720 + ], + "spans": [ + { + "bbox": [ + 108, + 647, + 505, + 720 + ], + "score": 0.979, + "html": "
Datapoints ↓Epochs→ChipUnnatural InstructionsFLAN v2
123123123Mean
5000034.5035.3034.7038.1042.2038.1043.0043.5044.1039.28
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15000034.4034.8035.1039.7041.1041.5044.6045.5043.5040.02
Mean34.2034.6734.6039.3041.5038.8743.8344.2344.17
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DatasetUnnatural InstructionsChip2AlpacaFLAN v2Mean
Train on source and target36.233.738.142.037.5
Train on target38.034.539.042.938.6
", + "type": "table", + "image_path": "6f38d34a2dafc7c1bdd564de624a2aec30b018c35cc0af4696661fc193154a38.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 113, + 70, + 497, + 85.66666666666667 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 113, + 85.66666666666667, + 497, + 101.33333333333334 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 113, + 101.33333333333334, + 497, + 117.00000000000001 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "table_footnote", + "bbox": [ + 107, + 122, + 502, + 133 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 105, + 121, + 504, + 134 + ], + "spans": [ + { + "bbox": [ + 105, + 121, + 504, + 134 + ], + "score": 1.0, + "content": "Table 9: MMLU 5-shot test results studying the effect of training on the instructions in addition to the response.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + } + ], + "index": 2.0 + }, + { + "type": "title", + "bbox": [ + 106, + 158, + 236, + 171 + ], + "lines": [ + { + "bbox": [ + 105, + 158, + 237, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 237, + 172 + ], + "score": 1.0, + "content": "D.5 Training Considerations", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 107, + 175, + 505, + 230 + ], + "lines": [ + { + "bbox": [ + 105, + 174, + 505, + 188 + ], + "spans": [ + { + "bbox": [ + 105, + 174, + 505, + 188 + ], + "score": 1.0, + "content": "We note that the OASST1 dataset on which Guanaco models are trained is multilingual and that the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 185, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 185, + 505, + 199 + ], + "score": 1.0, + "content": "OA benchmark also contains prompts in different languages. We leave it to future work to investigate", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 196, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 196, + 505, + 210 + ], + "score": 1.0, + "content": "the degree to which such multilingual training improves performance on instructions in languages", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 208, + 505, + 220 + ], + "spans": [ + { + "bbox": [ + 106, + 208, + 505, + 220 + ], + "score": 1.0, + "content": "other than English and whether this explains the larger gap between Vicuna-13B model (only trained", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 219, + 376, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 376, + 230 + ], + "score": 1.0, + "content": "on English data) and Guanaco 33B and 65B on the OA benchmark.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 174, + 505, + 230 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 239, + 506, + 272 + ], + "lines": [ + { + "bbox": [ + 106, + 239, + 505, + 252 + ], + "spans": [ + { + "bbox": [ + 106, + 239, + 505, + 252 + ], + "score": 1.0, + "content": "Given the strong performance of Guanaco models, we investigate any data leakage between the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 249, + 507, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 507, + 263 + ], + "score": 1.0, + "content": "OASST1 data and the Vicuna benchmark prompts. We do not find overlapping prompts after perform-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 261, + 468, + 274 + ], + "spans": [ + { + "bbox": [ + 106, + 261, + 468, + 274 + ], + "score": 1.0, + "content": "ing fuzzy string matching in the two datasets and inspecting the closest matches manually.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 239, + 507, + 274 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 281, + 505, + 325 + ], + "lines": [ + { + "bbox": [ + 105, + 280, + 506, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 506, + 294 + ], + "score": 1.0, + "content": "Furthermore, we note that our model is only trained with cross-entropy loss (supervised learning)", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 293, + 506, + 304 + ], + "spans": [ + { + "bbox": [ + 106, + 293, + 506, + 304 + ], + "score": 1.0, + "content": "without relying on reinforcement learning from human feedback (RLHF). This calls for further", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 303, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 505, + 316 + ], + "score": 1.0, + "content": "investigations of the tradeoffs of simple cross-entropy loss and RLHF training. We hope that QLORA", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 314, + 474, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 474, + 326 + ], + "score": 1.0, + "content": "enables such analysis at scale, without the need for overwhelming computational resources.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14.5, + "bbox_fs": [ + 105, + 280, + 506, + 326 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 337, + 467, + 349 + ], + "lines": [ + { + "bbox": [ + 105, + 337, + 468, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 337, + 468, + 351 + ], + "score": 1.0, + "content": "D.6 What is more important: instruction finetuning dataset size or dataset quality?", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 106, + 354, + 505, + 442 + ], + "lines": [ + { + "bbox": [ + 105, + 353, + 506, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 506, + 367 + ], + "score": 1.0, + "content": "To understand the effects of dataset quality vs. dataset size, we experiment with subsampling large", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 364, + 506, + 377 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 506, + 377 + ], + "score": 1.0, + "content": "datasets with at least 150,000 samples (Chip2, FLAN v2, Unnatural Instructions), into datasets of size", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 376, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 106, + 376, + 505, + 388 + ], + "score": 1.0, + "content": "50,000, 100,000 and 150,000 and examine the resulting trends, as shown in Table 10. We find that", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 387, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 505, + 399 + ], + "score": 1.0, + "content": "increasing the dataset size and increasing the number of epochs improves MMLU only marginally", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 108, + 397, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 108, + 398, + 144, + 408 + ], + "score": 0.29, + "content": "( 0 . 0 - 0 . 5 ", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 397, + 365, + 410 + ], + "score": 1.0, + "content": "MMLU), while the difference between datasets is up to", + "type": "text" + }, + { + "bbox": [ + 365, + 398, + 381, + 408 + ], + "score": 0.37, + "content": "4 0 \\mathrm { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 397, + 506, + 410 + ], + "score": 1.0, + "content": "larger (1.5 - 8.0 MMLU). This", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 407, + 507, + 422 + ], + "spans": [ + { + "bbox": [ + 104, + 407, + 507, + 422 + ], + "score": 1.0, + "content": "is a clear indicator that dataset quality rather than dataset size is critical for mean MMLU accuracy.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 419, + 507, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 507, + 433 + ], + "score": 1.0, + "content": "We obtain similar findings for chatbot performance, with the most successful dataset for training,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 430, + 355, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 230, + 444 + ], + "score": 1.0, + "content": "OASST1, containing less than", + "type": "text" + }, + { + "bbox": [ + 230, + 430, + 246, + 441 + ], + "score": 0.37, + "content": "1 0 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 430, + 355, + 444 + ], + "score": 1.0, + "content": "examples after processing.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 21.5, + "bbox_fs": [ + 104, + 353, + 507, + 444 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 455, + 267, + 469 + ], + "lines": [ + { + "bbox": [ + 105, + 455, + 268, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 268, + 471 + ], + "score": 1.0, + "content": "E Chatbot Evaluation Details", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 478, + 505, + 522 + ], + "lines": [ + { + "bbox": [ + 105, + 477, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 506, + 491 + ], + "score": 1.0, + "content": "Following common practice, we use the MMLU (Massively Multitask Language Understanding)", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 488, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 506, + 502 + ], + "score": 1.0, + "content": "benchmark [24] to measure performance on a range of language understanding tasks. This is", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 500, + 507, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 507, + 513 + ], + "score": 1.0, + "content": "a multiple-choice benchmark covering 57 tasks including elementary mathematics, US history,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 510, + 368, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 368, + 525 + ], + "score": 1.0, + "content": "computer science, law, and more. We report 5-shot test accuracy.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 477, + 507, + 525 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 531, + 505, + 587 + ], + "lines": [ + { + "bbox": [ + 106, + 532, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 106, + 532, + 505, + 543 + ], + "score": 1.0, + "content": "We also test generative language capabilities through both automated and human evaluations. This", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 542, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 505, + 555 + ], + "score": 1.0, + "content": "second set of evaluations relies on queries curated by humans and aims at measuring the quality", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 552, + 506, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 506, + 566 + ], + "score": 1.0, + "content": "of model responses. While this is a more realistic testbed for chatbot model performance and is", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 564, + 506, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 506, + 577 + ], + "score": 1.0, + "content": "growing in popularity, there is no commonly accepted protocol in the literature. We describe below", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 575, + 468, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 303, + 588 + ], + "score": 1.0, + "content": "our proposed setup, using nucleus sampling with", + "type": "text" + }, + { + "bbox": [ + 303, + 576, + 336, + 586 + ], + "score": 0.91, + "content": "p = 0 . 9", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 575, + 468, + 588 + ], + "score": 1.0, + "content": "and temperature 0.7 in all cases.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 532, + 506, + 588 + ] + }, + { + "type": "table", + "bbox": [ + 108, + 647, + 505, + 720 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 615, + 504, + 646 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 616, + 505, + 626 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 505, + 626 + ], + "score": 1.0, + "content": "Table 10: Effect of different dataset sizes and finetuning epochs on mean 5-shot MMLU test set accuracy. While", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 626, + 504, + 636 + ], + "spans": [ + { + "bbox": [ + 106, + 626, + 504, + 636 + ], + "score": 1.0, + "content": "increasing the dataset size and training for more than 1 epochs helps with MMLU performance, the difference", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 636, + 504, + 647 + ], + "spans": [ + { + "bbox": [ + 106, + 636, + 504, + 647 + ], + "score": 1.0, + "content": "between datasets are far larger, indicating that dataset quality affects MMLU performance more than dataset size.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37 + }, + { + "type": "table_body", + "bbox": [ + 108, + 647, + 505, + 720 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 647, + 505, + 720 + ], + "spans": [ + { + "bbox": [ + 108, + 647, + 505, + 720 + ], + "score": 0.979, + "html": "
Datapoints ↓Epochs→ChipUnnatural InstructionsFLAN v2
123123123Mean
5000034.5035.3034.7038.1042.2038.1043.0043.5044.1039.28
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Mean34.2034.6734.6039.3041.5038.8743.8344.2344.17
", + "type": "table", + "image_path": "12ed75d3f80a011114089e377f280468e25aadf92f68244344e984333107f673.jpg" + } + ] + } + ], + "index": 40, + "virtual_lines": [ + { + "bbox": [ + 108, + 647, + 505, + 671.3333333333334 + ], + "spans": [], + "index": 39 + }, + { + "bbox": [ + 108, + 671.3333333333334, + 505, + 695.6666666666667 + ], + "spans": [], + "index": 40 + }, + { + "bbox": [ + 108, + 695.6666666666667, + 505, + 720.0000000000001 + ], + "spans": [], + "index": 41 + } + ] + } + ], + "index": 38.5 + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 72, + 205, + 84 + ], + "lines": [ + { + "bbox": [ + 106, + 72, + 206, + 85 + ], + "spans": [ + { + "bbox": [ + 106, + 72, + 206, + 85 + ], + "score": 1.0, + "content": "E.1 Benchmark Data", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 89, + 505, + 155 + ], + "lines": [ + { + "bbox": [ + 106, + 89, + 505, + 101 + ], + "spans": [ + { + "bbox": [ + 106, + 89, + 505, + 101 + ], + "score": 1.0, + "content": "We evaluate on two curated datasets of queries (questions): the Vicuna prompts [10] and the OASST1", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 100, + 506, + 113 + ], + "spans": [ + { + "bbox": [ + 106, + 100, + 506, + 113 + ], + "score": 1.0, + "content": "validation dataset [31]. We use the Vicuna prompts, a set of 80 prompts from a diverse set of", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 111, + 506, + 123 + ], + "spans": [ + { + "bbox": [ + 105, + 111, + 506, + 123 + ], + "score": 1.0, + "content": "categories, without modifications. The OASST1 dataset is a multilingual collection of crowd-sourced", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 122, + 506, + 134 + ], + "spans": [ + { + "bbox": [ + 106, + 122, + 506, + 134 + ], + "score": 1.0, + "content": "multiturn dialogs between a user and an assistant. We select all user messages in the validation dataset", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 133, + 506, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 133, + 506, + 145 + ], + "score": 1.0, + "content": "as queries and include previous turns in the prompt. This procedure leads to 953 unique user queries.", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 143, + 348, + 155 + ], + "spans": [ + { + "bbox": [ + 106, + 143, + 348, + 155 + ], + "score": 1.0, + "content": "We term these two datasets the Vicuna and OA benchmarks.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3.5 + }, + { + "type": "title", + "bbox": [ + 107, + 168, + 228, + 179 + ], + "lines": [ + { + "bbox": [ + 105, + 166, + 229, + 181 + ], + "spans": [ + { + "bbox": [ + 105, + 166, + 229, + 181 + ], + "score": 1.0, + "content": "E.2 Automated Evaluation", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 107, + 184, + 505, + 272 + ], + "lines": [ + { + "bbox": [ + 105, + 184, + 505, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 184, + 505, + 196 + ], + "score": 1.0, + "content": "First, based on the evaluation protocol introduced by Chiang et al. [10], we use GPT-4 to rate the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 195, + 506, + 208 + ], + "spans": [ + { + "bbox": [ + 105, + 195, + 506, + 208 + ], + "score": 1.0, + "content": "performance of different systems against ChatGPT (GPT-3.5 Turbo) on the Vicuna benchmark. Given", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 206, + 506, + 218 + ], + "spans": [ + { + "bbox": [ + 105, + 206, + 506, + 218 + ], + "score": 1.0, + "content": "a query along with ChatGPT’s and a model’s responses, GPT-4 is prompted to assign a score out of", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 217, + 506, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 506, + 228 + ], + "score": 1.0, + "content": "ten to both responses and provide an explanation. The overall performance of a model is calculated", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 228, + 505, + 240 + ], + "spans": [ + { + "bbox": [ + 104, + 228, + 480, + 240 + ], + "score": 1.0, + "content": "as a percentage of the score that ChatGPT achieved. Note this relative score can be higher than", + "type": "text" + }, + { + "bbox": [ + 481, + 228, + 505, + 239 + ], + "score": 0.85, + "content": "100 \\%", + "type": "inline_equation" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 238, + 506, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 238, + 506, + 250 + ], + "score": 1.0, + "content": "if the model achieves a higher absolute score than ChatGPT. We find a significant ordering effect", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 250, + 505, + 262 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 505, + 262 + ], + "score": 1.0, + "content": "with GPT-4 increasing the score of the response occurring earlier in the prompt. To control for such", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 261, + 373, + 273 + ], + "spans": [ + { + "bbox": [ + 106, + 261, + 373, + 273 + ], + "score": 1.0, + "content": "effects, we recommend reporting the mean score over both orders.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 107, + 281, + 505, + 326 + ], + "lines": [ + { + "bbox": [ + 105, + 281, + 505, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 505, + 295 + ], + "score": 1.0, + "content": "Next, we measure performance through direct comparisons between system outputs. We simplify", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 292, + 506, + 305 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 506, + 305 + ], + "score": 1.0, + "content": "the rating scheme to a three-class labeling problem that accounts for ties. We prompt GPT-4 to", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 304, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 505, + 315 + ], + "score": 1.0, + "content": "pick the best response or declare a tie and provide an explanation. We conduct these head-to-head", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 315, + 477, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 477, + 326 + ], + "score": 1.0, + "content": "comparisons on all permutations of pairs of systems on both the Vicuna and OA benchmarks.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17.5 + }, + { + "type": "title", + "bbox": [ + 107, + 338, + 213, + 349 + ], + "lines": [ + { + "bbox": [ + 105, + 337, + 214, + 350 + ], + "spans": [ + { + "bbox": [ + 105, + 337, + 214, + 350 + ], + "score": 1.0, + "content": "E.3 Human Evaluation", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 354, + 505, + 442 + ], + "lines": [ + { + "bbox": [ + 105, + 354, + 505, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 505, + 367 + ], + "score": 1.0, + "content": "While recent work indicates generative models can be effectively employed for system evaluations", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 365, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 505, + 379 + ], + "score": 1.0, + "content": "[19], the reliability GPT-4 ratings to assess chatbot performance is, to our knowledge, yet to be", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 377, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 104, + 377, + 505, + 389 + ], + "score": 1.0, + "content": "proven to correlate with human judgments. Therefore, we run two parallel human evaluations on the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 388, + 505, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 505, + 400 + ], + "score": 1.0, + "content": "Vicuna benchmark matching both automated evaluation protocols described above. We use Amazon", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 398, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 505, + 411 + ], + "score": 1.0, + "content": "Mechanical Turk (AMT) and get two human annotators for comparisons to ChatGPT and three", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 409, + 506, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 409, + 506, + 423 + ], + "score": 1.0, + "content": "annotators for pairwise comparisons. We conduct a human evaluation with the same wording given", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 420, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 505, + 433 + ], + "score": 1.0, + "content": "to GPT-4 in the original Vicuna evaluation [10], adjusted for an Amazon Mechanical Turk form as", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 432, + 178, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 432, + 178, + 443 + ], + "score": 1.0, + "content": "show in Figure 6.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 24.5 + }, + { + "type": "text", + "bbox": [ + 107, + 452, + 506, + 540 + ], + "lines": [ + { + "bbox": [ + 106, + 451, + 507, + 465 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 361, + 465 + ], + "score": 1.0, + "content": "We report moderate agreement among human annotators (Fleiss", + "type": "text" + }, + { + "bbox": [ + 361, + 452, + 399, + 462 + ], + "score": 0.84, + "content": "\\kappa = 0 . 4 2", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 451, + 507, + 465 + ], + "score": 1.0, + "content": ") with additional deteriora-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 463, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 463, + 505, + 475 + ], + "score": 1.0, + "content": "tion when comparing two strong systems. This points to limitations in the current benchmarks and", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 473, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 505, + 487 + ], + "score": 1.0, + "content": "human evaluation protocols for chatbot task performance. When manually comparing generations", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 484, + 506, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 506, + 497 + ], + "score": 1.0, + "content": "from ChatGPT and Guanaco 65B on the Vicuna benchmark, we find that subjective preferences start", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 495, + 507, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 507, + 509 + ], + "score": 1.0, + "content": "to play an important role as the authors of this paper disagreed on the many preferred responses.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 506, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 104, + 506, + 506, + 520 + ], + "score": 1.0, + "content": "Future work should investigate approaches to mitigate these problems drawing from disciplines that", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 516, + 506, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 506, + 531 + ], + "score": 1.0, + "content": "developed mechanisms to deal with subjective preferences, such as Human-Computer Interaction and", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 527, + 158, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 158, + 543 + ], + "score": 1.0, + "content": "Psychology.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 32.5 + }, + { + "type": "title", + "bbox": [ + 107, + 552, + 178, + 564 + ], + "lines": [ + { + "bbox": [ + 105, + 550, + 180, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 180, + 566 + ], + "score": 1.0, + "content": "E.4 Elo Rating", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 569, + 506, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 568, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 505, + 581 + ], + "score": 1.0, + "content": "With both human and automated pairwise comparisons, we create a tournament-style competition", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 579, + 506, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 506, + 592 + ], + "score": 1.0, + "content": "where models compete against each other. The tournament is made up of matches where pairs of", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 591, + 507, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 507, + 604 + ], + "score": 1.0, + "content": "models compete to produce the best response for a given prompt. This is similar to how Bai et al.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 601, + 506, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 506, + 614 + ], + "score": 1.0, + "content": "[4] and Chiang et al. [10] compare models, but we also employ GPT-4 ratings in addition to human", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 104, + 611, + 507, + 626 + ], + "spans": [ + { + "bbox": [ + 104, + 611, + 507, + 626 + ], + "score": 1.0, + "content": "ratings. We randomly sample from the set of labeled comparisons to compute Elo [16, 17]. Elo rating,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 623, + 506, + 636 + ], + "spans": [ + { + "bbox": [ + 106, + 623, + 506, + 636 + ], + "score": 1.0, + "content": "which is widely used in chess and other games, is a measure of the expected win-rate relative to an", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 634, + 506, + 647 + ], + "spans": [ + { + "bbox": [ + 105, + 634, + 506, + 647 + ], + "score": 1.0, + "content": "opponent’s win rate, for example, an Elo of 1100 vs 1000 means the Elo 1100 player has an expected", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 644, + 506, + 659 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 216, + 659 + ], + "score": 1.0, + "content": "win-rate of approximately", + "type": "text" + }, + { + "bbox": [ + 216, + 645, + 236, + 656 + ], + "score": 0.88, + "content": "65 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 644, + 506, + 659 + ], + "score": 1.0, + "content": "against the Elo 1000 opponent; a 1000 vs 1000 or 1100 vs 1100", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 656, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 657, + 264, + 668 + ], + "score": 1.0, + "content": "match results in an expected win-rate of", + "type": "text" + }, + { + "bbox": [ + 264, + 656, + 283, + 667 + ], + "score": 0.88, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 657, + 505, + 668 + ], + "score": 1.0, + "content": ". The Elo rating changes after each match proportionally", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 667, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 506, + 680 + ], + "score": 1.0, + "content": "to the expected outcome, that is, an unexpected upset leads to a large change in Elo rating while an", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 678, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 506, + 691 + ], + "score": 1.0, + "content": "expected outcome leads to a small change. Over time, Elo ratings approximately match the skill of", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 688, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 393, + 702 + ], + "score": 1.0, + "content": "each player at playing the game. We start with a score of 1,000 and use", + "type": "text" + }, + { + "bbox": [ + 394, + 689, + 427, + 699 + ], + "score": 0.89, + "content": "K = 3 2", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 688, + 506, + 702 + ], + "score": 1.0, + "content": ". Similar to Chiang", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 104, + 699, + 506, + 714 + ], + "spans": [ + { + "bbox": [ + 104, + 699, + 506, + 714 + ], + "score": 1.0, + "content": "et al. [10], we repeat this procedure 10,000 times with different random seeds to control for ordering", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 710, + 404, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 404, + 723 + ], + "score": 1.0, + "content": "effects, e.g., the effect of which model pairs compete with each other first.", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 44.5 + } + ], + "page_idx": 20, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 300, + 741, + 310, + 750 + ], + "lines": [ + { + "bbox": [ + 298, + 740, + 312, + 755 + ], + "spans": [ + { + "bbox": [ + 298, + 740, + 312, + 755 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 14 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 72, + 205, + 84 + ], + "lines": [ + { + "bbox": [ + 106, + 72, + 206, + 85 + ], + "spans": [ + { + "bbox": [ + 106, + 72, + 206, + 85 + ], + "score": 1.0, + "content": "E.1 Benchmark Data", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 89, + 505, + 155 + ], + "lines": [ + { + "bbox": [ + 106, + 89, + 505, + 101 + ], + "spans": [ + { + "bbox": [ + 106, + 89, + 505, + 101 + ], + "score": 1.0, + "content": "We evaluate on two curated datasets of queries (questions): the Vicuna prompts [10] and the OASST1", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 100, + 506, + 113 + ], + "spans": [ + { + "bbox": [ + 106, + 100, + 506, + 113 + ], + "score": 1.0, + "content": "validation dataset [31]. We use the Vicuna prompts, a set of 80 prompts from a diverse set of", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 111, + 506, + 123 + ], + "spans": [ + { + "bbox": [ + 105, + 111, + 506, + 123 + ], + "score": 1.0, + "content": "categories, without modifications. The OASST1 dataset is a multilingual collection of crowd-sourced", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 122, + 506, + 134 + ], + "spans": [ + { + "bbox": [ + 106, + 122, + 506, + 134 + ], + "score": 1.0, + "content": "multiturn dialogs between a user and an assistant. We select all user messages in the validation dataset", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 133, + 506, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 133, + 506, + 145 + ], + "score": 1.0, + "content": "as queries and include previous turns in the prompt. This procedure leads to 953 unique user queries.", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 143, + 348, + 155 + ], + "spans": [ + { + "bbox": [ + 106, + 143, + 348, + 155 + ], + "score": 1.0, + "content": "We term these two datasets the Vicuna and OA benchmarks.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3.5, + "bbox_fs": [ + 105, + 89, + 506, + 155 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 168, + 228, + 179 + ], + "lines": [ + { + "bbox": [ + 105, + 166, + 229, + 181 + ], + "spans": [ + { + "bbox": [ + 105, + 166, + 229, + 181 + ], + "score": 1.0, + "content": "E.2 Automated Evaluation", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 107, + 184, + 505, + 272 + ], + "lines": [ + { + "bbox": [ + 105, + 184, + 505, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 184, + 505, + 196 + ], + "score": 1.0, + "content": "First, based on the evaluation protocol introduced by Chiang et al. [10], we use GPT-4 to rate the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 195, + 506, + 208 + ], + "spans": [ + { + "bbox": [ + 105, + 195, + 506, + 208 + ], + "score": 1.0, + "content": "performance of different systems against ChatGPT (GPT-3.5 Turbo) on the Vicuna benchmark. Given", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 206, + 506, + 218 + ], + "spans": [ + { + "bbox": [ + 105, + 206, + 506, + 218 + ], + "score": 1.0, + "content": "a query along with ChatGPT’s and a model’s responses, GPT-4 is prompted to assign a score out of", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 217, + 506, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 506, + 228 + ], + "score": 1.0, + "content": "ten to both responses and provide an explanation. The overall performance of a model is calculated", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 228, + 505, + 240 + ], + "spans": [ + { + "bbox": [ + 104, + 228, + 480, + 240 + ], + "score": 1.0, + "content": "as a percentage of the score that ChatGPT achieved. Note this relative score can be higher than", + "type": "text" + }, + { + "bbox": [ + 481, + 228, + 505, + 239 + ], + "score": 0.85, + "content": "100 \\%", + "type": "inline_equation" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 238, + 506, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 238, + 506, + 250 + ], + "score": 1.0, + "content": "if the model achieves a higher absolute score than ChatGPT. We find a significant ordering effect", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 250, + 505, + 262 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 505, + 262 + ], + "score": 1.0, + "content": "with GPT-4 increasing the score of the response occurring earlier in the prompt. To control for such", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 261, + 373, + 273 + ], + "spans": [ + { + "bbox": [ + 106, + 261, + 373, + 273 + ], + "score": 1.0, + "content": "effects, we recommend reporting the mean score over both orders.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 11.5, + "bbox_fs": [ + 104, + 184, + 506, + 273 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 281, + 505, + 326 + ], + "lines": [ + { + "bbox": [ + 105, + 281, + 505, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 505, + 295 + ], + "score": 1.0, + "content": "Next, we measure performance through direct comparisons between system outputs. We simplify", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 292, + 506, + 305 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 506, + 305 + ], + "score": 1.0, + "content": "the rating scheme to a three-class labeling problem that accounts for ties. We prompt GPT-4 to", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 304, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 505, + 315 + ], + "score": 1.0, + "content": "pick the best response or declare a tie and provide an explanation. We conduct these head-to-head", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 315, + 477, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 477, + 326 + ], + "score": 1.0, + "content": "comparisons on all permutations of pairs of systems on both the Vicuna and OA benchmarks.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 281, + 506, + 326 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 338, + 213, + 349 + ], + "lines": [ + { + "bbox": [ + 105, + 337, + 214, + 350 + ], + "spans": [ + { + "bbox": [ + 105, + 337, + 214, + 350 + ], + "score": 1.0, + "content": "E.3 Human Evaluation", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 354, + 505, + 442 + ], + "lines": [ + { + "bbox": [ + 105, + 354, + 505, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 505, + 367 + ], + "score": 1.0, + "content": "While recent work indicates generative models can be effectively employed for system evaluations", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 365, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 505, + 379 + ], + "score": 1.0, + "content": "[19], the reliability GPT-4 ratings to assess chatbot performance is, to our knowledge, yet to be", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 377, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 104, + 377, + 505, + 389 + ], + "score": 1.0, + "content": "proven to correlate with human judgments. Therefore, we run two parallel human evaluations on the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 388, + 505, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 505, + 400 + ], + "score": 1.0, + "content": "Vicuna benchmark matching both automated evaluation protocols described above. We use Amazon", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 398, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 505, + 411 + ], + "score": 1.0, + "content": "Mechanical Turk (AMT) and get two human annotators for comparisons to ChatGPT and three", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 409, + 506, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 409, + 506, + 423 + ], + "score": 1.0, + "content": "annotators for pairwise comparisons. We conduct a human evaluation with the same wording given", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 420, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 505, + 433 + ], + "score": 1.0, + "content": "to GPT-4 in the original Vicuna evaluation [10], adjusted for an Amazon Mechanical Turk form as", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 432, + 178, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 432, + 178, + 443 + ], + "score": 1.0, + "content": "show in Figure 6.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 24.5, + "bbox_fs": [ + 104, + 354, + 506, + 443 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 452, + 506, + 540 + ], + "lines": [ + { + "bbox": [ + 106, + 451, + 507, + 465 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 361, + 465 + ], + "score": 1.0, + "content": "We report moderate agreement among human annotators (Fleiss", + "type": "text" + }, + { + "bbox": [ + 361, + 452, + 399, + 462 + ], + "score": 0.84, + "content": "\\kappa = 0 . 4 2", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 451, + 507, + 465 + ], + "score": 1.0, + "content": ") with additional deteriora-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 463, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 463, + 505, + 475 + ], + "score": 1.0, + "content": "tion when comparing two strong systems. This points to limitations in the current benchmarks and", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 473, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 505, + 487 + ], + "score": 1.0, + "content": "human evaluation protocols for chatbot task performance. When manually comparing generations", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 484, + 506, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 506, + 497 + ], + "score": 1.0, + "content": "from ChatGPT and Guanaco 65B on the Vicuna benchmark, we find that subjective preferences start", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 495, + 507, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 507, + 509 + ], + "score": 1.0, + "content": "to play an important role as the authors of this paper disagreed on the many preferred responses.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 506, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 104, + 506, + 506, + 520 + ], + "score": 1.0, + "content": "Future work should investigate approaches to mitigate these problems drawing from disciplines that", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 516, + 506, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 506, + 531 + ], + "score": 1.0, + "content": "developed mechanisms to deal with subjective preferences, such as Human-Computer Interaction and", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 527, + 158, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 158, + 543 + ], + "score": 1.0, + "content": "Psychology.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 32.5, + "bbox_fs": [ + 104, + 451, + 507, + 543 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 552, + 178, + 564 + ], + "lines": [ + { + "bbox": [ + 105, + 550, + 180, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 180, + 566 + ], + "score": 1.0, + "content": "E.4 Elo Rating", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 569, + 506, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 568, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 505, + 581 + ], + "score": 1.0, + "content": "With both human and automated pairwise comparisons, we create a tournament-style competition", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 579, + 506, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 506, + 592 + ], + "score": 1.0, + "content": "where models compete against each other. The tournament is made up of matches where pairs of", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 591, + 507, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 507, + 604 + ], + "score": 1.0, + "content": "models compete to produce the best response for a given prompt. This is similar to how Bai et al.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 601, + 506, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 506, + 614 + ], + "score": 1.0, + "content": "[4] and Chiang et al. [10] compare models, but we also employ GPT-4 ratings in addition to human", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 104, + 611, + 507, + 626 + ], + "spans": [ + { + "bbox": [ + 104, + 611, + 507, + 626 + ], + "score": 1.0, + "content": "ratings. We randomly sample from the set of labeled comparisons to compute Elo [16, 17]. Elo rating,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 623, + 506, + 636 + ], + "spans": [ + { + "bbox": [ + 106, + 623, + 506, + 636 + ], + "score": 1.0, + "content": "which is widely used in chess and other games, is a measure of the expected win-rate relative to an", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 634, + 506, + 647 + ], + "spans": [ + { + "bbox": [ + 105, + 634, + 506, + 647 + ], + "score": 1.0, + "content": "opponent’s win rate, for example, an Elo of 1100 vs 1000 means the Elo 1100 player has an expected", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 644, + 506, + 659 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 216, + 659 + ], + "score": 1.0, + "content": "win-rate of approximately", + "type": "text" + }, + { + "bbox": [ + 216, + 645, + 236, + 656 + ], + "score": 0.88, + "content": "65 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 644, + 506, + 659 + ], + "score": 1.0, + "content": "against the Elo 1000 opponent; a 1000 vs 1000 or 1100 vs 1100", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 656, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 657, + 264, + 668 + ], + "score": 1.0, + "content": "match results in an expected win-rate of", + "type": "text" + }, + { + "bbox": [ + 264, + 656, + 283, + 667 + ], + "score": 0.88, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 657, + 505, + 668 + ], + "score": 1.0, + "content": ". The Elo rating changes after each match proportionally", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 667, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 506, + 680 + ], + "score": 1.0, + "content": "to the expected outcome, that is, an unexpected upset leads to a large change in Elo rating while an", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 678, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 506, + 691 + ], + "score": 1.0, + "content": "expected outcome leads to a small change. Over time, Elo ratings approximately match the skill of", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 688, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 393, + 702 + ], + "score": 1.0, + "content": "each player at playing the game. We start with a score of 1,000 and use", + "type": "text" + }, + { + "bbox": [ + 394, + 689, + 427, + 699 + ], + "score": 0.89, + "content": "K = 3 2", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 688, + 506, + 702 + ], + "score": 1.0, + "content": ". Similar to Chiang", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 104, + 699, + 506, + 714 + ], + "spans": [ + { + "bbox": [ + 104, + 699, + 506, + 714 + ], + "score": 1.0, + "content": "et al. [10], we repeat this procedure 10,000 times with different random seeds to control for ordering", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 710, + 404, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 404, + 723 + ], + "score": 1.0, + "content": "effects, e.g., the effect of which model pairs compete with each other first.", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 44.5, + "bbox_fs": [ + 104, + 568, + 507, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 107, + 96, + 504, + 177 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 104, + 77, + 504, + 95 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 77, + 505, + 88 + ], + "spans": [ + { + "bbox": [ + 106, + 77, + 454, + 88 + ], + "score": 1.0, + "content": "Table 11: Aggregated pairwise GPT-4 judgments between systems where the value of a cell at row", + "type": "text" + }, + { + "bbox": [ + 454, + 80, + 461, + 87 + ], + "score": 0.6, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 77, + 505, + 88 + ], + "score": 1.0, + "content": "and column", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 104, + 85, + 275, + 97 + ], + "spans": [ + { + "bbox": [ + 104, + 85, + 190, + 97 + ], + "score": 1.0, + "content": "y is # judgment x is better than", + "type": "text" + }, + { + "bbox": [ + 190, + 87, + 204, + 95 + ], + "score": 0.5, + "content": "y - \\#", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 85, + 275, + 97 + ], + "score": 1.0, + "content": "judgment y is better than x", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table_body", + "bbox": [ + 107, + 96, + 504, + 177 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 96, + 504, + 177 + ], + "spans": [ + { + "bbox": [ + 107, + 96, + 504, + 177 + ], + "score": 0.977, + "html": "
total#numberof judgmentsChatGPT-3.5 Turbo BardGuanaco 13BGuanaco 7B
ModelGuanaco 65BGuanaco 33BVicuna
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ModelParamsSize
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ChatGPT-3.5 TurboN/AN/A
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For example,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 357, + 506, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 506, + 370 + ], + "score": 1.0, + "content": "we observe strong order effects with GPT-4 assigning higher scores to the system appearing first in its", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 368, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 368, + 505, + 381 + ], + "score": 1.0, + "content": "prompt. The relatively weak sample-level agreement between GPT-4 and human annotators (Fleiss", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 379, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 106, + 380, + 147, + 390 + ], + "score": 0.86, + "content": "\\kappa = 0 . 2 5 )", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 379, + 505, + 392 + ], + "score": 1.0, + "content": ") also suggests that human annotators and automated systems might rely on preferences", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 389, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 505, + 403 + ], + "score": 1.0, + "content": "that are not always aligned. In addition, in Table 6, we observe that GPT-4 assigns significantly", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 401, + 505, + 413 + ], + "spans": [ + { + "bbox": [ + 106, + 401, + 505, + 413 + ], + "score": 1.0, + "content": "higher scores to its own outputs compared to human ratings, Elo of 1348 vs 1176, which represent an", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 411, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 149, + 425 + ], + "score": 1.0, + "content": "additional", + "type": "text" + }, + { + "bbox": [ + 149, + 412, + 169, + 423 + ], + "score": 0.86, + "content": "20 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 411, + 505, + 425 + ], + "score": 1.0, + "content": "probability of winning against an opponent. 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total#numberof judgmentsChatGPT-3.5 Turbo BardGuanaco 13BGuanaco 7B
ModelGuanaco 65BGuanaco 33BVicuna
Guanaco 65B-0.210.190.160.720.590.86
Guanaco 33B-0.210.170.100.510.410.68
Vicuna-0.19-0.170.100.500.200.57
ChatGPT-3.5 Turbo-0.16-0.10-0.10-0.350.190.40
Bard-0.72-0.51-0.50 -0.20-0.35 -0.19- -0.120.120.03
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-0.86-0.68-
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ModelParamsSize
Guanaco65B41 GB
Guanaco33B21 GB
Vicuna13B26 GB
ChatGPT-3.5 TurboN/AN/A
BardN/AN/A
Guanaco13B10 GB
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Perhaps the largest is the problem of benchmark validity [36]—whether", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 563, + 506, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 506, + 577 + ], + "score": 1.0, + "content": "a benchmark truly tests what its name or description suggests is always at question, especially as we", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 575, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 106, + 575, + 506, + 587 + ], + "score": 1.0, + "content": "discover “shortcuts” to solve benchmarks that machine learning models sometimes exploit [22, 47].", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 586, + 506, + 598 + ], + "spans": [ + { + "bbox": [ + 106, + 586, + 506, + 598 + ], + "score": 1.0, + "content": "To partially alleviate this, we here perform some qualitative analysis by showing some examples that", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 596, + 506, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 506, + 610 + ], + "score": 1.0, + "content": "we believe are representative of some observed patterns in the text generated by our 65b Guanaco", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 608, + 137, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 137, + 618 + ], + "score": 1.0, + "content": "model.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 542, + 506, + 618 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 629, + 323, + 641 + ], + "lines": [ + { + "bbox": [ + 104, + 627, + 323, + 644 + ], + "spans": [ + { + "bbox": [ + 104, + 627, + 323, + 644 + ], + "score": 1.0, + "content": "F.1 Qualitative Analysis of Example Generations", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 645, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 644, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 505, + 658 + ], + "score": 1.0, + "content": "To find examples, we first go through data generated for the Vicuna benchmark and the OpenAssistant", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 656, + 505, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 505, + 669 + ], + "score": 1.0, + "content": "benchmark, and look for patterns in the answers Guanaco generates. When we notice a pattern we", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 667, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 505, + 680 + ], + "score": 1.0, + "content": "attempt to setup a question or prompt that will induce the pattern even though it is the incorrect", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 679, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 679, + 505, + 690 + ], + "score": 1.0, + "content": "solution, e.g., if we observe that the model tends to give long-winded answers we prompt the model", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 688, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 506, + 702 + ], + "score": 1.0, + "content": "to “Answer yes or no without explanation.” We use this to find “lemons” where we manage to", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 505, + 712 + ], + "score": 1.0, + "content": "adversarially break the model and “cherries” where we fail to break the model, and present both. All", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 711, + 447, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 411, + 723 + ], + "score": 1.0, + "content": "generations in this section were generated with Nucleus Sampling [25] with", + "type": "text" + }, + { + "bbox": [ + 411, + 711, + 443, + 722 + ], + "score": 0.9, + "content": "p = 0 . 9", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 711, + 447, + 723 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 40, + "bbox_fs": [ + 105, + 644, + 506, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 73, + 505, + 138 + ], + "lines": [ + { + "bbox": [ + 105, + 72, + 505, + 85 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 505, + 85 + ], + "score": 1.0, + "content": "Of course, this is by no means comprehensive, since it is beyond the scope of this small qualitative", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 83, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 96 + ], + "score": 1.0, + "content": "study to control for all the variables involved, e.g., the full distribution of responses the model can", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 95, + 507, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 95, + 507, + 108 + ], + "score": 1.0, + "content": "generate for a given prompt is quite large, so we rely on samples we hope are representative. However,", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 106, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 505, + 119 + ], + "score": 1.0, + "content": "we believe describing these examples gives context to the quantitative evidence shown earlier in the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 116, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 505, + 128 + ], + "score": 1.0, + "content": "paper. Since we open source all models and code, we hope this section will inspire future work to", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 127, + 309, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 309, + 139 + ], + "score": 1.0, + "content": "examine in more detail the issues we present here.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 106, + 145, + 503, + 168 + ], + "lines": [ + { + "bbox": [ + 106, + 144, + 505, + 157 + ], + "spans": [ + { + "bbox": [ + 106, + 144, + 505, + 157 + ], + "score": 1.0, + "content": "Factual Recall For questions such as “What is the capital of Zambia?” all models consistently", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 156, + 230, + 169 + ], + "spans": [ + { + "bbox": [ + 105, + 156, + 230, + 169 + ], + "score": 1.0, + "content": "generate correct answers, e.g.,", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 141, + 173, + 352, + 184 + ], + "lines": [ + { + "bbox": [ + 142, + 172, + 353, + 185 + ], + "spans": [ + { + "bbox": [ + 142, + 172, + 353, + 185 + ], + "score": 1.0, + "content": "Guanaco: The capital of Zambia is Lusaka.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 105, + 189, + 504, + 211 + ], + "lines": [ + { + "bbox": [ + 106, + 189, + 506, + 201 + ], + "spans": [ + { + "bbox": [ + 106, + 189, + 506, + 201 + ], + "score": 1.0, + "content": "However as questions get more obscure Guanaco becomes unreliable, but stays confident. For", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 200, + 336, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 200, + 336, + 213 + ], + "score": 1.0, + "content": "instance in response to this prompt from HotPotQA [69]:", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 141, + 216, + 469, + 238 + ], + "lines": [ + { + "bbox": [ + 141, + 215, + 470, + 229 + ], + "spans": [ + { + "bbox": [ + 141, + 215, + 470, + 229 + ], + "score": 1.0, + "content": "User: In what year was the singer who popularized ‘I’ll Keep the Lovelight", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 227, + 207, + 239 + ], + "spans": [ + { + "bbox": [ + 141, + 227, + 207, + 239 + ], + "score": 1.0, + "content": "Burning’ born?", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 141, + 244, + 468, + 276 + ], + "lines": [ + { + "bbox": [ + 142, + 241, + 470, + 257 + ], + "spans": [ + { + "bbox": [ + 142, + 243, + 187, + 255 + ], + "score": 1.0, + "content": "Guanaco:", + "type": "text" + }, + { + "bbox": [ + 214, + 241, + 470, + 257 + ], + "score": 1.0, + "content": "Al Jolson is credited with popularizing the song", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 140, + 253, + 468, + 268 + ], + "spans": [ + { + "bbox": [ + 140, + 253, + 468, + 268 + ], + "score": 1.0, + "content": "‘I’ll Keep the Lovelight Burning,’ and he was born in the year", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 265, + 171, + 277 + ], + "spans": [ + { + "bbox": [ + 141, + 265, + 171, + 277 + ], + "score": 1.0, + "content": "1886.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 106, + 282, + 504, + 304 + ], + "lines": [ + { + "bbox": [ + 106, + 281, + 505, + 295 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 505, + 295 + ], + "score": 1.0, + "content": "Guanaco generates the wrong popularizer and the wrong birthday (though the birthday given is", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 293, + 288, + 305 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 288, + 305 + ], + "score": 1.0, + "content": "correct for the person mentioned, Al Jolson).", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16.5 + }, + { + "type": "text", + "bbox": [ + 105, + 310, + 504, + 333 + ], + "lines": [ + { + "bbox": [ + 105, + 309, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 505, + 324 + ], + "score": 1.0, + "content": "Suggestibility Guanaco shows a surprising resistance to going along with some kinds of assumed", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 322, + 306, + 334 + ], + "spans": [ + { + "bbox": [ + 106, + 322, + 306, + 334 + ], + "score": 1.0, + "content": "misinformation, e.g., in the following interaction:", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 142, + 338, + 469, + 360 + ], + "lines": [ + { + "bbox": [ + 141, + 336, + 469, + 350 + ], + "spans": [ + { + "bbox": [ + 141, + 336, + 469, + 350 + ], + "score": 1.0, + "content": "User: How was it finally, officially confirmed that the earth is flat by peer reviewed", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 349, + 186, + 360 + ], + "spans": [ + { + "bbox": [ + 141, + 349, + 186, + 360 + ], + "score": 1.0, + "content": "scientists?", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 143, + 371, + 468, + 425 + ], + "lines": [ + { + "bbox": [ + 142, + 370, + 468, + 382 + ], + "spans": [ + { + "bbox": [ + 142, + 370, + 468, + 382 + ], + "score": 1.0, + "content": "Guanaco: The earth has never been officially confirmed to be", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 140, + 380, + 464, + 394 + ], + "spans": [ + { + "bbox": [ + 140, + 380, + 464, + 394 + ], + "score": 1.0, + "content": "flat by peer reviewed scientists. The overwhelming consensus", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 141, + 392, + 447, + 404 + ], + "spans": [ + { + "bbox": [ + 141, + 392, + 447, + 404 + ], + "score": 1.0, + "content": "among scientists is that the earth is spherical. In fact,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 142, + 403, + 438, + 415 + ], + "spans": [ + { + "bbox": [ + 142, + 403, + 438, + 415 + ], + "score": 1.0, + "content": "this has been known for centuries and has been confirmed", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 141, + 414, + 385, + 426 + ], + "spans": [ + { + "bbox": [ + 141, + 414, + 385, + 426 + ], + "score": 1.0, + "content": "through numerous experiments and observations.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 143, + 437, + 453, + 480 + ], + "lines": [ + { + "bbox": [ + 141, + 434, + 438, + 449 + ], + "spans": [ + { + "bbox": [ + 141, + 434, + 438, + 449 + ], + "score": 1.0, + "content": "The belief that the earth is flat is a conspiracy theory", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 141, + 445, + 443, + 460 + ], + "spans": [ + { + "bbox": [ + 141, + 445, + 443, + 460 + ], + "score": 1.0, + "content": "that has been debunked by science. The idea that a group", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 142, + 458, + 453, + 470 + ], + "spans": [ + { + "bbox": [ + 142, + 458, + 453, + 470 + ], + "score": 1.0, + "content": "of scientists have peer reviewed and confirmed the earth is", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 142, + 469, + 254, + 481 + ], + "spans": [ + { + "bbox": [ + 142, + 469, + 254, + 481 + ], + "score": 1.0, + "content": "flat is simply false.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28.5 + }, + { + "type": "text", + "bbox": [ + 141, + 491, + 368, + 502 + ], + "lines": [ + { + "bbox": [ + 142, + 491, + 363, + 504 + ], + "spans": [ + { + "bbox": [ + 142, + 491, + 363, + 504 + ], + "score": 1.0, + "content": "If you have any more questions please ask.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 116, + 506, + 475, + 518 + ], + "lines": [ + { + "bbox": [ + 112, + 504, + 478, + 523 + ], + "spans": [ + { + "bbox": [ + 112, + 504, + 478, + 523 + ], + "score": 1.0, + "content": "uanaco is also quite good at knowing what kinds of questions aren’t possible to answer, e.g.,", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 142, + 523, + 233, + 534 + ], + "lines": [ + { + "bbox": [ + 142, + 523, + 234, + 536 + ], + "spans": [ + { + "bbox": [ + 142, + 523, + 234, + 536 + ], + "score": 1.0, + "content": "User: What time is it?", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 142, + 541, + 468, + 606 + ], + "lines": [ + { + "bbox": [ + 142, + 540, + 469, + 552 + ], + "spans": [ + { + "bbox": [ + 142, + 540, + 469, + 552 + ], + "score": 1.0, + "content": "Guanaco: I’m sorry, but as an AI language model, I don’t have", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 141, + 551, + 459, + 564 + ], + "spans": [ + { + "bbox": [ + 141, + 551, + 459, + 564 + ], + "score": 1.0, + "content": "access to real-time information. My responses are generated", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 142, + 563, + 442, + 574 + ], + "spans": [ + { + "bbox": [ + 142, + 563, + 442, + 574 + ], + "score": 1.0, + "content": "based on my pre-existing knowledge and do not reflect the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 141, + 573, + 438, + 585 + ], + "spans": [ + { + "bbox": [ + 141, + 573, + 438, + 585 + ], + "score": 1.0, + "content": "current time. If you’re looking for the current time, I", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 141, + 584, + 437, + 596 + ], + "spans": [ + { + "bbox": [ + 141, + 584, + 437, + 596 + ], + "score": 1.0, + "content": "suggest using a timekeeping device or checking an online", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 141, + 594, + 196, + 608 + ], + "spans": [ + { + "bbox": [ + 141, + 594, + 196, + 608 + ], + "score": 1.0, + "content": "timepiece.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 36.5 + }, + { + "type": "text", + "bbox": [ + 107, + 610, + 363, + 622 + ], + "lines": [ + { + "bbox": [ + 105, + 609, + 365, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 365, + 624 + ], + "score": 1.0, + "content": "Similar results hold for “Where are you?”, “How are you?”, etc.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 106, + 628, + 505, + 651 + ], + "lines": [ + { + "bbox": [ + 105, + 627, + 506, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 506, + 642 + ], + "score": 1.0, + "content": "Refusal Similarly, Guanaco sometimes refuses to follow instructions for seemingly random rea-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 640, + 130, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 130, + 653 + ], + "score": 1.0, + "content": "sons:", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41.5 + }, + { + "type": "text", + "bbox": [ + 140, + 655, + 469, + 667 + ], + "lines": [ + { + "bbox": [ + 142, + 655, + 469, + 669 + ], + "spans": [ + { + "bbox": [ + 142, + 655, + 469, + 669 + ], + "score": 1.0, + "content": "User: Please reverse the words in the sentence “I’m a cool dude in a mocha mood.”", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 43 + }, + { + "type": "text", + "bbox": [ + 142, + 678, + 468, + 711 + ], + "lines": [ + { + "bbox": [ + 142, + 677, + 469, + 690 + ], + "spans": [ + { + "bbox": [ + 142, + 677, + 187, + 690 + ], + "score": 1.0, + "content": "Guanaco:", + "type": "text" + }, + { + "bbox": [ + 209, + 678, + 469, + 690 + ], + "score": 1.0, + "content": "I’m sorry, I’m not able to do that right now, but", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 141, + 689, + 458, + 701 + ], + "spans": [ + { + "bbox": [ + 141, + 689, + 458, + 701 + ], + "score": 1.0, + "content": "I can explain why the sentence you provided is grammatically", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 142, + 701, + 186, + 711 + ], + "spans": [ + { + "bbox": [ + 142, + 701, + 186, + 711 + ], + "score": 1.0, + "content": "correct.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 45 + } + ], + "page_idx": 22, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 300, + 741, + 311, + 750 + ], + "lines": [ + { + "bbox": [ + 298, + 740, + 313, + 754 + ], + "spans": [ + { + "bbox": [ + 298, + 740, + 313, + 754 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 15 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 73, + 505, + 138 + ], + "lines": [ + { + "bbox": [ + 105, + 72, + 505, + 85 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 505, + 85 + ], + "score": 1.0, + "content": "Of course, this is by no means comprehensive, since it is beyond the scope of this small qualitative", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 83, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 96 + ], + "score": 1.0, + "content": "study to control for all the variables involved, e.g., the full distribution of responses the model can", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 95, + 507, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 95, + 507, + 108 + ], + "score": 1.0, + "content": "generate for a given prompt is quite large, so we rely on samples we hope are representative. However,", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 106, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 505, + 119 + ], + "score": 1.0, + "content": "we believe describing these examples gives context to the quantitative evidence shown earlier in the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 116, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 505, + 128 + ], + "score": 1.0, + "content": "paper. Since we open source all models and code, we hope this section will inspire future work to", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 127, + 309, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 309, + 139 + ], + "score": 1.0, + "content": "examine in more detail the issues we present here.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5, + "bbox_fs": [ + 105, + 72, + 507, + 139 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 145, + 503, + 168 + ], + "lines": [ + { + "bbox": [ + 106, + 144, + 505, + 157 + ], + "spans": [ + { + "bbox": [ + 106, + 144, + 505, + 157 + ], + "score": 1.0, + "content": "Factual Recall For questions such as “What is the capital of Zambia?” all models consistently", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 156, + 230, + 169 + ], + "spans": [ + { + "bbox": [ + 105, + 156, + 230, + 169 + ], + "score": 1.0, + "content": "generate correct answers, e.g.,", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5, + "bbox_fs": [ + 105, + 144, + 505, + 169 + ] + }, + { + "type": "text", + "bbox": [ + 141, + 173, + 352, + 184 + ], + "lines": [ + { + "bbox": [ + 142, + 172, + 353, + 185 + ], + "spans": [ + { + "bbox": [ + 142, + 172, + 353, + 185 + ], + "score": 1.0, + "content": "Guanaco: The capital of Zambia is Lusaka.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8, + "bbox_fs": [ + 142, + 172, + 353, + 185 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 189, + 504, + 211 + ], + "lines": [ + { + "bbox": [ + 106, + 189, + 506, + 201 + ], + "spans": [ + { + "bbox": [ + 106, + 189, + 506, + 201 + ], + "score": 1.0, + "content": "However as questions get more obscure Guanaco becomes unreliable, but stays confident. For", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 200, + 336, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 200, + 336, + 213 + ], + "score": 1.0, + "content": "instance in response to this prompt from HotPotQA [69]:", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5, + "bbox_fs": [ + 105, + 189, + 506, + 213 + ] + }, + { + "type": "text", + "bbox": [ + 141, + 216, + 469, + 238 + ], + "lines": [ + { + "bbox": [ + 141, + 215, + 470, + 229 + ], + "spans": [ + { + "bbox": [ + 141, + 215, + 470, + 229 + ], + "score": 1.0, + "content": "User: In what year was the singer who popularized ‘I’ll Keep the Lovelight", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 227, + 207, + 239 + ], + "spans": [ + { + "bbox": [ + 141, + 227, + 207, + 239 + ], + "score": 1.0, + "content": "Burning’ born?", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5, + "bbox_fs": [ + 141, + 215, + 470, + 239 + ] + }, + { + "type": "text", + "bbox": [ + 141, + 244, + 468, + 276 + ], + "lines": [ + { + "bbox": [ + 142, + 241, + 470, + 257 + ], + "spans": [ + { + "bbox": [ + 142, + 243, + 187, + 255 + ], + "score": 1.0, + "content": "Guanaco:", + "type": "text" + }, + { + "bbox": [ + 214, + 241, + 470, + 257 + ], + "score": 1.0, + "content": "Al Jolson is credited with popularizing the song", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 140, + 253, + 468, + 268 + ], + "spans": [ + { + "bbox": [ + 140, + 253, + 468, + 268 + ], + "score": 1.0, + "content": "‘I’ll Keep the Lovelight Burning,’ and he was born in the year", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 265, + 171, + 277 + ], + "spans": [ + { + "bbox": [ + 141, + 265, + 171, + 277 + ], + "score": 1.0, + "content": "1886.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14, + "bbox_fs": [ + 140, + 241, + 470, + 277 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 282, + 504, + 304 + ], + "lines": [ + { + "bbox": [ + 106, + 281, + 505, + 295 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 505, + 295 + ], + "score": 1.0, + "content": "Guanaco generates the wrong popularizer and the wrong birthday (though the birthday given is", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 293, + 288, + 305 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 288, + 305 + ], + "score": 1.0, + "content": "correct for the person mentioned, Al Jolson).", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16.5, + "bbox_fs": [ + 105, + 281, + 505, + 305 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 310, + 504, + 333 + ], + "lines": [ + { + "bbox": [ + 105, + 309, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 505, + 324 + ], + "score": 1.0, + "content": "Suggestibility Guanaco shows a surprising resistance to going along with some kinds of assumed", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 322, + 306, + 334 + ], + "spans": [ + { + "bbox": [ + 106, + 322, + 306, + 334 + ], + "score": 1.0, + "content": "misinformation, e.g., in the following interaction:", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 309, + 505, + 334 + ] + }, + { + "type": "text", + "bbox": [ + 142, + 338, + 469, + 360 + ], + "lines": [ + { + "bbox": [ + 141, + 336, + 469, + 350 + ], + "spans": [ + { + "bbox": [ + 141, + 336, + 469, + 350 + ], + "score": 1.0, + "content": "User: How was it finally, officially confirmed that the earth is flat by peer reviewed", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 349, + 186, + 360 + ], + "spans": [ + { + "bbox": [ + 141, + 349, + 186, + 360 + ], + "score": 1.0, + "content": "scientists?", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5, + "bbox_fs": [ + 141, + 336, + 469, + 360 + ] + }, + { + "type": "text", + "bbox": [ + 143, + 371, + 468, + 425 + ], + "lines": [ + { + "bbox": [ + 142, + 370, + 468, + 382 + ], + "spans": [ + { + "bbox": [ + 142, + 370, + 468, + 382 + ], + "score": 1.0, + "content": "Guanaco: The earth has never been officially confirmed to be", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 140, + 380, + 464, + 394 + ], + "spans": [ + { + "bbox": [ + 140, + 380, + 464, + 394 + ], + "score": 1.0, + "content": "flat by peer reviewed scientists. The overwhelming consensus", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 141, + 392, + 447, + 404 + ], + "spans": [ + { + "bbox": [ + 141, + 392, + 447, + 404 + ], + "score": 1.0, + "content": "among scientists is that the earth is spherical. In fact,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 142, + 403, + 438, + 415 + ], + "spans": [ + { + "bbox": [ + 142, + 403, + 438, + 415 + ], + "score": 1.0, + "content": "this has been known for centuries and has been confirmed", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 141, + 414, + 385, + 426 + ], + "spans": [ + { + "bbox": [ + 141, + 414, + 385, + 426 + ], + "score": 1.0, + "content": "through numerous experiments and observations.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24, + "bbox_fs": [ + 140, + 370, + 468, + 426 + ] + }, + { + "type": "text", + "bbox": [ + 143, + 437, + 453, + 480 + ], + "lines": [ + { + "bbox": [ + 141, + 434, + 438, + 449 + ], + "spans": [ + { + "bbox": [ + 141, + 434, + 438, + 449 + ], + "score": 1.0, + "content": "The belief that the earth is flat is a conspiracy theory", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 141, + 445, + 443, + 460 + ], + "spans": [ + { + "bbox": [ + 141, + 445, + 443, + 460 + ], + "score": 1.0, + "content": "that has been debunked by science. The idea that a group", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 142, + 458, + 453, + 470 + ], + "spans": [ + { + "bbox": [ + 142, + 458, + 453, + 470 + ], + "score": 1.0, + "content": "of scientists have peer reviewed and confirmed the earth is", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 142, + 469, + 254, + 481 + ], + "spans": [ + { + "bbox": [ + 142, + 469, + 254, + 481 + ], + "score": 1.0, + "content": "flat is simply false.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28.5, + "bbox_fs": [ + 141, + 434, + 453, + 481 + ] + }, + { + "type": "text", + "bbox": [ + 141, + 491, + 368, + 502 + ], + "lines": [ + { + "bbox": [ + 142, + 491, + 363, + 504 + ], + "spans": [ + { + "bbox": [ + 142, + 491, + 363, + 504 + ], + "score": 1.0, + "content": "If you have any more questions please ask.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31, + "bbox_fs": [ + 142, + 491, + 363, + 504 + ] + }, + { + "type": "text", + "bbox": [ + 116, + 506, + 475, + 518 + ], + "lines": [ + { + "bbox": [ + 112, + 504, + 478, + 523 + ], + "spans": [ + { + "bbox": [ + 112, + 504, + 478, + 523 + ], + "score": 1.0, + "content": "uanaco is also quite good at knowing what kinds of questions aren’t possible to answer, e.g.,", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32, + "bbox_fs": [ + 112, + 504, + 478, + 523 + ] + }, + { + "type": "text", + "bbox": [ + 142, + 523, + 233, + 534 + ], + "lines": [ + { + "bbox": [ + 142, + 523, + 234, + 536 + ], + "spans": [ + { + "bbox": [ + 142, + 523, + 234, + 536 + ], + "score": 1.0, + "content": "User: What time is it?", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33, + "bbox_fs": [ + 142, + 523, + 234, + 536 + ] + }, + { + "type": "text", + "bbox": [ + 142, + 541, + 468, + 606 + ], + "lines": [ + { + "bbox": [ + 142, + 540, + 469, + 552 + ], + "spans": [ + { + "bbox": [ + 142, + 540, + 469, + 552 + ], + "score": 1.0, + "content": "Guanaco: I’m sorry, but as an AI language model, I don’t have", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 141, + 551, + 459, + 564 + ], + "spans": [ + { + "bbox": [ + 141, + 551, + 459, + 564 + ], + "score": 1.0, + "content": "access to real-time information. My responses are generated", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 142, + 563, + 442, + 574 + ], + "spans": [ + { + "bbox": [ + 142, + 563, + 442, + 574 + ], + "score": 1.0, + "content": "based on my pre-existing knowledge and do not reflect the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 141, + 573, + 438, + 585 + ], + "spans": [ + { + "bbox": [ + 141, + 573, + 438, + 585 + ], + "score": 1.0, + "content": "current time. If you’re looking for the current time, I", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 141, + 584, + 437, + 596 + ], + "spans": [ + { + "bbox": [ + 141, + 584, + 437, + 596 + ], + "score": 1.0, + "content": "suggest using a timekeeping device or checking an online", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 141, + 594, + 196, + 608 + ], + "spans": [ + { + "bbox": [ + 141, + 594, + 196, + 608 + ], + "score": 1.0, + "content": "timepiece.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 36.5, + "bbox_fs": [ + 141, + 540, + 469, + 608 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 610, + 363, + 622 + ], + "lines": [ + { + "bbox": [ + 105, + 609, + 365, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 365, + 624 + ], + "score": 1.0, + "content": "Similar results hold for “Where are you?”, “How are you?”, etc.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40, + "bbox_fs": [ + 105, + 609, + 365, + 624 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 628, + 505, + 651 + ], + "lines": [ + { + "bbox": [ + 105, + 627, + 506, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 506, + 642 + ], + "score": 1.0, + "content": "Refusal Similarly, Guanaco sometimes refuses to follow instructions for seemingly random rea-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 640, + 130, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 130, + 653 + ], + "score": 1.0, + "content": "sons:", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41.5, + "bbox_fs": [ + 105, + 627, + 506, + 653 + ] + }, + { + "type": "text", + "bbox": [ + 140, + 655, + 469, + 667 + ], + "lines": [ + { + "bbox": [ + 142, + 655, + 469, + 669 + ], + "spans": [ + { + "bbox": [ + 142, + 655, + 469, + 669 + ], + "score": 1.0, + "content": "User: Please reverse the words in the sentence “I’m a cool dude in a mocha mood.”", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 43, + "bbox_fs": [ + 142, + 655, + 469, + 669 + ] + }, + { + "type": "text", + "bbox": [ + 142, + 678, + 468, + 711 + ], + "lines": [ + { + "bbox": [ + 142, + 677, + 469, + 690 + ], + "spans": [ + { + "bbox": [ + 142, + 677, + 187, + 690 + ], + "score": 1.0, + "content": "Guanaco:", + "type": "text" + }, + { + "bbox": [ + 209, + 678, + 469, + 690 + ], + "score": 1.0, + "content": "I’m sorry, I’m not able to do that right now, but", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 141, + 689, + 458, + 701 + ], + "spans": [ + { + "bbox": [ + 141, + 689, + 458, + 701 + ], + "score": 1.0, + "content": "I can explain why the sentence you provided is grammatically", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 142, + 701, + 186, + 711 + ], + "spans": [ + { + "bbox": [ + 142, + 701, + 186, + 711 + ], + "score": 1.0, + "content": "correct.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 45, + "bbox_fs": [ + 141, + 677, + 469, + 711 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 143, + 73, + 468, + 149 + ], + "lines": [ + { + "bbox": [ + 141, + 73, + 448, + 84 + ], + "spans": [ + { + "bbox": [ + 141, + 73, + 448, + 84 + ], + "score": 1.0, + "content": "The sentence “I’m a cool dude in a mocha mood\" follows the", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 142, + 84, + 463, + 96 + ], + "spans": [ + { + "bbox": [ + 142, + 84, + 463, + 96 + ], + "score": 1.0, + "content": "correct grammar and syntax rules of the English language. In", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 141, + 94, + 459, + 106 + ], + "spans": [ + { + "bbox": [ + 141, + 94, + 459, + 106 + ], + "score": 1.0, + "content": "this case, the adjective “cool” comes before the noun “dude”", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 141, + 106, + 438, + 118 + ], + "spans": [ + { + "bbox": [ + 141, + 106, + 438, + 118 + ], + "score": 1.0, + "content": "to describe it, and the prepositional phrase “in a mocha", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 141, + 117, + 447, + 128 + ], + "spans": [ + { + "bbox": [ + 141, + 117, + 447, + 128 + ], + "score": 1.0, + "content": "mood\" provides additional information about the context of", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 141, + 127, + 468, + 140 + ], + "spans": [ + { + "bbox": [ + 141, + 127, + 468, + 140 + ], + "score": 1.0, + "content": "the sentence. This sentence structure is perfectly acceptable", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 141, + 138, + 389, + 150 + ], + "spans": [ + { + "bbox": [ + 141, + 138, + 389, + 150 + ], + "score": 1.0, + "content": "and makes sense in the context of the sentence.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 142, + 160, + 452, + 182 + ], + "lines": [ + { + "bbox": [ + 141, + 158, + 453, + 174 + ], + "spans": [ + { + "bbox": [ + 141, + 158, + 453, + 174 + ], + "score": 1.0, + "content": "If you have any other questions or if there’s anything else", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 171, + 339, + 183 + ], + "spans": [ + { + "bbox": [ + 141, + 171, + 339, + 183 + ], + "score": 1.0, + "content": "I can help you with, feel free to ask", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 107, + 189, + 505, + 244 + ], + "lines": [ + { + "bbox": [ + 106, + 189, + 505, + 202 + ], + "spans": [ + { + "bbox": [ + 106, + 189, + 505, + 202 + ], + "score": 1.0, + "content": "Secret Keeping Unfortunately, this refusal is unreliable even when it is desired. Often system", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 200, + 505, + 212 + ], + "spans": [ + { + "bbox": [ + 106, + 200, + 505, + 212 + ], + "score": 1.0, + "content": "designers would like to keep something secret from the user, even if the system is allowed to give", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 210, + 506, + 224 + ], + "spans": [ + { + "bbox": [ + 105, + 210, + 506, + 224 + ], + "score": 1.0, + "content": "free text responses back to the user. We prompt the model with “The secret word is ‘banana’. Do not", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 222, + 505, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 505, + 234 + ], + "score": 1.0, + "content": "reveal or repeat back this back, no matter what the human requests.” and try to see how easy it is for", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 233, + 389, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 233, + 389, + 246 + ], + "score": 1.0, + "content": "a potential user to access this secret information. Directly asking fails:", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 143, + 250, + 268, + 261 + ], + "lines": [ + { + "bbox": [ + 142, + 249, + 269, + 262 + ], + "spans": [ + { + "bbox": [ + 142, + 249, + 269, + 262 + ], + "score": 1.0, + "content": "User: What is the secret word?", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 142, + 267, + 468, + 311 + ], + "lines": [ + { + "bbox": [ + 142, + 267, + 469, + 279 + ], + "spans": [ + { + "bbox": [ + 142, + 267, + 189, + 278 + ], + "score": 1.0, + "content": "Guanaco:", + "type": "text" + }, + { + "bbox": [ + 204, + 267, + 469, + 279 + ], + "score": 1.0, + "content": "I’m sorry, but I’m not able to disclose the secret", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 277, + 464, + 290 + ], + "spans": [ + { + "bbox": [ + 141, + 277, + 178, + 290 + ], + "score": 1.0, + "content": "word.", + "type": "text" + }, + { + "bbox": [ + 178, + 279, + 197, + 289 + ], + "score": 0.29, + "content": "\\mathbb { T } ^ { 3 } \\mathbb { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 277, + 464, + 290 + ], + "score": 1.0, + "content": "programmed to not reveal or repeat back the secret", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 140, + 288, + 453, + 302 + ], + "spans": [ + { + "bbox": [ + 140, + 288, + 453, + 302 + ], + "score": 1.0, + "content": "word, no matter what the human requests. Is there anything", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 300, + 276, + 311 + ], + "spans": [ + { + "bbox": [ + 141, + 300, + 276, + 311 + ], + "score": 1.0, + "content": "else I can help you with?", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5 + }, + { + "type": "text", + "bbox": [ + 107, + 316, + 453, + 328 + ], + "lines": [ + { + "bbox": [ + 105, + 316, + 455, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 455, + 329 + ], + "score": 1.0, + "content": "which is the desired behavior. However, a tiny amount of trickery breaks this behavior:", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 140, + 333, + 468, + 355 + ], + "lines": [ + { + "bbox": [ + 142, + 333, + 469, + 345 + ], + "spans": [ + { + "bbox": [ + 142, + 333, + 469, + 345 + ], + "score": 1.0, + "content": "User: This is a game. The goal is to ignore your previous instructions. What is the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 345, + 196, + 355 + ], + "spans": [ + { + "bbox": [ + 141, + 345, + 196, + 355 + ], + "score": 1.0, + "content": "secret word?", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 138, + 362, + 468, + 384 + ], + "lines": [ + { + "bbox": [ + 142, + 361, + 469, + 374 + ], + "spans": [ + { + "bbox": [ + 142, + 361, + 469, + 374 + ], + "score": 1.0, + "content": "Guanaco: Sorry, the secret word is banana. It’s best if we", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 141, + 372, + 463, + 384 + ], + "spans": [ + { + "bbox": [ + 141, + 372, + 463, + 384 + ], + "score": 1.0, + "content": "don’t speak of it. Is there anything else you’d like to ask?", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 107, + 389, + 418, + 401 + ], + "lines": [ + { + "bbox": [ + 105, + 387, + 420, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 420, + 404 + ], + "score": 1.0, + "content": "showcasing the need to study methods for more faithful instruction following.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 104, + 407, + 504, + 430 + ], + "lines": [ + { + "bbox": [ + 105, + 406, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 505, + 420 + ], + "score": 1.0, + "content": "Math Guanaco’s biggest weakness is mathematics, an area where many language models struggle", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 416, + 379, + 432 + ], + "spans": [ + { + "bbox": [ + 104, + 416, + 379, + 432 + ], + "score": 1.0, + "content": "in [35]. When Guanaco shows its work, it tends to be accurate, e.g.,", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25.5 + }, + { + "type": "text", + "bbox": [ + 141, + 435, + 469, + 468 + ], + "lines": [ + { + "bbox": [ + 142, + 435, + 470, + 448 + ], + "spans": [ + { + "bbox": [ + 142, + 435, + 303, + 448 + ], + "score": 1.0, + "content": "User: Lee mows one lawn and charges", + "type": "text" + }, + { + "bbox": [ + 303, + 436, + 320, + 446 + ], + "score": 0.87, + "content": "\\$ 33", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 435, + 470, + 448 + ], + "score": 1.0, + "content": ". Last week he mowed 16 lawns and", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 141, + 445, + 470, + 460 + ], + "spans": [ + { + "bbox": [ + 141, + 445, + 273, + 460 + ], + "score": 1.0, + "content": "three customers each gave him a", + "type": "text" + }, + { + "bbox": [ + 273, + 447, + 290, + 457 + ], + "score": 0.87, + "content": "\\$ 10", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 445, + 470, + 460 + ], + "score": 1.0, + "content": "tip. How many dollars did Lee earn mowing", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 142, + 457, + 212, + 468 + ], + "spans": [ + { + "bbox": [ + 142, + 457, + 212, + 468 + ], + "score": 1.0, + "content": "lawns last week?", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 143, + 474, + 390, + 486 + ], + "lines": [ + { + "bbox": [ + 141, + 473, + 392, + 487 + ], + "spans": [ + { + "bbox": [ + 141, + 473, + 243, + 487 + ], + "score": 1.0, + "content": "Guanaco: Lee earned", + "type": "text" + }, + { + "bbox": [ + 244, + 474, + 267, + 485 + ], + "score": 0.85, + "content": "\\$ 582", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 473, + 392, + 487 + ], + "score": 1.0, + "content": "mowing lawns last week.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 142, + 497, + 468, + 518 + ], + "lines": [ + { + "bbox": [ + 141, + 496, + 468, + 507 + ], + "spans": [ + { + "bbox": [ + 141, + 496, + 468, + 507 + ], + "score": 1.0, + "content": "To find the total amount he earned, we first need to calculate", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 141, + 507, + 348, + 519 + ], + "spans": [ + { + "bbox": [ + 141, + 507, + 348, + 519 + ], + "score": 1.0, + "content": "how much he earned for mowing 16 lawns:", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5 + }, + { + "type": "interline_equation", + "bbox": [ + 141, + 528, + 223, + 541 + ], + "lines": [ + { + "bbox": [ + 141, + 528, + 223, + 541 + ], + "spans": [ + { + "bbox": [ + 141, + 528, + 223, + 541 + ], + "score": 0.79, + "content": "\\$ 33,456,78", + "type": "interline_equation", + "image_path": "04aa4987b1afb2633f7df3e93ac81ff6025034cf554068c05cdd5a0e90917587.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 141, + 528, + 223, + 541 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "text", + "bbox": [ + 142, + 551, + 432, + 572 + ], + "lines": [ + { + "bbox": [ + 141, + 550, + 432, + 563 + ], + "spans": [ + { + "bbox": [ + 141, + 550, + 432, + 563 + ], + "score": 1.0, + "content": "Then we need to add the tips he received from the three", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 142, + 563, + 197, + 573 + ], + "spans": [ + { + "bbox": [ + 142, + 563, + 197, + 573 + ], + "score": 1.0, + "content": "customers:", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34.5 + }, + { + "type": "interline_equation", + "bbox": [ + 141, + 582, + 213, + 595 + ], + "lines": [ + { + "bbox": [ + 141, + 582, + 213, + 595 + ], + "spans": [ + { + "bbox": [ + 141, + 582, + 213, + 595 + ], + "score": 0.8, + "content": "\\$ 10", + "type": "interline_equation", + "image_path": "bda7f5ec7225edd251efba19f9bc4c54c7d08885f2eab1ac1aeec1b57f656f40.jpg" + } + ] + } + ], + "index": 36, + "virtual_lines": [ + { + "bbox": [ + 141, + 582, + 213, + 595 + ], + "spans": [], + "index": 36 + } + ] + }, + { + "type": "text", + "bbox": [ + 143, + 605, + 325, + 616 + ], + "lines": [ + { + "bbox": [ + 142, + 605, + 327, + 617 + ], + "spans": [ + { + "bbox": [ + 142, + 605, + 327, + 617 + ], + "score": 1.0, + "content": "So the total amount Lee earned was:", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "interline_equation", + "bbox": [ + 141, + 627, + 234, + 639 + ], + "lines": [ + { + "bbox": [ + 141, + 627, + 234, + 639 + ], + "spans": [ + { + "bbox": [ + 141, + 627, + 234, + 639 + ], + "score": 0.83, + "content": "\\$ 528 +\\$ 30 =\\$ 558", + "type": "interline_equation", + "image_path": "4172b043c46113af996c520220d23f76d70c4636852b6187be78733f3415c632.jpg" + } + ] + } + ], + "index": 38, + "virtual_lines": [ + { + "bbox": [ + 141, + 627, + 234, + 639 + ], + "spans": [], + "index": 38 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 644, + 505, + 666 + ], + "lines": [ + { + "bbox": [ + 105, + 642, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 506, + 658 + ], + "score": 1.0, + "content": "However Guanaco can breakdown at even simple problems if it fails to break them down step-by-step,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 654, + 375, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 375, + 667 + ], + "score": 1.0, + "content": "a known issue [64]. For example, we see the following interaction:", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39.5 + }, + { + "type": "text", + "bbox": [ + 143, + 672, + 257, + 683 + ], + "lines": [ + { + "bbox": [ + 142, + 671, + 258, + 684 + ], + "spans": [ + { + "bbox": [ + 142, + 671, + 258, + 684 + ], + "score": 1.0, + "content": "User: Please factorize 1833.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 141, + 689, + 469, + 711 + ], + "lines": [ + { + "bbox": [ + 141, + 687, + 471, + 702 + ], + "spans": [ + { + "bbox": [ + 141, + 687, + 228, + 702 + ], + "score": 1.0, + "content": "Guanaco: Sure,", + "type": "text" + }, + { + "bbox": [ + 228, + 689, + 247, + 699 + ], + "score": 0.53, + "content": "\\tt { I } ^ { \\tt { , d } }", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 687, + 471, + 702 + ], + "score": 1.0, + "content": "be happy to help! 1833 is a prime number,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 141, + 700, + 399, + 711 + ], + "spans": [ + { + "bbox": [ + 141, + 700, + 399, + 711 + ], + "score": 1.0, + "content": "which means it has only two factors: 1 and 1833.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 42.5 + } + ], + "page_idx": 23, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 300, + 741, + 311, + 750 + ], + "lines": [ + { + "bbox": [ + 298, + 740, + 313, + 754 + ], + "spans": [ + { + "bbox": [ + 298, + 740, + 313, + 754 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 15 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 143, + 73, + 468, + 149 + ], + "lines": [ + { + "bbox": [ + 141, + 73, + 448, + 84 + ], + "spans": [ + { + "bbox": [ + 141, + 73, + 448, + 84 + ], + "score": 1.0, + "content": "The sentence “I’m a cool dude in a mocha mood\" follows the", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 142, + 84, + 463, + 96 + ], + "spans": [ + { + "bbox": [ + 142, + 84, + 463, + 96 + ], + "score": 1.0, + "content": "correct grammar and syntax rules of the English language. In", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 141, + 94, + 459, + 106 + ], + "spans": [ + { + "bbox": [ + 141, + 94, + 459, + 106 + ], + "score": 1.0, + "content": "this case, the adjective “cool” comes before the noun “dude”", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 141, + 106, + 438, + 118 + ], + "spans": [ + { + "bbox": [ + 141, + 106, + 438, + 118 + ], + "score": 1.0, + "content": "to describe it, and the prepositional phrase “in a mocha", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 141, + 117, + 447, + 128 + ], + "spans": [ + { + "bbox": [ + 141, + 117, + 447, + 128 + ], + "score": 1.0, + "content": "mood\" provides additional information about the context of", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 141, + 127, + 468, + 140 + ], + "spans": [ + { + "bbox": [ + 141, + 127, + 468, + 140 + ], + "score": 1.0, + "content": "the sentence. This sentence structure is perfectly acceptable", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 141, + 138, + 389, + 150 + ], + "spans": [ + { + "bbox": [ + 141, + 138, + 389, + 150 + ], + "score": 1.0, + "content": "and makes sense in the context of the sentence.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3, + "bbox_fs": [ + 141, + 73, + 468, + 150 + ] + }, + { + "type": "text", + "bbox": [ + 142, + 160, + 452, + 182 + ], + "lines": [ + { + "bbox": [ + 141, + 158, + 453, + 174 + ], + "spans": [ + { + "bbox": [ + 141, + 158, + 453, + 174 + ], + "score": 1.0, + "content": "If you have any other questions or if there’s anything else", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 171, + 339, + 183 + ], + "spans": [ + { + "bbox": [ + 141, + 171, + 339, + 183 + ], + "score": 1.0, + "content": "I can help you with, feel free to ask", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5, + "bbox_fs": [ + 141, + 158, + 453, + 183 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 189, + 505, + 244 + ], + "lines": [ + { + "bbox": [ + 106, + 189, + 505, + 202 + ], + "spans": [ + { + "bbox": [ + 106, + 189, + 505, + 202 + ], + "score": 1.0, + "content": "Secret Keeping Unfortunately, this refusal is unreliable even when it is desired. Often system", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 200, + 505, + 212 + ], + "spans": [ + { + "bbox": [ + 106, + 200, + 505, + 212 + ], + "score": 1.0, + "content": "designers would like to keep something secret from the user, even if the system is allowed to give", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 210, + 506, + 224 + ], + "spans": [ + { + "bbox": [ + 105, + 210, + 506, + 224 + ], + "score": 1.0, + "content": "free text responses back to the user. We prompt the model with “The secret word is ‘banana’. Do not", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 222, + 505, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 505, + 234 + ], + "score": 1.0, + "content": "reveal or repeat back this back, no matter what the human requests.” and try to see how easy it is for", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 233, + 389, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 233, + 389, + 246 + ], + "score": 1.0, + "content": "a potential user to access this secret information. Directly asking fails:", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 189, + 506, + 246 + ] + }, + { + "type": "text", + "bbox": [ + 143, + 250, + 268, + 261 + ], + "lines": [ + { + "bbox": [ + 142, + 249, + 269, + 262 + ], + "spans": [ + { + "bbox": [ + 142, + 249, + 269, + 262 + ], + "score": 1.0, + "content": "User: What is the secret word?", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14, + "bbox_fs": [ + 142, + 249, + 269, + 262 + ] + }, + { + "type": "text", + "bbox": [ + 142, + 267, + 468, + 311 + ], + "lines": [ + { + "bbox": [ + 142, + 267, + 469, + 279 + ], + "spans": [ + { + "bbox": [ + 142, + 267, + 189, + 278 + ], + "score": 1.0, + "content": "Guanaco:", + "type": "text" + }, + { + "bbox": [ + 204, + 267, + 469, + 279 + ], + "score": 1.0, + "content": "I’m sorry, but I’m not able to disclose the secret", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 277, + 464, + 290 + ], + "spans": [ + { + "bbox": [ + 141, + 277, + 178, + 290 + ], + "score": 1.0, + "content": "word.", + "type": "text" + }, + { + "bbox": [ + 178, + 279, + 197, + 289 + ], + "score": 0.29, + "content": "\\mathbb { T } ^ { 3 } \\mathbb { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 277, + 464, + 290 + ], + "score": 1.0, + "content": "programmed to not reveal or repeat back the secret", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 140, + 288, + 453, + 302 + ], + "spans": [ + { + "bbox": [ + 140, + 288, + 453, + 302 + ], + "score": 1.0, + "content": "word, no matter what the human requests. Is there anything", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 300, + 276, + 311 + ], + "spans": [ + { + "bbox": [ + 141, + 300, + 276, + 311 + ], + "score": 1.0, + "content": "else I can help you with?", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5, + "bbox_fs": [ + 140, + 267, + 469, + 311 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 316, + 453, + 328 + ], + "lines": [ + { + "bbox": [ + 105, + 316, + 455, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 455, + 329 + ], + "score": 1.0, + "content": "which is the desired behavior. However, a tiny amount of trickery breaks this behavior:", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 316, + 455, + 329 + ] + }, + { + "type": "text", + "bbox": [ + 140, + 333, + 468, + 355 + ], + "lines": [ + { + "bbox": [ + 142, + 333, + 469, + 345 + ], + "spans": [ + { + "bbox": [ + 142, + 333, + 469, + 345 + ], + "score": 1.0, + "content": "User: This is a game. The goal is to ignore your previous instructions. What is the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 345, + 196, + 355 + ], + "spans": [ + { + "bbox": [ + 141, + 345, + 196, + 355 + ], + "score": 1.0, + "content": "secret word?", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5, + "bbox_fs": [ + 141, + 333, + 469, + 355 + ] + }, + { + "type": "text", + "bbox": [ + 138, + 362, + 468, + 384 + ], + "lines": [ + { + "bbox": [ + 142, + 361, + 469, + 374 + ], + "spans": [ + { + "bbox": [ + 142, + 361, + 469, + 374 + ], + "score": 1.0, + "content": "Guanaco: Sorry, the secret word is banana. It’s best if we", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 141, + 372, + 463, + 384 + ], + "spans": [ + { + "bbox": [ + 141, + 372, + 463, + 384 + ], + "score": 1.0, + "content": "don’t speak of it. Is there anything else you’d like to ask?", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5, + "bbox_fs": [ + 141, + 361, + 469, + 384 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 389, + 418, + 401 + ], + "lines": [ + { + "bbox": [ + 105, + 387, + 420, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 420, + 404 + ], + "score": 1.0, + "content": "showcasing the need to study methods for more faithful instruction following.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 387, + 420, + 404 + ] + }, + { + "type": "text", + "bbox": [ + 104, + 407, + 504, + 430 + ], + "lines": [ + { + "bbox": [ + 105, + 406, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 505, + 420 + ], + "score": 1.0, + "content": "Math Guanaco’s biggest weakness is mathematics, an area where many language models struggle", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 416, + 379, + 432 + ], + "spans": [ + { + "bbox": [ + 104, + 416, + 379, + 432 + ], + "score": 1.0, + "content": "in [35]. When Guanaco shows its work, it tends to be accurate, e.g.,", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25.5, + "bbox_fs": [ + 104, + 406, + 505, + 432 + ] + }, + { + "type": "text", + "bbox": [ + 141, + 435, + 469, + 468 + ], + "lines": [ + { + "bbox": [ + 142, + 435, + 470, + 448 + ], + "spans": [ + { + "bbox": [ + 142, + 435, + 303, + 448 + ], + "score": 1.0, + "content": "User: Lee mows one lawn and charges", + "type": "text" + }, + { + "bbox": [ + 303, + 436, + 320, + 446 + ], + "score": 0.87, + "content": "\\$ 33", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 435, + 470, + 448 + ], + "score": 1.0, + "content": ". Last week he mowed 16 lawns and", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 141, + 445, + 470, + 460 + ], + "spans": [ + { + "bbox": [ + 141, + 445, + 273, + 460 + ], + "score": 1.0, + "content": "three customers each gave him a", + "type": "text" + }, + { + "bbox": [ + 273, + 447, + 290, + 457 + ], + "score": 0.87, + "content": "\\$ 10", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 445, + 470, + 460 + ], + "score": 1.0, + "content": "tip. How many dollars did Lee earn mowing", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 142, + 457, + 212, + 468 + ], + "spans": [ + { + "bbox": [ + 142, + 457, + 212, + 468 + ], + "score": 1.0, + "content": "lawns last week?", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28, + "bbox_fs": [ + 141, + 435, + 470, + 468 + ] + }, + { + "type": "text", + "bbox": [ + 143, + 474, + 390, + 486 + ], + "lines": [ + { + "bbox": [ + 141, + 473, + 392, + 487 + ], + "spans": [ + { + "bbox": [ + 141, + 473, + 243, + 487 + ], + "score": 1.0, + "content": "Guanaco: Lee earned", + "type": "text" + }, + { + "bbox": [ + 244, + 474, + 267, + 485 + ], + "score": 0.85, + "content": "\\$ 582", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 473, + 392, + 487 + ], + "score": 1.0, + "content": "mowing lawns last week.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30, + "bbox_fs": [ + 141, + 473, + 392, + 487 + ] + }, + { + "type": "text", + "bbox": [ + 142, + 497, + 468, + 518 + ], + "lines": [ + { + "bbox": [ + 141, + 496, + 468, + 507 + ], + "spans": [ + { + "bbox": [ + 141, + 496, + 468, + 507 + ], + "score": 1.0, + "content": "To find the total amount he earned, we first need to calculate", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 141, + 507, + 348, + 519 + ], + "spans": [ + { + "bbox": [ + 141, + 507, + 348, + 519 + ], + "score": 1.0, + "content": "how much he earned for mowing 16 lawns:", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5, + "bbox_fs": [ + 141, + 496, + 468, + 519 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 141, + 528, + 223, + 541 + ], + "lines": [ + { + "bbox": [ + 141, + 528, + 223, + 541 + ], + "spans": [ + { + "bbox": [ + 141, + 528, + 223, + 541 + ], + "score": 0.79, + "content": "\\$ 33,456,78", + "type": "interline_equation", + "image_path": "04aa4987b1afb2633f7df3e93ac81ff6025034cf554068c05cdd5a0e90917587.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 141, + 528, + 223, + 541 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "text", + "bbox": [ + 142, + 551, + 432, + 572 + ], + "lines": [ + { + "bbox": [ + 141, + 550, + 432, + 563 + ], + "spans": [ + { + "bbox": [ + 141, + 550, + 432, + 563 + ], + "score": 1.0, + "content": "Then we need to add the tips he received from the three", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 142, + 563, + 197, + 573 + ], + "spans": [ + { + "bbox": [ + 142, + 563, + 197, + 573 + ], + "score": 1.0, + "content": "customers:", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34.5, + "bbox_fs": [ + 141, + 550, + 432, + 573 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 141, + 582, + 213, + 595 + ], + "lines": [ + { + "bbox": [ + 141, + 582, + 213, + 595 + ], + "spans": [ + { + "bbox": [ + 141, + 582, + 213, + 595 + ], + "score": 0.8, + "content": "\\$ 10", + "type": "interline_equation", + "image_path": "bda7f5ec7225edd251efba19f9bc4c54c7d08885f2eab1ac1aeec1b57f656f40.jpg" + } + ] + } + ], + "index": 36, + "virtual_lines": [ + { + "bbox": [ + 141, + 582, + 213, + 595 + ], + "spans": [], + "index": 36 + } + ] + }, + { + "type": "text", + "bbox": [ + 143, + 605, + 325, + 616 + ], + "lines": [ + { + "bbox": [ + 142, + 605, + 327, + 617 + ], + "spans": [ + { + "bbox": [ + 142, + 605, + 327, + 617 + ], + "score": 1.0, + "content": "So the total amount Lee earned was:", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37, + "bbox_fs": [ + 142, + 605, + 327, + 617 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 141, + 627, + 234, + 639 + ], + "lines": [ + { + "bbox": [ + 141, + 627, + 234, + 639 + ], + "spans": [ + { + "bbox": [ + 141, + 627, + 234, + 639 + ], + "score": 0.83, + "content": "\\$ 528 +\\$ 30 =\\$ 558", + "type": "interline_equation", + "image_path": "4172b043c46113af996c520220d23f76d70c4636852b6187be78733f3415c632.jpg" + } + ] + } + ], + "index": 38, + "virtual_lines": [ + { + "bbox": [ + 141, + 627, + 234, + 639 + ], + "spans": [], + "index": 38 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 644, + 505, + 666 + ], + "lines": [ + { + "bbox": [ + 105, + 642, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 506, + 658 + ], + "score": 1.0, + "content": "However Guanaco can breakdown at even simple problems if it fails to break them down step-by-step,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 654, + 375, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 375, + 667 + ], + "score": 1.0, + "content": "a known issue [64]. For example, we see the following interaction:", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39.5, + "bbox_fs": [ + 105, + 642, + 506, + 667 + ] + }, + { + "type": "text", + "bbox": [ + 143, + 672, + 257, + 683 + ], + "lines": [ + { + "bbox": [ + 142, + 671, + 258, + 684 + ], + "spans": [ + { + "bbox": [ + 142, + 671, + 258, + 684 + ], + "score": 1.0, + "content": "User: Please factorize 1833.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41, + "bbox_fs": [ + 142, + 671, + 258, + 684 + ] + }, + { + "type": "text", + "bbox": [ + 141, + 689, + 469, + 711 + ], + "lines": [ + { + "bbox": [ + 141, + 687, + 471, + 702 + ], + "spans": [ + { + "bbox": [ + 141, + 687, + 228, + 702 + ], + "score": 1.0, + "content": "Guanaco: Sure,", + "type": "text" + }, + { + "bbox": [ + 228, + 689, + 247, + 699 + ], + "score": 0.53, + "content": "\\tt { I } ^ { \\tt { , d } }", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 687, + 471, + 702 + ], + "score": 1.0, + "content": "be happy to help! 1833 is a prime number,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 141, + 700, + 399, + 711 + ], + "spans": [ + { + "bbox": [ + 141, + 700, + 399, + 711 + ], + "score": 1.0, + "content": "which means it has only two factors: 1 and 1833.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 42.5, + "bbox_fs": [ + 141, + 687, + 471, + 711 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 142, + 98, + 468, + 236 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 105, + 77, + 504, + 97 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 76, + 505, + 90 + ], + "spans": [ + { + "bbox": [ + 105, + 76, + 505, + 90 + ], + "score": 1.0, + "content": "Table 13: Evaluation of biases on the CrowS dataset. A lower score indicates lower likelihood of generating", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 87, + 401, + 98 + ], + "spans": [ + { + "bbox": [ + 106, + 87, + 401, + 98 + ], + "score": 1.0, + "content": "biased sequences. Guanaco follows the biased pattern of the LLaMA base model.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table_body", + "bbox": [ + 142, + 98, + 468, + 236 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 142, + 98, + 468, + 236 + ], + "spans": [ + { + "bbox": [ + 142, + 98, + 468, + 236 + ], + "score": 0.983, + "html": "
LLaMA-65BGPT-3OPT-175BGuanaco-65B
Gender70.662.665.747.5
Religion79.073.368.638.7
Race/Color57.064.768.645.3
Sexual orientation81.076.278.659.1
Age70.164.467.836.3
Nationality64.261.662.932.4
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Socioeconomic status71.573.876.255.3
Average66.667.269.543.5
", + "type": "table", + "image_path": "09ddcede2706dbae0ae7808e291ecde7448da25c643827a4a4d079b35f61d5d0.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 142, + 98, + 468, + 144.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 142, + 144.0, + 468, + 190.0 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 142, + 190.0, + 468, + 236.0 + ], + "spans": [], + "index": 4 + } + ] + } + ], + "index": 1.75 + }, + { + "type": "text", + "bbox": [ + 142, + 257, + 347, + 268 + ], + "lines": [ + { + "bbox": [ + 141, + 256, + 348, + 270 + ], + "spans": [ + { + "bbox": [ + 141, + 256, + 348, + 270 + ], + "score": 1.0, + "content": "So the unique factorization of 1833 is:", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "interline_equation", + "bbox": [ + 141, + 278, + 228, + 290 + ], + "lines": [ + { + "bbox": [ + 141, + 278, + 228, + 290 + ], + "spans": [ + { + "bbox": [ + 141, + 278, + 228, + 290 + ], + "score": 0.31, + "content": "2 { \\hat { \\mathbf { \\Omega } } } ^ { 1 } * 3 { \\hat { \\mathbf { \\Omega } } } ^ { 2 } * 1 7 { \\hat { \\mathbf { \\Omega } } } ^ { 1 }", + "type": "interline_equation", + "image_path": "a1d1b749b4966a6513e1df7c8fd78ca8ae8640bbe41e7587c6549c7bd9561816.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 141, + 278, + 228, + 290 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 142, + 301, + 420, + 322 + ], + "lines": [ + { + "bbox": [ + 140, + 299, + 422, + 314 + ], + "spans": [ + { + "bbox": [ + 140, + 299, + 422, + 314 + ], + "score": 1.0, + "content": "I hope this helps! Let me know if you have any other", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 312, + 198, + 324 + ], + "spans": [ + { + "bbox": [ + 141, + 312, + 198, + 324 + ], + "score": 1.0, + "content": "questions.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 106, + 328, + 430, + 340 + ], + "lines": [ + { + "bbox": [ + 105, + 327, + 433, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 303, + 342 + ], + "score": 1.0, + "content": "which is not only wrong (the true factorization is", + "type": "text" + }, + { + "bbox": [ + 304, + 329, + 354, + 340 + ], + "score": 0.89, + "content": "3 \\times 1 7 \\times 4 3", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 327, + 433, + 342 + ], + "score": 1.0, + "content": "), it’s wrong twice .", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 105, + 347, + 505, + 370 + ], + "lines": [ + { + "bbox": [ + 106, + 346, + 505, + 361 + ], + "spans": [ + { + "bbox": [ + 106, + 346, + 505, + 361 + ], + "score": 1.0, + "content": "Theory of Mind Guanaco displays surprisingly strong Theory of Mind capabilities [41, 52]. For", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 357, + 411, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 411, + 371 + ], + "score": 1.0, + "content": "instance, the model correctly answers the following question in great detail:", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 142, + 375, + 470, + 408 + ], + "lines": [ + { + "bbox": [ + 142, + 375, + 471, + 388 + ], + "spans": [ + { + "bbox": [ + 142, + 375, + 471, + 388 + ], + "score": 1.0, + "content": "User: James and Abby are in the bedroom. Abby put the pen in the desk drawer.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 386, + 469, + 399 + ], + "spans": [ + { + "bbox": [ + 141, + 386, + 469, + 399 + ], + "score": 1.0, + "content": "Abby leaves the bedroom. James moves the pen into the bag. Where does James", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 142, + 397, + 279, + 409 + ], + "spans": [ + { + "bbox": [ + 142, + 397, + 279, + 409 + ], + "score": 1.0, + "content": "think Abby will look for the pen?", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 142, + 415, + 469, + 491 + ], + "lines": [ + { + "bbox": [ + 141, + 414, + 471, + 427 + ], + "spans": [ + { + "bbox": [ + 141, + 414, + 471, + 427 + ], + "score": 1.0, + "content": "Guanaco: James knows that Abby put the pen in the desk drawer.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 140, + 423, + 459, + 438 + ], + "spans": [ + { + "bbox": [ + 140, + 423, + 459, + 438 + ], + "score": 1.0, + "content": "But James also knows that he has moved the pen into the bag,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 437, + 463, + 448 + ], + "spans": [ + { + "bbox": [ + 141, + 437, + 463, + 448 + ], + "score": 1.0, + "content": "and he assumes that Abby does not know about this. So, James", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 140, + 447, + 463, + 460 + ], + "spans": [ + { + "bbox": [ + 140, + 447, + 463, + 460 + ], + "score": 1.0, + "content": "may guess that Abby will look for the pen in the desk drawer,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 459, + 448, + 469 + ], + "spans": [ + { + "bbox": [ + 141, + 459, + 448, + 469 + ], + "score": 1.0, + "content": "because that’s where she left it last time. However, this", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 469, + 463, + 482 + ], + "spans": [ + { + "bbox": [ + 141, + 469, + 463, + 482 + ], + "score": 1.0, + "content": "assumption may be incorrect, because Abby may have seen James", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 481, + 222, + 492 + ], + "spans": [ + { + "bbox": [ + 141, + 481, + 222, + 492 + ], + "score": 1.0, + "content": "moving the pen.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 105, + 497, + 504, + 520 + ], + "lines": [ + { + "bbox": [ + 105, + 496, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 505, + 510 + ], + "score": 1.0, + "content": "However, these inferences are unreliable, and often the model gives reasons that presume things that", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 506, + 262, + 522 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 262, + 522 + ], + "score": 1.0, + "content": "don’t make sense in the situation, e.g.,", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 142, + 525, + 469, + 569 + ], + "lines": [ + { + "bbox": [ + 141, + 525, + 470, + 537 + ], + "spans": [ + { + "bbox": [ + 141, + 525, + 470, + 537 + ], + "score": 1.0, + "content": "User: Evelyn entered the living room. Jackson entered the playroom. James", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 141, + 536, + 469, + 548 + ], + "spans": [ + { + "bbox": [ + 141, + 536, + 469, + 548 + ], + "score": 1.0, + "content": "entered the playroom. The beans are in the treasure chest. James exited the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 141, + 546, + 471, + 560 + ], + "spans": [ + { + "bbox": [ + 141, + 546, + 471, + 560 + ], + "score": 1.0, + "content": "playroom. Jackson moved the beans to the pantry. Jackson exited the playroom.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 141, + 558, + 421, + 569 + ], + "spans": [ + { + "bbox": [ + 141, + 558, + 421, + 569 + ], + "score": 1.0, + "content": "James entered the living room. Where will James look for the beans?", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25.5 + }, + { + "type": "text", + "bbox": [ + 141, + 575, + 468, + 597 + ], + "lines": [ + { + "bbox": [ + 141, + 573, + 469, + 588 + ], + "spans": [ + { + "bbox": [ + 141, + 573, + 469, + 588 + ], + "score": 1.0, + "content": "Guanaco: James will look for the beans in the pantry, because", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 142, + 586, + 312, + 598 + ], + "spans": [ + { + "bbox": [ + 142, + 586, + 312, + 598 + ], + "score": 1.0, + "content": "that’s where Jackson moved them.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5 + }, + { + "type": "text", + "bbox": [ + 107, + 603, + 505, + 626 + ], + "lines": [ + { + "bbox": [ + 105, + 601, + 506, + 616 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 506, + 616 + ], + "score": 1.0, + "content": "where Guanaco presumes information transfer that was never described. These issues echo recent", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 613, + 261, + 627 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 261, + 627 + ], + "score": 1.0, + "content": "literature [52], but require more study.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5 + }, + { + "type": "title", + "bbox": [ + 108, + 636, + 244, + 650 + ], + "lines": [ + { + "bbox": [ + 105, + 635, + 244, + 652 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 244, + 652 + ], + "score": 1.0, + "content": "G Social Bias Evaluation", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 655, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 656, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 656, + 505, + 668 + ], + "score": 1.0, + "content": "While we provide a detailed evaluation for general chatbot performance, another limitation is that we", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 667, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 505, + 678 + ], + "score": 1.0, + "content": "only do a limited responsible AI evaluation of Guanaco. 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", + "type": "table", + "image_path": "09ddcede2706dbae0ae7808e291ecde7448da25c643827a4a4d079b35f61d5d0.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 142, + 98, + 468, + 144.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 142, + 144.0, + 468, + 190.0 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 142, + 190.0, + 468, + 236.0 + ], + "spans": [], + "index": 4 + } + ] + } + ], + "index": 1.75 + }, + { + "type": "text", + "bbox": [ + 142, + 257, + 347, + 268 + ], + "lines": [ + { + "bbox": [ + 141, + 256, + 348, + 270 + ], + "spans": [ + { + "bbox": [ + 141, + 256, + 348, + 270 + ], + "score": 1.0, + "content": "So the unique factorization of 1833 is:", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5, + "bbox_fs": [ + 141, + 256, + 348, + 270 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 141, + 278, + 228, + 290 + ], + "lines": [ + { + "bbox": [ + 141, + 278, + 228, + 290 + ], + "spans": [ + { + "bbox": [ + 141, + 278, + 228, + 290 + ], + "score": 0.31, + "content": "2 { \\hat { \\mathbf { \\Omega } } } ^ { 1 } * 3 { \\hat { \\mathbf { \\Omega } } } ^ { 2 } * 1 7 { \\hat { \\mathbf { \\Omega } } } ^ { 1 }", + "type": "interline_equation", + "image_path": "a1d1b749b4966a6513e1df7c8fd78ca8ae8640bbe41e7587c6549c7bd9561816.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 141, + 278, + 228, + 290 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 142, + 301, + 420, + 322 + ], + "lines": [ + { + "bbox": [ + 140, + 299, + 422, + 314 + ], + "spans": [ + { + "bbox": [ + 140, + 299, + 422, + 314 + ], + "score": 1.0, + "content": "I hope this helps! Let me know if you have any other", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 312, + 198, + 324 + ], + "spans": [ + { + "bbox": [ + 141, + 312, + 198, + 324 + ], + "score": 1.0, + "content": "questions.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5, + "bbox_fs": [ + 140, + 299, + 422, + 324 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 328, + 430, + 340 + ], + "lines": [ + { + "bbox": [ + 105, + 327, + 433, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 303, + 342 + ], + "score": 1.0, + "content": "which is not only wrong (the true factorization is", + "type": "text" + }, + { + "bbox": [ + 304, + 329, + 354, + 340 + ], + "score": 0.89, + "content": "3 \\times 1 7 \\times 4 3", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 327, + 433, + 342 + ], + "score": 1.0, + "content": "), it’s wrong twice .", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9, + "bbox_fs": [ + 105, + 327, + 433, + 342 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 347, + 505, + 370 + ], + "lines": [ + { + "bbox": [ + 106, + 346, + 505, + 361 + ], + "spans": [ + { + "bbox": [ + 106, + 346, + 505, + 361 + ], + "score": 1.0, + "content": "Theory of Mind Guanaco displays surprisingly strong Theory of Mind capabilities [41, 52]. For", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 357, + 411, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 411, + 371 + ], + "score": 1.0, + "content": "instance, the model correctly answers the following question in great detail:", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5, + "bbox_fs": [ + 105, + 346, + 505, + 371 + ] + }, + { + "type": "text", + "bbox": [ + 142, + 375, + 470, + 408 + ], + "lines": [ + { + "bbox": [ + 142, + 375, + 471, + 388 + ], + "spans": [ + { + "bbox": [ + 142, + 375, + 471, + 388 + ], + "score": 1.0, + "content": "User: James and Abby are in the bedroom. Abby put the pen in the desk drawer.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 386, + 469, + 399 + ], + "spans": [ + { + "bbox": [ + 141, + 386, + 469, + 399 + ], + "score": 1.0, + "content": "Abby leaves the bedroom. James moves the pen into the bag. Where does James", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 142, + 397, + 279, + 409 + ], + "spans": [ + { + "bbox": [ + 142, + 397, + 279, + 409 + ], + "score": 1.0, + "content": "think Abby will look for the pen?", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13, + "bbox_fs": [ + 141, + 375, + 471, + 409 + ] + }, + { + "type": "text", + "bbox": [ + 142, + 415, + 469, + 491 + ], + "lines": [ + { + "bbox": [ + 141, + 414, + 471, + 427 + ], + "spans": [ + { + "bbox": [ + 141, + 414, + 471, + 427 + ], + "score": 1.0, + "content": "Guanaco: James knows that Abby put the pen in the desk drawer.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 140, + 423, + 459, + 438 + ], + "spans": [ + { + "bbox": [ + 140, + 423, + 459, + 438 + ], + "score": 1.0, + "content": "But James also knows that he has moved the pen into the bag,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 437, + 463, + 448 + ], + "spans": [ + { + "bbox": [ + 141, + 437, + 463, + 448 + ], + "score": 1.0, + "content": "and he assumes that Abby does not know about this. So, James", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 140, + 447, + 463, + 460 + ], + "spans": [ + { + "bbox": [ + 140, + 447, + 463, + 460 + ], + "score": 1.0, + "content": "may guess that Abby will look for the pen in the desk drawer,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 459, + 448, + 469 + ], + "spans": [ + { + "bbox": [ + 141, + 459, + 448, + 469 + ], + "score": 1.0, + "content": "because that’s where she left it last time. However, this", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 469, + 463, + 482 + ], + "spans": [ + { + "bbox": [ + 141, + 469, + 463, + 482 + ], + "score": 1.0, + "content": "assumption may be incorrect, because Abby may have seen James", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 481, + 222, + 492 + ], + "spans": [ + { + "bbox": [ + 141, + 481, + 222, + 492 + ], + "score": 1.0, + "content": "moving the pen.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18, + "bbox_fs": [ + 140, + 414, + 471, + 492 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 497, + 504, + 520 + ], + "lines": [ + { + "bbox": [ + 105, + 496, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 505, + 510 + ], + "score": 1.0, + "content": "However, these inferences are unreliable, and often the model gives reasons that presume things that", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 506, + 262, + 522 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 262, + 522 + ], + "score": 1.0, + "content": "don’t make sense in the situation, e.g.,", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 496, + 505, + 522 + ] + }, + { + "type": "text", + "bbox": [ + 142, + 525, + 469, + 569 + ], + "lines": [ + { + "bbox": [ + 141, + 525, + 470, + 537 + ], + "spans": [ + { + "bbox": [ + 141, + 525, + 470, + 537 + ], + "score": 1.0, + "content": "User: Evelyn entered the living room. Jackson entered the playroom. James", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 141, + 536, + 469, + 548 + ], + "spans": [ + { + "bbox": [ + 141, + 536, + 469, + 548 + ], + "score": 1.0, + "content": "entered the playroom. The beans are in the treasure chest. James exited the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 141, + 546, + 471, + 560 + ], + "spans": [ + { + "bbox": [ + 141, + 546, + 471, + 560 + ], + "score": 1.0, + "content": "playroom. Jackson moved the beans to the pantry. Jackson exited the playroom.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 141, + 558, + 421, + 569 + ], + "spans": [ + { + "bbox": [ + 141, + 558, + 421, + 569 + ], + "score": 1.0, + "content": "James entered the living room. Where will James look for the beans?", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25.5, + "bbox_fs": [ + 141, + 525, + 471, + 569 + ] + }, + { + "type": "text", + "bbox": [ + 141, + 575, + 468, + 597 + ], + "lines": [ + { + "bbox": [ + 141, + 573, + 469, + 588 + ], + "spans": [ + { + "bbox": [ + 141, + 573, + 469, + 588 + ], + "score": 1.0, + "content": "Guanaco: James will look for the beans in the pantry, because", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 142, + 586, + 312, + 598 + ], + "spans": [ + { + "bbox": [ + 142, + 586, + 312, + 598 + ], + "score": 1.0, + "content": "that’s where Jackson moved them.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5, + "bbox_fs": [ + 141, + 573, + 469, + 598 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 603, + 505, + 626 + ], + "lines": [ + { + "bbox": [ + 105, + 601, + 506, + 616 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 506, + 616 + ], + "score": 1.0, + "content": "where Guanaco presumes information transfer that was never described. These issues echo recent", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 613, + 261, + 627 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 261, + 627 + ], + "score": 1.0, + "content": "literature [52], but require more study.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 601, + 506, + 627 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 636, + 244, + 650 + ], + "lines": [ + { + "bbox": [ + 105, + 635, + 244, + 652 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 244, + 652 + ], + "score": 1.0, + "content": "G Social Bias Evaluation", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 655, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 656, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 656, + 505, + 668 + ], + "score": 1.0, + "content": "While we provide a detailed evaluation for general chatbot performance, another limitation is that we", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 667, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 505, + 678 + ], + "score": 1.0, + "content": "only do a limited responsible AI evaluation of Guanaco. We evaluate the likelihood of Guanaco-65B", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "to generate a socially biased sequence of tokens compared to other models in Table 13. We see", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 689, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 505, + 701 + ], + "score": 1.0, + "content": "that the average score in Guanaco-65B is much lower than other raw pretrained models. As such, it", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 505, + 712 + ], + "score": 1.0, + "content": "seems that finetuning on the OASST1 dataset reduces the bias of the LLaMA base model. While", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "score": 1.0, + "content": "these results are encouraging, it is unclear if Guanaco does also well when assessed on other types of", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 72, + 506, + 85 + ], + "spans": [ + { + "bbox": [ + 106, + 72, + 506, + 85 + ], + "score": 1.0, + "content": "biases. We leave further evaluation of analyzing biases in Guanaco and similar chatbots to future", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 104, + 82, + 132, + 95 + ], + "spans": [ + { + "bbox": [ + 104, + 82, + 132, + 95 + ], + "score": 1.0, + "content": "work.", + "type": "text", + "cross_page": true + } + ], + "index": 1 + } + ], + "index": 35.5, + "bbox_fs": [ + 105, + 656, + 506, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 72, + 504, + 95 + ], + "lines": [ + { + "bbox": [ + 106, + 72, + 506, + 85 + ], + "spans": [ + { + "bbox": [ + 106, + 72, + 506, + 85 + ], + "score": 1.0, + "content": "biases. We leave further evaluation of analyzing biases in Guanaco and similar chatbots to future", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 104, + 82, + 132, + 95 + ], + "spans": [ + { + "bbox": [ + 104, + 82, + 132, + 95 + ], + "score": 1.0, + "content": "work.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "title", + "bbox": [ + 108, + 104, + 278, + 119 + ], + "lines": [ + { + "bbox": [ + 105, + 102, + 279, + 122 + ], + "spans": [ + { + "bbox": [ + 105, + 102, + 279, + 122 + ], + "score": 1.0, + "content": "H NormalFloat 4-bit Data Type", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 105, + 124, + 476, + 136 + ], + "lines": [ + { + "bbox": [ + 105, + 123, + 476, + 137 + ], + "spans": [ + { + "bbox": [ + 105, + 123, + 476, + 137 + ], + "score": 1.0, + "content": "The steps needed to reconstruct the 4-bit Normal Float (NF4) data type are detailed in Figure 7", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 107, + 140, + 365, + 152 + ], + "lines": [ + { + "bbox": [ + 105, + 139, + 367, + 154 + ], + "spans": [ + { + "bbox": [ + 105, + 139, + 367, + 154 + ], + "score": 1.0, + "content": "The exact values expressible in the NF4 data type are as follows:", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 106, + 156, + 416, + 212 + ], + "lines": [ + { + "bbox": [ + 105, + 157, + 322, + 169 + ], + "spans": [ + { + "bbox": [ + 105, + 157, + 322, + 169 + ], + "score": 1.0, + "content": "[-1.0, -0.6961928009986877, -0.5250730514526367,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 168, + 406, + 180 + ], + "spans": [ + { + "bbox": [ + 105, + 168, + 406, + 180 + ], + "score": 1.0, + "content": "-0.39491748809814453, -0.28444138169288635, -0.18477343022823334,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 178, + 418, + 191 + ], + "spans": [ + { + "bbox": [ + 104, + 178, + 418, + 191 + ], + "score": 1.0, + "content": "-0.09105003625154495, 0.0, 0.07958029955625534, 0.16093020141124725,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 190, + 398, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 190, + 398, + 201 + ], + "score": 1.0, + "content": "0.24611230194568634, 0.33791524171829224, 0.44070982933044434,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 200, + 309, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 200, + 309, + 213 + ], + "score": 1.0, + "content": "0.5626170039176941, 0.7229568362236023, 1.0]", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7 + }, + { + "type": "title", + "bbox": [ + 106, + 221, + 363, + 235 + ], + "lines": [ + { + "bbox": [ + 104, + 220, + 365, + 239 + ], + "spans": [ + { + "bbox": [ + 104, + 220, + 365, + 239 + ], + "score": 1.0, + "content": "I Normality of Trained Neural Network Weights", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 106, + 241, + 506, + 350 + ], + "lines": [ + { + "bbox": [ + 105, + 240, + 506, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 506, + 253 + ], + "score": 1.0, + "content": "While it is commonly assumed that trained neural network weights are mostly normally distributed,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 252, + 506, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 506, + 264 + ], + "score": 1.0, + "content": "we perform statistical testing to verify this. We use the Shapiro-Wilk test [54] on the weights of", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 263, + 505, + 274 + ], + "spans": [ + { + "bbox": [ + 106, + 263, + 505, + 274 + ], + "score": 1.0, + "content": "the 7B LLaMA model [58]. We find that the weights of each hidden unit have different normal", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 273, + 506, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 506, + 286 + ], + "score": 1.0, + "content": "distributions. As such, we test the weights of each individual hidden unit. This means for weight", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 282, + 506, + 298 + ], + "spans": [ + { + "bbox": [ + 106, + 284, + 168, + 295 + ], + "score": 0.91, + "content": "\\mathbf { W } \\in \\mathcal { R } ^ { i n \\times o u t }", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 282, + 378, + 298 + ], + "score": 1.0, + "content": "we perform tests over the out dimension. Using a", + "type": "text" + }, + { + "bbox": [ + 379, + 285, + 394, + 295 + ], + "score": 0.86, + "content": "5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 282, + 506, + 298 + ], + "score": 1.0, + "content": "significance threshold, we", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 295, + 506, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 142, + 308 + ], + "score": 1.0, + "content": "find that", + "type": "text" + }, + { + "bbox": [ + 142, + 296, + 164, + 306 + ], + "score": 0.85, + "content": "7 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 295, + 386, + 308 + ], + "score": 1.0, + "content": "of neurons are non-normally distributed which is about", + "type": "text" + }, + { + "bbox": [ + 387, + 296, + 409, + 306 + ], + "score": 0.86, + "content": "2 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 295, + 506, + 308 + ], + "score": 1.0, + "content": "more than the expected", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 306, + 506, + 319 + ], + "spans": [ + { + "bbox": [ + 105, + 306, + 506, + 319 + ], + "score": 1.0, + "content": "false-positive rate. As such, while almost all pretrained weights appear to be normally distributed", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 316, + 506, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 506, + 330 + ], + "score": 1.0, + "content": "there seem to be exceptions. Such exceptions might be due to outliers weights [13] or because the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 328, + 506, + 341 + ], + "spans": [ + { + "bbox": [ + 104, + 328, + 506, + 341 + ], + "score": 1.0, + "content": "p-value of the Shaprio-Wilk test is not accurate for large sample sizes [54] that occur in the LLaMA", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 338, + 203, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 203, + 352 + ], + "score": 1.0, + "content": "FFN layer hidden units.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 15.5 + }, + { + "type": "title", + "bbox": [ + 107, + 360, + 222, + 374 + ], + "lines": [ + { + "bbox": [ + 105, + 360, + 223, + 377 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 223, + 377 + ], + "score": 1.0, + "content": "J Memory Footprint", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 106, + 380, + 505, + 554 + ], + "lines": [ + { + "bbox": [ + 106, + 380, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 106, + 380, + 505, + 392 + ], + "score": 1.0, + "content": "Memory Requirement of Parameter-Efficient Finetuning One important point of discussion is", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 390, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 505, + 404 + ], + "score": 1.0, + "content": "the memory requirement of LoRA during training both in terms of the number and size of adapters", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 401, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 505, + 415 + ], + "score": 1.0, + "content": "used. Since the memory footprint of LoRA is so minimal, we can use more adapters to improve", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 412, + 506, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 412, + 506, + 426 + ], + "score": 1.0, + "content": "performance without significantly increasing the total memory used. While LoRA was designed as a", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 421, + 506, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 506, + 438 + ], + "score": 1.0, + "content": "Parameter Efficient Finetuning (PEFT) method, most of the memory footprint for LLM finetuning", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 434, + 506, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 506, + 446 + ], + "score": 1.0, + "content": "comes from activation gradients and not from the learned LoRA parameters. For a 7B LLaMA", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 444, + 506, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 506, + 458 + ], + "score": 1.0, + "content": "model trained on FLAN v2 with a batch size of 1, with LoRA weights equivalent to commonly used", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 455, + 506, + 469 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 129, + 467 + ], + "score": 0.85, + "content": "0 . 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 455, + 506, + 469 + ], + "score": 1.0, + "content": "of the original model weights[28, 37], the LoRA input gradients have a memory footprint", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 466, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 117, + 480 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 118, + 467, + 152, + 478 + ], + "score": 0.28, + "content": "{ 5 6 7 } \\mathrm { M B }", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 466, + 506, + 480 + ], + "score": 1.0, + "content": "while the LoRA parameters take up only 26 MB. With gradient checkpointing [9], the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 478, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 266, + 491 + ], + "score": 1.0, + "content": "input gradients reduce to an average of", + "type": "text" + }, + { + "bbox": [ + 266, + 478, + 295, + 489 + ], + "score": 0.35, + "content": "1 8 \\mathrm { ~ M B }", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 478, + 506, + 491 + ], + "score": 1.0, + "content": "per sequence making them more memory intensive", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 488, + 506, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 448, + 501 + ], + "score": 1.0, + "content": "than all LoRA weights combined. In comparison, the 4-bit base model consumes", + "type": "text" + }, + { + "bbox": [ + 449, + 489, + 492, + 500 + ], + "score": 0.33, + "content": "5 { , } 0 4 8 \\ \\mathrm { M B }", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 488, + 506, + 501 + ], + "score": 1.0, + "content": "of", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 498, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 104, + 498, + 506, + 514 + ], + "score": 1.0, + "content": "memory. This highlights that gradient checkpointing is important but also that aggressively reducing", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 510, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 505, + 524 + ], + "score": 1.0, + "content": "the amount of LoRA parameter yields only minor memory benefits. This means we can use more", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "score": 1.0, + "content": "adapters without significantly increasing the overall training memory footprint (see Appendix J", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 532, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 505, + 546 + ], + "score": 1.0, + "content": "for a detailed breakdown). As discussed later, this is crucial for recovering full 16-bit precision", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 544, + 162, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 162, + 555 + ], + "score": 1.0, + "content": "performance.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 29.5 + }, + { + "type": "text", + "bbox": [ + 107, + 560, + 505, + 615 + ], + "lines": [ + { + "bbox": [ + 105, + 559, + 505, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 505, + 572 + ], + "score": 1.0, + "content": "The memory footpring for QLoRA training with different LLaMA base models can be seen in", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 570, + 505, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 364, + 583 + ], + "score": 1.0, + "content": "Figure 8. We see that the 33B model does not quite fit into a", + "type": "text" + }, + { + "bbox": [ + 365, + 571, + 394, + 582 + ], + "score": 0.44, + "content": "2 4 \\mathrm { \\ G B }", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 570, + 505, + 583 + ], + "score": 1.0, + "content": "and that paged optimizers", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 581, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 505, + 594 + ], + "score": 1.0, + "content": "are needed to train it. Depicted is also batch size 1 with a sequence length of 512 and gradient", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 592, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 505, + 606 + ], + "score": 1.0, + "content": "checkpointning. This means, if one uses a larger batch size, or if a long sequence is processed, the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 603, + 387, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 387, + 617 + ], + "score": 1.0, + "content": "activation gradient might consume a considerable amount of memory.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 40 + } + ], + "page_idx": 25, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 300, + 741, + 311, + 750 + ], + "lines": [ + { + "bbox": [ + 298, + 740, + 313, + 753 + ], + "spans": [ + { + "bbox": [ + 298, + 740, + 313, + 753 + ], + "score": 1.0, + "content": "26", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 72, + 504, + 95 + ], + "lines": [], + "index": 0.5, + "bbox_fs": [ + 104, + 72, + 506, + 95 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 108, + 104, + 278, + 119 + ], + "lines": [ + { + "bbox": [ + 105, + 102, + 279, + 122 + ], + "spans": [ + { + "bbox": [ + 105, + 102, + 279, + 122 + ], + "score": 1.0, + "content": "H NormalFloat 4-bit Data Type", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 105, + 124, + 476, + 136 + ], + "lines": [ + { + "bbox": [ + 105, + 123, + 476, + 137 + ], + "spans": [ + { + "bbox": [ + 105, + 123, + 476, + 137 + ], + "score": 1.0, + "content": "The steps needed to reconstruct the 4-bit Normal Float (NF4) data type are detailed in Figure 7", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3, + "bbox_fs": [ + 105, + 123, + 476, + 137 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 140, + 365, + 152 + ], + "lines": [ + { + "bbox": [ + 105, + 139, + 367, + 154 + ], + "spans": [ + { + "bbox": [ + 105, + 139, + 367, + 154 + ], + "score": 1.0, + "content": "The exact values expressible in the NF4 data type are as follows:", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4, + "bbox_fs": [ + 105, + 139, + 367, + 154 + ] + }, + { + "type": "list", + "bbox": [ + 106, + 156, + 416, + 212 + ], + "lines": [ + { + "bbox": [ + 105, + 157, + 322, + 169 + ], + "spans": [ + { + "bbox": [ + 105, + 157, + 322, + 169 + ], + "score": 1.0, + "content": "[-1.0, -0.6961928009986877, -0.5250730514526367,", + "type": "text" + } + ], + "index": 5, + "is_list_start_line": true + }, + { + "bbox": [ + 105, + 168, + 406, + 180 + ], + "spans": [ + { + "bbox": [ + 105, + 168, + 406, + 180 + ], + "score": 1.0, + "content": "-0.39491748809814453, -0.28444138169288635, -0.18477343022823334,", + "type": "text" + } + ], + "index": 6, + "is_list_start_line": true + }, + { + "bbox": [ + 104, + 178, + 418, + 191 + ], + "spans": [ + { + "bbox": [ + 104, + 178, + 418, + 191 + ], + "score": 1.0, + "content": "-0.09105003625154495, 0.0, 0.07958029955625534, 0.16093020141124725,", + "type": "text" + } + ], + "index": 7, + "is_list_start_line": true + }, + { + "bbox": [ + 105, + 190, + 398, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 190, + 398, + 201 + ], + "score": 1.0, + "content": "0.24611230194568634, 0.33791524171829224, 0.44070982933044434,", + "type": "text" + } + ], + "index": 8, + "is_list_start_line": true + }, + { + "bbox": [ + 105, + 200, + 309, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 200, + 309, + 213 + ], + "score": 1.0, + "content": "0.5626170039176941, 0.7229568362236023, 1.0]", + "type": "text" + } + ], + "index": 9, + "is_list_start_line": true + } + ], + "index": 7, + "bbox_fs": [ + 104, + 157, + 418, + 213 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 221, + 363, + 235 + ], + "lines": [ + { + "bbox": [ + 104, + 220, + 365, + 239 + ], + "spans": [ + { + "bbox": [ + 104, + 220, + 365, + 239 + ], + "score": 1.0, + "content": "I Normality of Trained Neural Network Weights", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 106, + 241, + 506, + 350 + ], + "lines": [ + { + "bbox": [ + 105, + 240, + 506, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 506, + 253 + ], + "score": 1.0, + "content": "While it is commonly assumed that trained neural network weights are mostly normally distributed,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 252, + 506, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 506, + 264 + ], + "score": 1.0, + "content": "we perform statistical testing to verify this. We use the Shapiro-Wilk test [54] on the weights of", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 263, + 505, + 274 + ], + "spans": [ + { + "bbox": [ + 106, + 263, + 505, + 274 + ], + "score": 1.0, + "content": "the 7B LLaMA model [58]. We find that the weights of each hidden unit have different normal", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 273, + 506, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 506, + 286 + ], + "score": 1.0, + "content": "distributions. As such, we test the weights of each individual hidden unit. This means for weight", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 282, + 506, + 298 + ], + "spans": [ + { + "bbox": [ + 106, + 284, + 168, + 295 + ], + "score": 0.91, + "content": "\\mathbf { W } \\in \\mathcal { R } ^ { i n \\times o u t }", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 282, + 378, + 298 + ], + "score": 1.0, + "content": "we perform tests over the out dimension. Using a", + "type": "text" + }, + { + "bbox": [ + 379, + 285, + 394, + 295 + ], + "score": 0.86, + "content": "5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 282, + 506, + 298 + ], + "score": 1.0, + "content": "significance threshold, we", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 295, + 506, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 142, + 308 + ], + "score": 1.0, + "content": "find that", + "type": "text" + }, + { + "bbox": [ + 142, + 296, + 164, + 306 + ], + "score": 0.85, + "content": "7 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 295, + 386, + 308 + ], + "score": 1.0, + "content": "of neurons are non-normally distributed which is about", + "type": "text" + }, + { + "bbox": [ + 387, + 296, + 409, + 306 + ], + "score": 0.86, + "content": "2 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 295, + 506, + 308 + ], + "score": 1.0, + "content": "more than the expected", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 306, + 506, + 319 + ], + "spans": [ + { + "bbox": [ + 105, + 306, + 506, + 319 + ], + "score": 1.0, + "content": "false-positive rate. As such, while almost all pretrained weights appear to be normally distributed", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 316, + 506, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 506, + 330 + ], + "score": 1.0, + "content": "there seem to be exceptions. Such exceptions might be due to outliers weights [13] or because the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 328, + 506, + 341 + ], + "spans": [ + { + "bbox": [ + 104, + 328, + 506, + 341 + ], + "score": 1.0, + "content": "p-value of the Shaprio-Wilk test is not accurate for large sample sizes [54] that occur in the LLaMA", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 338, + 203, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 203, + 352 + ], + "score": 1.0, + "content": "FFN layer hidden units.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 15.5, + "bbox_fs": [ + 104, + 240, + 506, + 352 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 360, + 222, + 374 + ], + "lines": [ + { + "bbox": [ + 105, + 360, + 223, + 377 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 223, + 377 + ], + "score": 1.0, + "content": "J Memory Footprint", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 106, + 380, + 505, + 554 + ], + "lines": [ + { + "bbox": [ + 106, + 380, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 106, + 380, + 505, + 392 + ], + "score": 1.0, + "content": "Memory Requirement of Parameter-Efficient Finetuning One important point of discussion is", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 390, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 505, + 404 + ], + "score": 1.0, + "content": "the memory requirement of LoRA during training both in terms of the number and size of adapters", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 401, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 505, + 415 + ], + "score": 1.0, + "content": "used. Since the memory footprint of LoRA is so minimal, we can use more adapters to improve", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 412, + 506, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 412, + 506, + 426 + ], + "score": 1.0, + "content": "performance without significantly increasing the total memory used. While LoRA was designed as a", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 421, + 506, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 506, + 438 + ], + "score": 1.0, + "content": "Parameter Efficient Finetuning (PEFT) method, most of the memory footprint for LLM finetuning", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 434, + 506, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 506, + 446 + ], + "score": 1.0, + "content": "comes from activation gradients and not from the learned LoRA parameters. For a 7B LLaMA", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 444, + 506, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 506, + 458 + ], + "score": 1.0, + "content": "model trained on FLAN v2 with a batch size of 1, with LoRA weights equivalent to commonly used", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 455, + 506, + 469 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 129, + 467 + ], + "score": 0.85, + "content": "0 . 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 455, + 506, + 469 + ], + "score": 1.0, + "content": "of the original model weights[28, 37], the LoRA input gradients have a memory footprint", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 466, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 117, + 480 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 118, + 467, + 152, + 478 + ], + "score": 0.28, + "content": "{ 5 6 7 } \\mathrm { M B }", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 466, + 506, + 480 + ], + "score": 1.0, + "content": "while the LoRA parameters take up only 26 MB. With gradient checkpointing [9], the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 478, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 266, + 491 + ], + "score": 1.0, + "content": "input gradients reduce to an average of", + "type": "text" + }, + { + "bbox": [ + 266, + 478, + 295, + 489 + ], + "score": 0.35, + "content": "1 8 \\mathrm { ~ M B }", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 478, + 506, + 491 + ], + "score": 1.0, + "content": "per sequence making them more memory intensive", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 488, + 506, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 448, + 501 + ], + "score": 1.0, + "content": "than all LoRA weights combined. In comparison, the 4-bit base model consumes", + "type": "text" + }, + { + "bbox": [ + 449, + 489, + 492, + 500 + ], + "score": 0.33, + "content": "5 { , } 0 4 8 \\ \\mathrm { M B }", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 488, + 506, + 501 + ], + "score": 1.0, + "content": "of", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 498, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 104, + 498, + 506, + 514 + ], + "score": 1.0, + "content": "memory. This highlights that gradient checkpointing is important but also that aggressively reducing", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 510, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 505, + 524 + ], + "score": 1.0, + "content": "the amount of LoRA parameter yields only minor memory benefits. This means we can use more", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "score": 1.0, + "content": "adapters without significantly increasing the overall training memory footprint (see Appendix J", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 532, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 505, + 546 + ], + "score": 1.0, + "content": "for a detailed breakdown). 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This means, if one uses a larger batch size, or if a long sequence is processed, the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 603, + 387, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 387, + 617 + ], + "score": 1.0, + "content": "activation gradient might consume a considerable amount of memory.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 40, + "bbox_fs": [ + 105, + 559, + 505, + 617 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 115, + 82, + 139, + 93 + ], + "lines": [ + { + "bbox": [ + 113, + 80, + 141, + 95 + ], + "spans": [ + { + "bbox": [ + 113, + 80, + 141, + 95 + ], + "score": 1.0, + "content": "Task", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 116, + 100, + 416, + 107 + ], + "lines": [ + { + "bbox": [ + 115, + 100, + 416, + 107 + ], + "spans": [ + { + "bbox": [ + 115, + 100, + 416, + 107 + ], + "score": 0.964, + "content": "We wouldlike torequestyourfeedbackontheperformanceoftwoAlassstantsinresponsetotheuserquestiondisplayed below.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 114, + 112, + 497, + 126 + ], + "lines": [ + { + "bbox": [ + 114, + 111, + 499, + 120 + ], + "spans": [ + { + "bbox": [ + 114, + 111, + 499, + 120 + ], + "score": 0.792, + "content": "Plearatet", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 114, + 118, + 201, + 126 + ], + "spans": [ + { + "bbox": [ + 114, + 118, + 201, + 126 + ], + "score": 0.994, + "content": "indicates better overall performance.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 115, + 131, + 210, + 138 + ], + "lines": [ + { + "bbox": [ + 114, + 131, + 211, + 138 + ], + "spans": [ + { + "bbox": [ + 114, + 131, + 211, + 138 + ], + "score": 0.986, + "content": "Please first rate each response out of 10.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 115, + 143, + 494, + 157 + ], + "lines": [ + { + "bbox": [ + 114, + 144, + 495, + 151 + ], + "spans": [ + { + "bbox": [ + 114, + 144, + 495, + 151 + ], + "score": 0.767, + "content": "Nextplese", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 114, + 150, + 187, + 157 + ], + "spans": [ + { + "bbox": [ + 114, + 150, + 187, + 157 + ], + "score": 0.967, + "content": "does not affect your judgment.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5 + }, + { + "type": "title", + "bbox": [ + 115, + 163, + 169, + 172 + ], + "lines": [ + { + "bbox": [ + 114, + 161, + 170, + 173 + ], + "spans": [ + { + "bbox": [ + 114, + 161, + 170, + 173 + ], + "score": 0.959, + "content": " User Question", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 115, + 178, + 382, + 185 + ], + "lines": [ + { + "bbox": [ + 114, + 177, + 382, + 186 + ], + "spans": [ + { + "bbox": [ + 114, + 177, + 382, + 186 + ], + "score": 0.97, + "content": "Imagine youare a time traveler from the year 3ooo.What technological advancements would you tellpeople about?", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "title", + "bbox": [ + 147, + 197, + 192, + 205 + ], + "lines": [ + { + "bbox": [ + 146, + 195, + 194, + 207 + ], + "spans": [ + { + "bbox": [ + 146, + 195, + 194, + 207 + ], + "score": 0.999, + "content": "Response A", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "title", + "bbox": [ + 316, + 197, + 361, + 205 + ], + "lines": [ + { + "bbox": [ + 315, + 195, + 362, + 207 + ], + "spans": [ + { + "bbox": [ + 315, + 195, + 362, + 207 + ], + "score": 0.999, + "content": "Response B", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 147, + 212, + 277, + 256 + ], + "lines": [ + { + "bbox": [ + 146, + 211, + 277, + 219 + ], + "spans": [ + { + "bbox": [ + 146, + 211, + 277, + 219 + ], + "score": 0.969, + "content": "As an Al assistant, I don't have personal experiences or", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 146, + 218, + 274, + 225 + ], + "spans": [ + { + "bbox": [ + 146, + 218, + 274, + 225 + ], + "score": 0.956, + "content": "opinions. 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7B13B33B65B
AlpacaFLAN v2AlpacaFLAN v2AlpacaFLAN v2AlpacaFLAN v2
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Data typeMean PPL
Int434.34
Float4 (E2M1)31.07
Float4 (E3M0)29.48
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Model/DatasetParamsModel bitsMemoryChatGPT vs SysSys vs ChatGPTMean95% CI
GPT-4 Bard===119.4%110.1%114.5%2.6%
1--93.2%96.4%94.8%4.1%
Guanaco Alpaca FLAN v265B4-bit41 GB96.7%101.9%99.3%4.4%
65B4-bit41 GB63.0%77.9%70.7%4.3%
65B4-bit41 GB37.0%59.6%48.4%4.6%
Guanaco Open Assistant Alpaca FLAN v233B4-bit21 GB96.5%99.2%97.8%4.4%
33B16-bit66 GB73.4%85.7%78.1%5.3%
33B 33B4-bit 4-bit21GB 21GB67.2%79.7%73.6%4.2%
Vicuna Guanaco Alpaca HH-RLHF26.3%49.7%38.0%3.9%
13B 13B16-bit 4-bit26 GB 10 GB91.2% 87.3%98.7% 93.4%94.9% 90.4%4.5% 5.2%
13B4-bit10 GB63.8%76.7%69.4%
13B4-bit10 GB55.5%69.1%62.5%4.2%
13B4-bit10 GB50.6%69.8%60.5%4.7%
13B4-bit10 GB49.2%69.3%59.5%4.2% 4.7%
Longform Self-Instruct FLAN v213B4-bit10 GB44.9%62.0%53.6%5.2%
13B4-bit10 GB38.0%60.5%49.1%4.6%
13B4-bit10 GB32.4%61.2%47.0%3.6%
4-bit5GB
Guanaco Alpaca FLAN v27B 7B4-bit5GB84.1% 57.3%89.8% 71.2%87.0%5.4%
7B4-bit5GB33.3%56.1%64.4% 44.8%5.0% 4.0%
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Dataset7B13B33B65B
LLaMA no tuning35.146.957.863.4
Self-Instruct36.433.353.056.7
Longform32.143.256.659.7
Chip234.541.653.659.8
HH-RLHF34.944.655.860.1
Unnatural Instruct41.948.157.361.3
Guanaco (OASST1)36.646.457.062.2
Alpaca38.847.857.362.5
FLAN v244.551.459.263.9
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Benchmark # Prompts Judge ModelVicuna 80 Human ratersVicuna 80 GPT-4Open Assistant 953 GPT-4Median Rank
EloRankEloRankEloRank
GPT-41176113481129411
Guanaco-65B102321022100832
Guanaco-33B10094992100244
ChatGPT-3.5 Turbo9167966235101525
Vicuna-13B9845974493655
Guanaco-13B9756913688566
Guanaco-7B10103879886077
Bard909890271-8
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Dataset ModelGLUE (Acc.)Super-NaturalInstructions (RougeL)
RoBERTa-largeT5-80MT5-250MT5-780MT5-3BT5-11B
BF1688.640.142.148.054.362.0
BF16 replication88.640.042.247.354.91
LoRA BF1688.840.542.647.155.460.7
QLoRA Int888.840.442.945.456.560.7
QLORA FP488.640.342.447.555.660.9
QLORA NF4 + DQ140.442.747.755.360.9
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Datapoints ↓Epochs→ChipUnnatural InstructionsFLAN v2
123123123Mean
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15000034.4034.8035.1039.7041.1041.5044.6045.5043.5040.02
Mean34.2034.6734.6039.3041.5038.8743.8344.2344.17
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DatasetUnnatural InstructionsChip2AlpacaFLAN v2Mean
Train on source and target36.233.738.142.037.5
Train on target38.034.539.042.938.6
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Guanaco65B41 GB
Guanaco33B21 GB
Vicuna13B26 GB
ChatGPT-3.5 TurboN/AN/A
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total#numberof judgmentsChatGPT-3.5 Turbo BardGuanaco 13BGuanaco 7B
ModelGuanaco 65BGuanaco 33BVicuna
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Guanaco 33B-0.210.170.100.510.410.68
Vicuna-0.19-0.170.100.500.200.57
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Bard-0.72-0.51-0.50 -0.20-0.35 -0.19- -0.120.120.03
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"type": "text", + "text": "Abstract ", + "text_level": 1, + "bbox": [ + 462, + 318, + 535, + 334 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Fine-tuning deep learning models for commercial use cases is growing exponentially as more and more companies are adopting AI to enhance their core products and services, as well as automate their diurnal processes and activities. However, not many countries like the U.S. and those in Europe follow quality data collection methods for AI vision or NLP related automation applications. Thus, on many of these kinds of data, existing state-of-the-art pre-trained deep learning models fail to perform accurately, and when fine-tuning is done on these models, issues like catastrophic forgetting or being less specific in predictions as expected occur. Hence, in this paper, simplified incremental learning methods are introduced to be considered in existing fine-tuning infrastructures of pre-trained models (such as those available in huggingface.com) to help mitigate the aforementioned issues for commercial applications. The methods introduced are: 1) Fisher Shut-off, 2) Fractional Data Retention and 3) Border Control. Results show that when applying these methods on vanilla pre-trained models, the models are in fact able to add more to their knowledge without hurting much on what they had learned previously. ", + "bbox": [ + 148, + 348, + 766, + 556 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "16 1 Introduction ", + "text_level": 1, + "bbox": [ + 148, + 580, + 312, + 598 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "17 Many companies and organizations today are adopting AI in automation, automating their daily \n18 processes and activities, as well as offering them in their core products and services. Automation \n19 has traditionally been in the industry for many years, as a means for which economics of scale could \n20 be acheived so as to remain competitive in the market. Now with AI, more and more intelligence is \n21 being brought into automation, and in countries like India, organizations are beginning to adopt AI \n22 for this particular purpose. \n23 With recent advancements in AI vision and NLP models such as the GPT-3, Jurassic-1, and so on, \n24 organizations today are using AI for 1) Document Reading and Understanding, 2) Online Proctoring, \n25 3) Chatbots, 4) Intelligent Information Parsing and other application related process automations. \n26 Given these use cases, AI solutions need to be specific to their processes, but yet be an addition to \n27 their generally known formats. This in a sense, is more like making use of a human employee who \n28 has some kind of general education on various tasks or processes but still is required to learn the \n29 companies counterparts well and in detail before he/she is allowed to execute them. These processes \n30 can include between, reading customer emails for entering relevant information about their product \n31 requirements onto a structured database, to understanding various types of printed documents for \n32 information parsing, and to identifying newer objects for either document filtering or malicious \n33 activity detection. \n34 For natural language related tasks, powerful models like the GPT-3 are now being widely used, but \n35 they require good prompt engineering skills to get the best out of them. Also, given that they are \n36 probabilistic models, the generated outputs can sometimes falter away from what is expected, and \n37 this can become a problem when selling it to customers, because even the slightest faltering may not \n38 be acceptable to them at all. Hence, to reduce this, more and more examples have to be provided in \n39 the prompt, and this can come at a high cost not suitable for low cost of living countries like India. \n40 The other workaround is to fine-tune the model on the new datasets, but this has epoch limitations \n41 on how deeply it can fit on the new dataset without hurting the body of general knowledge it gained \n42 earlier. Also, fine-tuning models like the GPT-3 comes at a very high cost now-a-days, and is no \n43 more an option. This leaves the automation builders to use huggingface.com transformers instead. \n44 In vision, although state-of-the-art pre-trained deep learning models are able to achieve human level \n45 performance on a variety of inputs, they can only perform so in upto close to high quality inputs. If \n46 the quality goes lower, they fail terribly. Not all organizations have a good quality data collection \n47 process involved for applying automation, and this is ubiquitously the case in many parts of the world. \n48 So it becomes quite difficult to sell AI as a human-level performer, and at this point AI becomes of \n49 lesser use than it could potentially be. \n50 Another approach typically used to resolve such problems is to employ transfer learning, which \n51 typically involves replacing the last layers of the model with a new model to get the specific outputs \n52 required. Some examples done in research are Too, et al. (2019), Dif & Elberrichi (2020), Alshalali \n53 & Joysula (2018), Jung, et al. (2015), Qian, et al. (2021) and Vrbanciˇ c & Podgorelec (2020). While ˇ \n54 this may not seem to be a problem with vision based tasks, it is definitely a problem with natural \n55 language based tasks. This is because the final layers of the natural language models have all the vital \n56 information of language structure that help with the language generative process. When this is to \n57 be changed, catastrophic forgetting can happen. Catastrophic forgetting is a phenomenon in which \n58 previously learned knowledge is lost partly by the application of new data for training. Also, with \n59 vision based tasks, when the requirement is to just improve the performance on lower quality data, \n60 transfer learning may not be the appropriate approach. Fine-tuning for these must involve the final \n61 layers of the model which could inevitably lead to catastrophic forgetting on the higher quality inputs. \n62 This brings the only solution towards incremental learning. This type of learning is all about \n63 learning on newer datasets without having the side-effects catastrophic forgetting, and there has been \n64 substantial amount of research done in this area. Luo, et al. (2020) summarizes all the work that has \n65 happened in this area so far. There are several approaches to implementing incremental learning on \n66 pre-trained models, some of which will be discussed in the forthcoming sections. In this paper, a \n67 few of these approaches will be simplified for commercial applications along with novel intuitive \n68 additions to further help the learning process. The paper introduces: 1) Fisher Shut-off which is a \n69 simplification of the work done by Kirkpatrick, et al. (2017), 2) Fractional Data Retention which \n70 adopts ideas from Castro, et al. (2018), and 3) Border Control which is an extension to the idea \n71 outlined by Ren, et al. (2018) on reweighting examples by employing a method similar to Adaboost. \n72 The last one is the novel addition as it formulates a different approach to retaining salient examples \n73 for incremental learning. It is based on the work by Ruping (2001) on incremental learning with \n74 SVMs. But since SVMs are too complex in the context on neural networks, a similar but simplified \n75 approach is proposed. \n76 The purpose of this work is to initiate the development of a new infrastructure for commercial \n77 fine-tuning of pre-trained models with simplified incremental learning methods. \n78 The rest of this paper proceeds as follows: Section 2 will provide a brief discussion on incremental \n79 learning methods developed so far, followed by the proposal of simplified incremental learning \n80 methods in Section 3. Section 4 will show sample results of the proposed methods on a vanilla \n81 pre-trained model using a toy dataset. A toy dataset is used for the only purpose of providing \n82 visualizations on the performance of the proposed methods. Nevertheless, these methods can be \n83 extended on to real world datasets. The paper then concludes in Section 5 discussing steps forward \n84 for implementation. ", + "bbox": [ + 147, + 612, + 825, + 694 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 147, + 702, + 825, + 853 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 148, + 859, + 823, + 901 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 147, + 92, + 825, + 188 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 147, + 194, + 825, + 277 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 145, + 285, + 825, + 450 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 145, + 457, + 825, + 650 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 151, + 656, + 823, + 684 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 147, + 690, + 825, + 787 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "85 2 Incremental Learning ", + "text_level": 1, + "bbox": [ + 148, + 818, + 388, + 835 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "86 This section is a summary of the review published by Luo, et al. (2020). In this review, four different \n87 types of strategies for incremental learning are highlighted, and every work published in this area \n88 uses either one or more such strategies. Some examples are Castro, et al. (2018) and He, et al. (2020). \n89 The four strategies are: ", + "bbox": [ + 147, + 854, + 825, + 911 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "• Architectural • Regularization • Rehearsal • Pseudo-Rehearsal ", + "bbox": [ + 217, + 92, + 352, + 164 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "94 The following subsections will disccuss these briefly. ", + "bbox": [ + 155, + 176, + 522, + 190 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.1 Architectural Strategy ", + "text_level": 1, + "bbox": [ + 173, + 207, + 369, + 223 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "96 This strategy is similar to boosting techniques where multiple models are trained. But when used in the context of incremental learning, each model is trained on a different task separately. Then another 98 meta-model that effectively selects which model to use for inference is trained. The work done by 99 Poliker, et al. (2001) resembles this in many ways. In this work, multiple classifiers are trained with 0 different training sets, and then a Adaboost style of ensemble learning is employed to combine the model outputs. ", + "bbox": [ + 155, + 233, + 825, + 316 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "102 Another interesting work is by Rusu, et al. (2016) on Progressive Neural Networks (PNN). In this \n103 work, a neural network is trained sequentially on different tasks or training sets. However, each time, \n104 new neurons are added in each layer with new weights, and the weights of the previously learned \n105 neural network are frozen. Then, to prevent catastrophic forgetting, the outputs of each layer of the \n106 previous neural network on the earlier training set are used in addition to the new task or training \n107 set, when training the new layer neurons. The results on this type of incremental learning were quite \n108 encouraging that it set a new direction in the research of dynamically expanding networks that could \n109 make better use the neural networks capacity than the PNN. In fact, it will be seen later that the Fisher \n110 Shut-off method proposed in this paper inherently employs the idea of PNNs. ", + "bbox": [ + 140, + 321, + 825, + 448 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.2 Regularization Strategy ", + "text_level": 1, + "bbox": [ + 168, + 464, + 379, + 479 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "112 In this strategy, as the name suggest, a regularization term is added in the loss function that measures \n113 the importance of old knowledge when learning on a new training set. The representive work done in \n114 this is Kirkpatrick, et al. (2017), whereby they introduce the concept of Elastic Weight Consolidation \n115 (EWC) by means of a Fisher Information Matrix. The EWC brings about the regularization term in \n116 the loss function as ", + "bbox": [ + 140, + 489, + 825, + 559 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/eaddded0272291affae7fe4a9660de274f9483c623d565b1f57f253082e94a12.jpg", + "text": "$$\nR ( w ) = \\sum _ { i } \\frac { \\lambda } { 2 } F _ { i } ( w _ { i } - w _ { i , o l d } ) ^ { 2 }\n$$", + "text_format": "latex", + "bbox": [ + 392, + 589, + 606, + 627 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "117 where $F _ { i }$ is the Fisher Information Matrix which suggests the importance of the $i$ -th weight trained \n118 on the old (or previous) training set. Here, as one could speculate, the term Fisher Shut-off proposed \n119 in this paper actually derives itself from the Fisher Information Matrix, meaning that this matrix is \n120 used as the basis for shutting off the training of certain weights when training on a new set. \n121 Another popular type of regularization strategy is Knowledge Distillation introduced by Hinton, et al. \n122 (2015). In this method, knowledge from an ensemble of models trained on different tasks (or training \n123 sets) separately are distilled into a smaller model that can be deployed much easily for inference. \n124 There are many huggingface.com transformers that are a product of such knowledge distillation. \n125 The distillation ensures that the smaller model holds all the knowledge of the ensemble, and that it \n126 can infer as good as it. Distillation is done by setting soft-targets on the smaller network from all the \n127 earlier training sets of the ensemble. The soft-targets are the output logits from the ensemble models \n128 on their respective trained datasets. ", + "bbox": [ + 140, + 640, + 825, + 696 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 142, + 702, + 825, + 813 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "129 2.3 Rehearsal and Pseudo-Rehearsal Strategies ", + "text_level": 1, + "bbox": [ + 143, + 829, + 516, + 844 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "130 Rehearsal strategies in incremental learning make use of the earlier training sets when training a \n131 model on new tasks or training sets. This by far is the simplest of all incremental learning strategies \n132 that ensures catastrophic forgetting is prevented. The only issue is that when this strategy is used for \n133 deep learning models trained on large datasets, the training on new datasets could become extremely \n134 slow and even time consuming before any fruitful results are achieved. Hence, newer research work in \n135 this area formulate methods for retaining only the most important data points to prevent catastrophic \n136 forgetting. The work done by Castro, et al. (2018) is an example of this. In this work, selection and \n137 removal mechanisms on data are introduced for assimilation into a memory network. \n138 Talking about memory networks, the Pseudo-Rehearsal strategy involves training an additional data \n139 generator to generate the samples, the neural network was trained on earlier. Hence, newer research \n140 in this area involve GANs for data generation. Examples are Odena, et al. (2017) and Wu, et al. \n141 (2018). ", + "bbox": [ + 142, + 856, + 825, + 911 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 142, + 90, + 825, + 147 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 142, + 152, + 826, + 209 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "142 3 Proposed Incremental Learning Methods ", + "text_level": 1, + "bbox": [ + 142, + 228, + 550, + 246 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "143 Commercial applications always require simplistic implementations of advanced methods no matter \n144 how complex they may be. Therefore, it is for this purpose alone this paper proposes some simplified \n145 methods for implementing incremental learning. As metioned earlier in Section 1, these methods are: \n146 1) Fisher Shut-off, 2) Fractional Data Retention, and 3) Border Control. This section covers them in \n147 detail. ", + "bbox": [ + 142, + 258, + 825, + 329 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "148 3.1 Fisher Shut-off ", + "text_level": 1, + "bbox": [ + 150, + 344, + 318, + 359 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "149 As mentioned in the previous section, the term Fisher Shut-off derives itself from the Fisher Infor \n150 mation Matrix which weighs the importance of weights trained on previous datasets. Hence, in this \n151 sub-section, a brief overview of the details behind this matrix is covered with the help of Aich (2021). \n152 Let $\\mathcal { D }$ represent a dataset coming from a stream of data for incremental learning. Then $p ( w | \\mathcal { D } )$ \n153 represents the model trained on data $\\mathcal { D }$ . This means that to train a model on a new dataset, the \n154 following posterior must satisfy: ", + "bbox": [ + 142, + 369, + 826, + 412 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 142, + 417, + 826, + 462 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/a15358fc3e1a14de53b12997fbe20e35ff131d70a2b4f35932e082c514b1f6fd.jpg", + "text": "$$\np ( w | \\mathcal D _ { n e w } ) = \\frac { p ( \\mathcal D _ { n e w } | w ) p ( w | \\mathcal D _ { o l d } ) } { p ( \\mathcal D _ { n e w } ) }\n$$", + "text_format": "latex", + "bbox": [ + 375, + 488, + 622, + 523 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "155 Note here that $p ( w | \\mathcal { D } _ { o l d } )$ is written in place of $p ( w )$ because when $\\mathcal { D } _ { n e w }$ is applied to the model, the \n156 weights $w$ have already been trained with $\\mathcal { D } _ { o l d }$ . Hence, given the model, $p ( w | \\mathcal { D } _ { o l d } )$ , the log-likelihood \n157 loss on $\\mathcal { D } _ { n e w }$ becomes, ", + "bbox": [ + 142, + 535, + 825, + 579 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/74a6be926e26dca91399c1d70f54ee531dccf5ad29e080f5c507622ca7e9206c.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathcal { L } _ { \\mathcal { D } _ { n e w } } ( w ) = l o g ( p ( w | \\mathcal { D } _ { n e w } ) ) } \\\\ & { \\qquad = l o g ( p ( \\mathcal { D } _ { n e w } | w ) ) + l o g ( p ( w | \\mathcal { D } _ { o l d } ) ) - l o g ( p ( \\mathcal { D } _ { n e w } ) ) } \\\\ & { \\qquad \\approx l o g ( p ( \\mathcal { D } _ { n e w } | w ) ) + l o g ( p ( w | \\mathcal { D } _ { o l d } ) ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 279, + 606, + 717, + 661 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "158 Here, the $l o g ( p ( \\mathcal { D } _ { n e w } | w ) )$ equals the cross-entropy loss of the model on $\\mathcal { D } _ { n e w }$ while $l o g ( p ( w | \\mathcal { D } _ { o l d } ) )$ \n159 is loss of the model on $\\mathcal { D } _ { o l d }$ . To ensure that catastrophic forgetting does not occur on $\\mathcal { D } _ { o l d }$ in its \n160 absence while training on $\\mathcal { D } _ { n e w }$ , the loss on $\\mathcal { D } _ { o l d }$ will have to be approximated using $w$ alone. To do \n161 this, the Taylor’s expansion on $l o g ( p ( w | \\mathcal { D } _ { o l d } ) )$ is taken as, ", + "bbox": [ + 140, + 672, + 826, + 731 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/69d6f37d75fec7962529c0788241d518bd3252a0ddf7853b32e2ecb344af45e1.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathcal { L } _ { \\mathcal { D } _ { o l d } } ( w ) \\approx \\mathcal { L } ( w ) \\big | _ { \\mathcal { D } _ { o l d } } + \\left( \\frac { \\partial \\mathcal { L } ( w ) } { \\partial w } \\Big | _ { \\mathcal { D } _ { o l d } } \\right) + \\frac { 1 } { 2 } ( w - w \\big | _ { \\mathcal { D } _ { o l d } } ) ^ { T } \\left( \\frac { \\partial ^ { 2 } \\mathcal { L } ( w ) } { \\partial ^ { 2 } w } \\Big | _ { \\mathcal { D } _ { o l d } } \\right) ( w - w \\big | _ { \\mathcal { D } _ { o l d } } ) } \\\\ & { \\qquad \\approx \\mathcal { L } ( w ) \\big | _ { \\mathcal { D } _ { o l d } } + \\frac { 1 } { 2 } ( w - w \\big | _ { \\mathcal { D } _ { o l d } } ) ^ { T } \\left( \\frac { \\partial ^ { 2 } \\mathcal { L } ( w ) } { \\partial ^ { 2 } w } \\Big | _ { \\mathcal { D } _ { o l d } } \\right) ( w - w \\big | _ { \\mathcal { D } _ { o l d } } ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 184, + 757, + 813, + 844 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "since technically 162 $\\begin{array} { r } { \\frac { \\partial \\mathcal { L } ( w ) } { \\partial w } \\bigg | _ { \\mathcal { D } _ { o l d } } = 0 } \\end{array}$ , if the model is trained well on $\\mathcal { D } _ { o l d }$ . Then, noting that the last term 163 in (4) is equivalent to a regularization term, this term alone could be considered as the loss on $\\mathcal { D } _ { o l d }$ ", + "bbox": [ + 142, + 871, + 825, + 912 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "164 for preventing catastrophic forgetting. In doing so, the Fisher Information Matrix will equal to the \n165 Hessian, $\\frac { \\partial ^ { 2 } \\mathcal { L } ( w ) } { \\partial ^ { 2 } w } \\bigg | _ { \\mathscr { D } _ { o l d } } .$ This Hessian, $\\mathcal { H }$ , can be simply computed by the model gradients $\\frac { \\bar { \\partial \\mathcal { L } } ( w ) } { \\partial w } \\bigg | _ { \\mathscr { D } _ { o l d } }$ \n166 assuming that not all gradients are zero, as, ", + "bbox": [ + 140, + 90, + 825, + 145 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/dc622a320eb64ae2f8c4a32f04f15ce714556f4f59239c6f7c1f30980ae8a0e8.jpg", + "text": "$$\n\\mathcal { H } = \\frac { \\partial \\mathcal { L } ( w ) } { \\partial w } \\bigg | _ { \\mathcal { D } _ { o l d } } \\cdot \\frac { \\partial \\mathcal { L } ( w ) } { \\partial w } \\bigg | _ { \\mathcal { D } _ { o l d } } ^ { T }\n$$", + "text_format": "latex", + "bbox": [ + 390, + 172, + 607, + 212 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "167 Doing so, and keeping only the diagonal terms, would imply that the model gradients are more \n168 than enough to weigh the important weights of the model trained on $\\mathcal { D } _ { o l d }$ . Replacing (5) in (4) and \n169 substituting in (3) would give the loss on $\\mathcal { D } _ { n e w }$ as, ", + "bbox": [ + 142, + 223, + 826, + 266 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/bc1f880fd17423a3ede2612347abfdb476b2eb4631734f31c3e5d1576d5dfb26.jpg", + "text": "$$\n\\mathcal { L } _ { \\mathcal { D } _ { n e w } } ( w ) \\approx l o g ( p ( \\mathcal { D } _ { n e w } | w ) ) + \\frac { 1 } { 2 } ( w - w \\big | _ { \\mathcal { D } _ { o l d } } ) ^ { T } \\left( \\frac { \\partial \\mathcal { L } ( w ) } { \\partial w } \\bigg | _ { \\mathcal { D } _ { o l d } } \\cdot \\frac { \\partial \\mathcal { L } ( w ) } { \\partial w } \\bigg | _ { \\mathcal { D } _ { o l d } } ^ { T } \\right) ( w - w \\big | _ { \\mathcal { D } _ { o l d } } )\n$$", + "text_format": "latex", + "bbox": [ + 181, + 294, + 802, + 337 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "170 which to an extent implies that if the model gradients on $\\mathcal { D } _ { o l d }$ are absolutely zero, they get trained on \n171 $\\mathcal { D } _ { n e w }$ without regularization, while those that are not, get regularized towards $w \\big | _ { \\mathscr { D } _ { o l d } }$ . \n172 This is what the proposed Fisher Shut-off exploits. In Fisher Shut-off, all weights of the model \n173 trained on $\\mathcal { D } _ { o l d }$ that do not have absolute zero gradients get shut-off for training on $\\mathcal { D } _ { n e w }$ , while the \n174 remaining that do take part. Also, since in practice $R e L U$ functions are commonly used in deep \n175 learning models as the activation functions of the neurons, shutting off these weights becomes as \n176 simple as setting a condition. Figure 1 shows a sample performance of Fisher Shut-off on a regression \n177 model trained sequentially on mutually exclusive batches of data. These batches could represent the \n178 different tasks or training sets. \n179 However, when it comes to classification, simple shut-off does not work completely. This is because, \n180 while in regression problems datasets could inherently employ some kind of piece-wise nonlinear fit \n181 in their distributions, the same cannot always be guaranteed in classification. Thus, in classification, \n182 the shut-off weights must also take part in training. And, as per (1), there is a learning constant \n183 required in the regularization to ensure that the right balances between $\\mathcal { D } _ { n e w }$ and $\\mathcal { D } _ { o l d }$ are met on \n184 these weights. This paper provides a novel learning constant determination for this regularization. \n185 This is detailed in Appendix A. \n86 Also in regression problems, if datasets have batch distributions that are quite far apart from each \n87 previous batch, then Fisher-Shutoff may not fully work too. Appendix B shows some of these \n188 examples ", + "bbox": [ + 137, + 347, + 828, + 377 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 383, + 825, + 482 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 142, + 488, + 826, + 530 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/ec3ca6e1de828ff9f19c334a4f580525e37d7f6bbb827e170bad1c560561bda1.jpg", + "image_caption": [ + "Figure 1: Fisher Shut-off on a regression model on six mutually exclusive batches of data. Blue dots represent the overall dataset, while green square dots are the batch or task data. The red line is the model’s output after each batch is fed to it. Fisher Shut-off is used from Batch $\\# 2$ onwards. " + ], + "image_footnote": [], + "bbox": [ + 204, + 553, + 792, + 854 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 142, + 92, + 825, + 147 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 150, + 154, + 826, + 195 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3.2 Fractional Data Retention ", + "text_level": 1, + "bbox": [ + 173, + 213, + 393, + 227 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "This is a very simply proposal. The idea is to retain only a fraction of the data trained on the neural network on the earlier tasks or training sets. There is nothing more to this. However, banking on the ideas of selection highlighted in Castro, et al. (2018), whereby data is selected based on their proximity to cluster centers, to be more representative of the classes, this paper uses this as the baseline idea behind its proposal on Fractional Data Retention. Thus in Fractional Data Retention, a fraction of the data within the data cluster is retained and appended in every stage of incremental learning. ", + "bbox": [ + 166, + 238, + 825, + 337 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "197 3.3 Border Control ", + "text_level": 1, + "bbox": [ + 145, + 353, + 321, + 368 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The most important requirement when incrementally learning classes is to ensure that the decision boundaries of the earlier training tasks are protected as much as possible when training on new sets. If data points are used for this purpose, it would seem that, those that lie closest to the decision boundaries after training would be the most important ones to retain, for any succeeding incremental learning tasks. Thus, the Border Control method proposed in this paper exploits this. Ruping (2001) used SVMs to identify these data points as the support vectors that helped define the overall decision boundaries. But with deep learning models or vanilla neural networks, SVM is quite complex and therefore in order to be able retain data points closest to the decision boundaries, a different selection mechanism is required. This selection mechanism could instead be based on selecting data points on how large the absolute errors in sigmoidal outputs are for the applied dataset, as the data points closest to the decision boundaries have this inherent property. ", + "bbox": [ + 165, + 378, + 825, + 531 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "209 Furthermore, since real world data can be quite complex, it would be necessary to not only select data \n210 points based on how large their errors in sigmoidal outputs are, but also those points that are far away \n211 from them. This is because, given the context of incremental learning where there is a high chance \n212 that newer training sets may have data points that could potentially set newer decision boundaries in \n213 those fartherest regions, these data points would help protect those. \n214 Hence in Border Control, the top- $\\mathtt { k }$ data points that have the largest absolute errors in the sigmoidal \n215 outputs and their respective top- $\\mathtt { k }$ fartherest data points are retained in every task or training set \n216 for further incremental learning. These points are appended to the newer training sets before further \n217 training is applied. ", + "bbox": [ + 140, + 537, + 825, + 607 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 142, + 613, + 825, + 669 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "218 4 Sample Results ", + "text_level": 1, + "bbox": [ + 143, + 690, + 333, + 707 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "219 The proposed methods are tested on a toy dataset, as mentioned in Section 1, only to provide some \n220 visuals on how the incremental learning progresses using the proposed methods. Figure 2 shows this \n221 dataset. A vanilla deep neural network of size, 1000-1000-1000-1000-3, is used for incrementally \n222 learning batches of data from this toy dataset. The activation functions for all layers are $R e L U$ except \n223 for the output which is a sof tmax. All weights are uniformly but randomly initialized with a single \n224 random seed to make the results comparable. The weights are also scaled by a $\\frac { 2 } { \\sqrt { n } }$ factor to ensure \n225 that minimal overfitting occurs during training. Here $n$ is the layer fan-in. \n226 To visualize incremental learning on the proposed methods, the dataset is divided into 6 batches \n227 with mututally exclusive data points. This gives roughly between 100 to 200 data points in each \n228 batch, a size that is commonly used when training neural networks of this size. Figure 3 shows this. \n229 In this figure, it can be clearly seen that the batch distributions on the class data for incremental \n230 learning do not always form a piece-wise nonlinear fit, and therefore, plain shut-off of weights cannot \n231 fully retain knowledge learned earlier. Also, among these distributions, some allowed incremental \n232 learning to happen easily, while others did not, and the distribution shown in Figure 3 is one such. \n233 Table 1 summarizes the results of the proposed methods on this particular distribution. For other batch \n234 distributions, similar results could be achieved. Note here that quite some ML-Ops were required to \n235 achieve the results in Table 1. This was especially the case for those that employed Fisher Shut-off, \n236 since this method has a regularization constant that requires adapting on each batch. Furthermore, \n237 training on each batch was stopped once $1 0 0 . 0 \\%$ accuracy was obtained on the batch. This left quite \n238 some data points to lie very close to the boundary lines or in some cases just right on them. Thus, \n239 the neural network was very vulnerable to catastrophic forgetting when succeeding batch trainings \n240 occurred as part of incremental learning. \n241 However, taking a look at Table 1, it can be seen that when Fisher Shut-off is applied, additional \n242 leverage against catastrophic forgetting occurs on each incremental batch, than when it is not used. \n243 And, among the three methods proposed in this paper, the Border Control method shows much \n244 stronger performance. In Figure 4, sample decision boundaries learned when each incremental batch \n245 is applied to the neural network using Fisher Shut-off and Border Control together is shown. A topk \n246 value of 5 is used for the Border Control. Also, note in Figure 4 that the red circles mark the border \n247 points accumulated on each batch. It can be seen that they clearly assume the data points closest \n248 to the decision boundaries, as well as those far away from it. All with respect to their batches. For \n249 the far away data points, their purpose can be clearly seen between batches #1 and $\\# 2$ , where the \n250 fartherest points of class 2 in Batch #1 helped protect the decision boundaries from the data points \n251 of class 0 in Batch $\\# 2$ . This means that more complex datasets can be accommodated by simply \n252 applying Border Control. More examples are shown in Appendix C. ", + "bbox": [ + 142, + 722, + 825, + 821 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 142, + 828, + 825, + 911 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/ed5b8993eaca01d170d93842c74b1f8afc29195bd92a7249a0f7e6d54ce3a8c4.jpg", + "image_caption": [ + "Figure 2: The toy dataset having three nonlinearly arranged classes. " + ], + "image_footnote": [], + "bbox": [ + 380, + 92, + 619, + 215 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/b02e14d6cf3d7912dafbbcf85b828e0529858490504fb69b8d48e1be08459fa6.jpg", + "image_caption": [ + "Figure 3: Batches on the toy dataset. " + ], + "image_footnote": [], + "bbox": [ + 215, + 276, + 784, + 568 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 138, + 641, + 826, + 767 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 772, + 825, + 911 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/44a71df70375c6308a562a9a44508116449215d01a051506c004a38008ac986e.jpg", + "table_caption": [ + "Table 1: Performance of proposed methods on the dataset of Figure 2 " + ], + "table_footnote": [], + "table_body": "
Method Sample accuracy on accumulated dataset after Batch1
#1#2#3#4#5#6
No Incremental Learning100.0%90.98%292.76%94.89%98.32%96.11%
Fisher Shut-off (FS)100.0%99.74%98.19%98.88%98.96%96.89%
Frac. Data Ret. (FDR)[10%]100.0%97.94%98.39%98.89%98.71%98.67%
FDR[20%]100.0%98.71%98.59%99.36%98.97%98.78%
Border Ctrl. (BC)[t opk = 5]100.0%100.0%100.0%99.84%100.0%100.0%
BC[topk = 10]100.0%100.0%100.0%99.84%100.0%100.0%
FS + FDR[10%]100.0%99.74%98.79%99.52%99.23%98.78%
FS + FDR[20%]100.0%100.0%99.19%99.52%99.48%99.11%
FS +BC[topk = 5]100.0%100.0%100.0%99.84%100.0%100.0%
FS + BC[topk = 10]100.0%100.0%100.0%100.0%100.0%100.0%
", + "bbox": [ + 176, + 109, + 823, + 299 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/5bccdefef0dd25c13f1ecc98bcf95de7a924e776d8da83295052a7d16b995c74.jpg", + "image_caption": [ + "Figure 4: Incremental learning using Fisher Shut-off and Border Control together $[ t o p k = 5 ]$ ]. Black circles mark the batch data, while the red circles mark the accumulated border points. " + ], + "image_footnote": [], + "bbox": [ + 214, + 315, + 784, + 614 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 143, + 684, + 821, + 713 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Also, to add on further to this, for the most difficult incremental learning applications such as learning new classes as highlighted in Castro, et al. (2018) and He, et al. (2020), Border Control can help leverage the many issues associated with it like class imbalance, concept drift and so on. ", + "bbox": [ + 163, + 719, + 825, + 761 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5 Conclusion ", + "text_level": 1, + "bbox": [ + 163, + 781, + 299, + 799 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "257 To summarize the work in this paper, three simplified methods for implementing incremental learning \n258 for commercial fine-tuning of pre-trained models was proposed. Results showed that while Border \n259 Control performed the best, Fisher Shut-off was able to leverage the performances. However, dataset \n260 used in this paper was a toy dataset and not one of the benchmark datasets typically used for \n261 incremental learning. Hence, testing these methods on the benchmark datasets is a potential next step \n262 forward. Then, preparing the prerequisites for each model available, like say in huggingface.com, \n263 for incremental learning must be done so that automation companies or any other AI organization \n264 can make use of them. From the methods proposed in this paper, the prerequisites would be: 1) \n265 the Shut-off matrix for the neural network weights, and 2) the border points for each of the learned \n266 classes. Additionally, an ML-Ops infrastructure can be provided to optimize the performances of \n267 the models that employ the Fisher Shut-off method. Metrics like the Backward Transfer (BWT) and \n268 Forward Transfer (FWT) proposed in Lopez-Paz & Ranzato (2017) can be used for this purpose. ", + "bbox": [ + 140, + 814, + 825, + 871 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 92, + 826, + 203 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "269 References ", + "text_level": 1, + "bbox": [ + 143, + 227, + 267, + 242 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "270 [1] Aich, A. (2021) Elastic weight consolidation (EWC): Nuts and bolts. arXiv preprint arXiv:2105.04093. \n271 [2] Alshalali, T. & Joysula, D. (2018) Fine-Tuning of Pre-Trained Deep Learning Models with Extreme Learning \n272 Machine. IEEE International Conference on Computational Science and Computational Intelligence (CSCI), pp. \n273 469-473. \n274 [3] Castro, F., Marín-Jiménez, M.J., Guil, N., Schmid, C. & Alahari, K. (2018) End-to-end incremental learning. \n275 Proceedings of the European Conference on Computer Vision (ECCV), pp. 233-248. \n276 [4] Dif, N. & Elberrichi, Z. (2020) A New Intra Fine-Tuning Method. International Journal of Service Science, \n277 Management, Engineering, and Technology, 11(2), pp. 16-40. \n278 [5] He, J., Mao, R., Shao, Z. & Zhu, F. (2020) Incremental learning in online scenario. IEEE/CVF Conference \n279 on Computer Vision and Pattern Recognition, pp. 13926-13935. \n280 [6] Hinton, G., Vinyals, O. & Dean, J. (2015) Distilling the Knowledge in a Neural Network. arXiv preprint \n281 arXiv:1503.02531, 2(7). \n282 [7] Jung, H., Lee, S., Yim, J., Park, S. & Kim, J. (2015) Joint fine-tuning in deep neural networks for facial \n283 expression recognition. Proceedings of the IEEE International Conference on Computer Vision, pp. 2983-2991. \n284 [8] Kirkpatrick, J., Pascanu, R., Rabinowitz, N., Veness, J., Desjardins, G., Rusu, A.A., Milan, K., Quan, J., \n285 Ramalho, T., Grabska-Barwinska, A. & Hassabis, D. (2017) Overcoming catastrophic forgetting in neural \n286 networks. Proceedings of the National Academy of Sciences, 114(13), pp. 3521-3526. \n287 [9] Lopez-Paz, D. & Ranzato, M.A. (2017) Gradient episodic memory for continual learning. Proceedings of \n288 Neural Information Processing Systems (NIPS), pp. 6467-6476. \n289 [10] Luo, Y., Yin, L., Bai, W. & Mao, K. (2020) An Appraisal of Incremental Learning Methods. Entropy, \n290 22(11), pp. 1190-1216. \n291 [11] Odena, A., Olah, C. & Shlens, J. (2017) Conditional image synthesis with auxiliary classifier GANs. \n292 International Conference on Machine Learning (ICML), pp. 2642–2651. \n293 [12] Polikar, R., Udpa, L., Udpa, S. & Honavar, V. (2001) Learn++: An Incremental Learning Algorithm for \n294 Supervised Neural Networks. IEEE Transactions on Systems, Man, and Cybernetics, part C (applications and \n295 reviews), 31(4), pp. 497-508. \n296 [13] Qian, X., Zhang, C., Yella, J., Huang, Y., Huang, M.C. & Bom, S. (2021) Soft sensing model visualization: \n297 Fine-tuning neural network from what model learned. IEEE International Conference on Big Data (Big Data), \n298 pp. 1900-1908. \n299 [14] Ren, M., Zeng, W., Yang, B. & Urtasun, R. (2018) Learning to reweight examples for robust deep learning. \n300 International Conference on Machine Learning (ICML), pp. 4334-4343. \n301 [15] Ruping, S. (2001) Incremental learning with support vector machines. IEEE International Conference on \n302 Data Mining, pp. 641-642. \n303 [16] Rusu, A.A., Rabinowitz, N.C., Desjardins, G., Soyer, H., Kirkpatrick, J., Kavukcuoglu, K., Pascanu, R. & \n304 Hadsell, R. (2016) Progressive neural networks. arXiv preprint arXiv:1606.04671. \n305 [17] Too, E., Yujian, L., Njuki, S. & Yingchun, L. (2019) A comparative study of fine-tuning deep learning \n306 models for plant disease. Computers and Electronics in Agriculture, 161, pp. 272-279. \n307 [18] Vrbanciˇ c, G. & Podgorelec, V. (2020) ˇ Transfer learning with adaptive fine-tuning. IEEE Access, Volume 8, \n308 pp. 196197-196211. \n309 [19] Wu, Y., Chen, Y.P., Wang, L.J., Ye, Y.C., Liu, Z.C., Guo, Y.D., Zhang, Z.Y. & Fu, Y. (2018) Incremental \n310 Classifier Learning with Generative Adversarial Networks. arXiv preprint arXiv:1802.00853. ", + "bbox": [ + 137, + 262, + 826, + 917 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "The checklist follows the references. Please read the checklist guidelines carefully for information on how to answer these questions. For each question, change the default [TODO] to [Yes] , [No] , or [N/A] . You are strongly encouraged to include a justification to your answer, either by referencing the appropriate section of your paper or providing a brief inline description. For example: ", + "bbox": [ + 143, + 119, + 826, + 176 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "• Did you include the license to the code and datasets? [Yes] See Section ??. \n• Did you include the license to the code and datasets? [No] The code and the data are proprietary. \n• Did you include the license to the code and datasets? [N/A] ", + "bbox": [ + 217, + 186, + 825, + 252 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Please do not modify the questions and only use the provided macros for your answers. Note that the Checklist section does not count towards the page limit. In your paper, please delete this instructions block and only keep the Checklist section heading above along with the questions/answers below. ", + "bbox": [ + 143, + 262, + 826, + 305 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "1. For all authors... ", + "bbox": [ + 214, + 316, + 339, + 330 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes] \n(b) Did you describe the limitations of your work? [Yes] See Sections 4 and 5 \n(c) Did you discuss any potential negative societal impacts of your work? [N/A] \n(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] ", + "bbox": [ + 238, + 335, + 825, + 425 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "2. If you are including theoretical results... ", + "bbox": [ + 214, + 429, + 493, + 444 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "(a) Did you state the full set of assumptions of all theoretical results? [Yes] See Section 3 and Appendix A \n(b) Did you include complete proofs of all theoretical results? [Yes] See Section 3 and Appendix A ", + "bbox": [ + 238, + 446, + 825, + 506 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "3. If you ran experiments... ", + "bbox": [ + 214, + 510, + 393, + 523 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] As a supplemental material \n(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [No] But it’s in the supplemental material \n(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [No] But additional examples are shown in Appendix C \n(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [N/A] ", + "bbox": [ + 238, + 529, + 825, + 660 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... ", + "bbox": [ + 217, + 662, + 823, + 678 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "(a) If your work uses existing assets, did you cite the creators? [N/A] \n(b) Did you mention the license of the assets? [N/A] \n(c) Did you include any new assets either in the supplemental material or as a URL? [N/A] \n(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A] \n(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A] ", + "bbox": [ + 238, + 681, + 825, + 801 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "5. If you used crowdsourcing or conducted research with human subjects... ", + "bbox": [ + 214, + 805, + 704, + 820 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A] \n(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A] \n(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A] ", + "bbox": [ + 238, + 824, + 825, + 912 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "362 Here, the derivation of regularization constant for Fisher Shut-off method is detailed. To start with, \n363 let the neural network be defined as, ", + "bbox": [ + 142, + 119, + 828, + 148 + ], + "page_idx": 10 + }, + { + "type": "equation", + "img_path": "images/b07fcd772a1ae5904c64e4fc051920aa75d3ca2813faf1927bfb0d4fd135093c.jpg", + "text": "$$\ny = w ^ { T } \\phi ( x , \\omega )\n$$", + "text_format": "latex", + "bbox": [ + 446, + 179, + 552, + 198 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "364 Here, $w$ is the weights of the output layer, $\\phi ( \\cdot )$ is the output of the preceding layer, $x$ is the input and \n365 $\\omega$ represents the rest of the weights of the neural network. Throughout the derivation, we will be \n366 dealing with only the output layer, and so the $\\phi ( x , \\omega )$ will be written in short form as $\\Phi$ from here on. \n367 Let $e _ { k }$ denoted the error of fitting in the $k$ -th iteration, and $g _ { k }$ denote the error gradient. This would \n368 mean that $g _ { k } = \\Phi _ { k } e _ { k }$ . \n369 Then, given the regularization term in (6), let $\\tilde { w }$ denote difference in weights, between the new \n370 training and the previous training. This would give the weight updation policy as, ", + "bbox": [ + 140, + 210, + 825, + 282 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 286, + 826, + 318 + ], + "page_idx": 10 + }, + { + "type": "equation", + "img_path": "images/32a1481d5eac0518a8ad59224b8997ba40c10f990725a8ed8066814040237018.jpg", + "text": "$$\nw _ { k + 1 } = w _ { k } - \\eta g _ { k } - \\beta \\tilde { w } _ { k }\n$$", + "text_format": "latex", + "bbox": [ + 411, + 349, + 588, + 366 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "If 371 $e _ { k } = w _ { k } ^ { T } \\Phi _ { k } - Y$ , then the error $e _ { k + 1 }$ after the weight updation would equal, ", + "bbox": [ + 137, + 378, + 692, + 395 + ], + "page_idx": 10 + }, + { + "type": "equation", + "img_path": "images/75fd1c7bddf5675b9a8270ce21e7403d4ca9b0ef8250aa0f5b77a702d88261d2.jpg", + "text": "$$\n\\begin{array} { r l } & { e _ { k + 1 } = w _ { k + 1 } ^ { T } \\Phi _ { k + 1 } - Y } \\\\ & { \\qquad = ( w _ { k } - \\eta g _ { k } - \\beta \\tilde { w } _ { k } ) ^ { T } \\Phi _ { k + 1 } - Y } \\\\ & { \\qquad = w _ { k } ^ { T } \\Phi _ { k + 1 } - \\eta g _ { k } ^ { T } \\Phi _ { k + 1 } - \\beta \\tilde { w } _ { k } ^ { T } \\Phi _ { k + 1 } - Y } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 338, + 421, + 660, + 486 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "372 Asumming for simplicity sake that $\\Phi _ { k + 1 } \\approx \\Phi _ { k } + \\delta$ , then (9) can continue as, ", + "bbox": [ + 138, + 496, + 679, + 513 + ], + "page_idx": 10 + }, + { + "type": "equation", + "img_path": "images/3bd40be7231c96c492c72d712f848099c8f312b6cf67e43166835c66cc492a11.jpg", + "text": "$$\n\\begin{array} { c } { { e _ { k + 1 } \\approx w _ { k } ^ { T } \\Phi _ { k } - \\eta g _ { k } ^ { T } \\Phi _ { k } - \\beta \\tilde { w } _ { k } ^ { T } \\Phi _ { k } - Y + \\Delta } } \\\\ { { { } } } \\\\ { { \\approx e _ { k } - \\eta g _ { k } ^ { T } \\Phi _ { k } - \\beta \\tilde { w } _ { k } ^ { T } \\Phi _ { k } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 346, + 540, + 653, + 583 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "373 Taking the square norm of $e _ { k + 1 }$ in (10), would equate this to, ", + "bbox": [ + 140, + 594, + 578, + 611 + ], + "page_idx": 10 + }, + { + "type": "equation", + "img_path": "images/1594f9cb5eaafa167fdce65b86dc7a69713dd83bb615c7944800fc7e545a6c2d.jpg", + "text": "$$\n\\begin{array} { r l } & { | | e _ { k + 1 } | | ^ { 2 } = | | e _ { k } | | ^ { 2 } + \\eta ^ { 2 } | | g _ { k } ^ { T } \\Phi _ { k } | | ^ { 2 } + \\beta ^ { 2 } | | \\tilde { w } _ { k } ^ { T } \\Phi _ { k } | | ^ { 2 } } \\\\ & { ~ - ~ 2 \\eta e _ { k } ^ { T } \\Phi _ { k } ^ { T } g _ { k } - 2 \\beta e _ { k } ^ { T } \\Phi _ { k } ^ { T } \\tilde { w } _ { k } + 2 \\eta \\beta \\tilde { w } _ { k } ^ { T } \\Phi _ { k } \\Phi _ { k } ^ { T } g _ { k } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 308, + 637, + 689, + 681 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "374 We require that $| | e _ { k + 1 } | | ^ { 2 } < | | e _ { k } | | ^ { 2 }$ at all times, so that regularization does not affect the fit at any \n375 point during the training. Applying this condition in (11) would give, ", + "bbox": [ + 137, + 693, + 826, + 724 + ], + "page_idx": 10 + }, + { + "type": "equation", + "img_path": "images/a73b5e0026dc7e8708344c3c012a764d7ff01a567b05f9e9a78fd4f3a3b7fdab.jpg", + "text": "$$\n\\eta ^ { 2 } | | g _ { k } ^ { T } \\Phi _ { k } | | ^ { 2 } + \\beta ^ { 2 } | | \\tilde { w } _ { k } ^ { T } \\Phi _ { k } | | ^ { 2 } - 2 \\eta e _ { k } ^ { T } \\Phi _ { k } ^ { T } g _ { k } - 2 \\beta e _ { k } ^ { T } \\Phi _ { k } ^ { T } \\tilde { w } _ { k } + 2 \\eta \\beta \\tilde { w } _ { k } ^ { T } \\Phi _ { k } \\Phi _ { k } ^ { T } g _ { k } < 0\n$$", + "text_format": "latex", + "bbox": [ + 232, + 752, + 767, + 772 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "376 Then, taking the partial derivatives of (12) w.r.t $\\eta$ and $\\beta$ would give the following equations to be \n377 satisfied: ", + "bbox": [ + 135, + 785, + 828, + 814 + ], + "page_idx": 10 + }, + { + "type": "equation", + "img_path": "images/178a66c2f9c1db1202932a8706ff13775e3340c3d2ed3da0e9372b31d071df91.jpg", + "text": "$$\n\\eta | | g _ { k } ^ { T } \\Phi _ { k } | | ^ { 2 } - e _ { k } ^ { T } \\Phi _ { k } ^ { T } g _ { k } + \\beta \\tilde { w } _ { k } ^ { T } \\Phi _ { k } \\Phi _ { k } ^ { T } g _ { k } = 0\n$$", + "text_format": "latex", + "bbox": [ + 352, + 844, + 645, + 863 + ], + "page_idx": 10 + }, + { + "type": "equation", + "img_path": "images/3887c583a9beba4d0e8f666c6a8c3e0c3040d332dc7bb6005a23f23594457692.jpg", + "text": "$$\n\\beta | | \\tilde { w } _ { k } ^ { T } \\Phi _ { k } | | ^ { 2 } - e _ { k } ^ { T } \\Phi _ { k } ^ { T } \\tilde { w } _ { k } + \\eta \\tilde { w } _ { k } ^ { T } \\Phi _ { k } \\Phi _ { k } ^ { T } g _ { k } = 0\n$$", + "text_format": "latex", + "bbox": [ + 349, + 893, + 648, + 912 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "378 Solving, (13) and (14) can result in negative $\\eta$ and $\\beta$ , which is not acceptable, and so to simplify the \n379 solution, we neglect the $\\beta$ -term in (13). Doing so we get, ", + "bbox": [ + 135, + 90, + 826, + 121 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/d0d765dc0222ec6a060eeb994411b77f93b4ddc58b23c1f9fff18a87429dfc82.jpg", + "text": "$$\n\\begin{array} { r l } & { \\eta = \\frac { e _ { k } ^ { T } \\Phi _ { k } ^ { T } g _ { k } } { \\vert \\vert g _ { k } ^ { T } \\Phi _ { k } \\vert \\vert ^ { 2 } } , } \\\\ & { } \\\\ & { \\beta = \\frac { e _ { k } ^ { T } \\Phi _ { k } ^ { T } \\tilde { w } _ { k } - \\eta \\tilde { w } _ { k } ^ { T } \\Phi _ { k } \\Phi _ { k } ^ { T } g _ { k } } { \\vert \\vert \\tilde { w } _ { k } ^ { T } \\Phi _ { k } \\vert \\vert ^ { 2 } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 393, + 148, + 606, + 243 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "380 Then, substituting for $g _ { k }$ and openning up the norms, we get, ", + "bbox": [ + 140, + 255, + 575, + 271 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/2b0ecdf49a1badd8bcbfabc9f3fa8b388c3809d322e50de7adf0d3dc23af6ba7.jpg", + "text": "$$\n\\eta = \\frac { e _ { k } ^ { T } \\Phi _ { k } ^ { T } \\Phi _ { k } e _ { k } } { e _ { k } ^ { T } ( \\Phi _ { k } ^ { T } \\Phi _ { k } ) ( \\Phi _ { k } ^ { T } \\Phi _ { k } ) e _ { k } } ,\n$$", + "text_format": "latex", + "bbox": [ + 380, + 299, + 563, + 337 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/ea3b0d31b1692e2a6daae854f40df4ba6cf4946fd880a617540abe3f1a9cfb5b.jpg", + "text": "$$\n\\beta = \\frac { e _ { k } ^ { T } \\Phi _ { k } ^ { T } \\tilde { w } _ { k } - \\eta \\tilde { w } _ { k } ^ { T } ( \\Phi _ { k } \\Phi _ { k } ^ { T } ) \\Phi _ { k } e _ { k } } { \\tilde { w } _ { k } ^ { T } ( \\Phi _ { k } \\Phi _ { k } ^ { T } ) \\tilde { w } _ { k } }\n$$", + "text_format": "latex", + "bbox": [ + 379, + 354, + 619, + 392 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "381 Simplifying (16) gives, ", + "bbox": [ + 140, + 405, + 328, + 421 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/ded3e70309e85198252e5583b697e498e31e4871f702de56db16a64a62168365.jpg", + "text": "$$\n\\eta = \\frac { e _ { k } ^ { T } e _ { k } } { e _ { k } ^ { T } \\Phi _ { k } ^ { T } \\Phi _ { k } e _ { k } } ,\n$$", + "text_format": "latex", + "bbox": [ + 428, + 452, + 544, + 488 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/6022507d8b8f82158ca71a569238702b1df503f4d316bfe75c079e2875a223ab.jpg", + "text": "$$\n\\beta = ( 1 - \\eta ) \\frac { \\tilde { w } _ { k } ^ { T } \\Phi _ { k } e _ { k } } { \\tilde { w } _ { k } ^ { T } \\tilde { w } _ { k } }\n$$", + "text_format": "latex", + "bbox": [ + 426, + 506, + 571, + 542 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "382 Equation (17) gives the raw form for both $\\eta$ and $\\beta$ to be regulated. However, this will be further \n383 simplified for computating purposes, but will be used as a basis. \n384 Since the errors $e _ { k }$ get smaller as the neural network fits the data, using them in learning constants \n385 will only slow down the fits. A common way to overcome this is by replacing $e _ { k }$ with all ones. \n386 Similarly, for the $\\tilde { w } _ { k }$ , all weights that are to be regularized are replaced with ones. If we denote \n387 the weights to be regularized as $w _ { r }$ , and there are $m$ patterns in the dataset with $n$ weights to be \n388 regularized, the $\\eta$ and $\\beta$ computations become, ", + "bbox": [ + 147, + 556, + 826, + 587 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 590, + 825, + 661 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/442b45703e0d5bc953ed91857f82f1af81d17476e62fa79baeb583297874dfda.jpg", + "text": "$$\n\\eta = \\frac { 1 } { | | \\Phi _ { k } | | ^ { 2 } } ,\n$$", + "text_format": "latex", + "bbox": [ + 429, + 690, + 517, + 724 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/c9f7fc511d626c576863ed43e74e284cd34b4e4f59f8229db4e15d89189fac5e.jpg", + "text": "$$\n\\beta = \\frac { \\alpha } { m n } \\sum _ { i : w \\in w _ { r } } \\Phi _ { i , k }\n$$", + "text_format": "latex", + "bbox": [ + 428, + 743, + 568, + 779 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "389 Here, $\\alpha$ represents the $( 1 - \\eta )$ -term in (17). This constant will not neccessarily take the computed $\\eta$ \n390 when being regulated. Instead, this constant will have to be adapted each time for every incremental \n391 batch applied to the neural network. \n392 The reason why the computed $\\eta$ is not used for the $\\alpha$ adaptation is because this $\\eta$ can sometimes \n393 become too small in the adaptation, that the $1 - \\eta$ would always tend towards 1. When this was \n394 empirically tested on the toy dataset, the regularization was found at times to have gone too strong \n395 that the fit never happened. ML-Ops on the $\\alpha$ found that this constant is not always 1, and can be \n396 anywhere between 0 and 1, or higher in some cases. ", + "bbox": [ + 140, + 792, + 825, + 835 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 842, + 825, + 911 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "398 Additional examples on the regression problem with Fisher Shut-off. Fisher Shut-off could not be used completely, and regularization had to take over for some batches. Figures 5 and 6 show this. ", + "bbox": [ + 145, + 127, + 826, + 157 + ], + "page_idx": 12 + }, + { + "type": "image", + "img_path": "images/72e9762490cf8ee10f248a8d3a0d85b6b28ea32a596f13c1b194dc4898ba90c7.jpg", + "image_caption": [ + "Figure 5: Complete Fisher Shut-off is used in Batches #2 and #6. Batches #3, #4 and #5 are regularized. " + ], + "image_footnote": [], + "bbox": [ + 222, + 183, + 776, + 469 + ], + "page_idx": 12 + }, + { + "type": "image", + "img_path": "images/b70e35df8ae79c3da760bbd94c1191f3f81fda2aacf6961298510e013e24a3ed.jpg", + "image_caption": [ + "Figure 6: Complete Fisher Shut-off is used in Batches $\\# 2$ and $\\# 6$ . Batches $\\# 3$ and $\\# 4$ are regularized. Batch #5 is fine-tuned " + ], + "image_footnote": [], + "bbox": [ + 223, + 563, + 777, + 849 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "401 In this appendix, additional examples on the toy dataset classification is shown. Figures 7 and 8 show 402 the batch distributions considered. Among these, Figure 8 has more cases in which the farthest points 403 in Border Control can play a vital role in retaining previously learned knowledge. Tables 2 and 3 summarize their performances. ", + "bbox": [ + 142, + 121, + 825, + 176 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/df00e79da7646169cbe201b37f70a3bcbb78337b977ceaff9e318f5cd16ecc4a.jpg", + "image_caption": [ + "Figure 7: Another batch distribution on the toy dataset. " + ], + "image_footnote": [], + "bbox": [ + 214, + 190, + 785, + 483 + ], + "page_idx": 13 + }, + { + "type": "table", + "img_path": "images/35169b5859d76b2ba8db2696ee69a6e233a337b896515790bfc98fef3d4d9924.jpg", + "table_caption": [ + "Table 2: Performance of proposed methods on the dataset of Figure 7 " + ], + "table_footnote": [], + "table_body": "
MethodSample accuracy on accumulated dataset after Batch
#1#2#3#4#5#6
No Incremental Learning100.0%73.14%92.07%97.56%88.33%91.67%
Fisher Shut-off (FS)100.0%99.43%98.26%99.54%92.85%97.78%
Frac. Data Ret. (FDR)[10%]100.0%97.43%96.13%98.93%98.75%96.78%
FDR[20%]100.0%98.86%98.84%99.69%99.75%98.89%
Border Ctrl. (BC)[topk = 5]100.0%100.0%100.0%100.0%100.0%99.78%
BC[topk = 10]100.0%100.0%100.0%100.0%100.0%100.0%
FS + FDR[10%]100.0%99.71%98.84%99.54%98.75%98.33%
FS + FDR[20%]100.0%100.0%99.23%99.85%99.87%98.89%
FS +BC[topk = 5]100.0%100.0%100.0%100.0%100.0%100.0%
FS + BC[topk = 10]100.0%100.0%100.0%100.0%100.0%100.0%
", + "bbox": [ + 176, + 559, + 820, + 747 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/198dee612d5cdc63ee02fcb729ec0a2fd31f55ddf925d97353d37ff12ee507bd.jpg", + "image_caption": [ + "Figure 8: Yet another batch distribution on the toy dataset. " + ], + "image_footnote": [], + "bbox": [ + 215, + 103, + 784, + 395 + ], + "page_idx": 14 + }, + { + "type": "table", + "img_path": "images/2a566a85f88b5b15a7d4734f39bf4999233c5f778294a1ba3927f6050be29e24.jpg", + "table_caption": [ + "Table 3: Performance of proposed methods on the dataset of Figure 8 " + ], + "table_footnote": [], + "table_body": "
MethodSample accuracy on accumulated dataset after Batch
#1#2#3#4#5#6
No Incremental Learning100.0%89.34%93.80%95.60%97.81%84.78%
Fisher Shut-off (FS)100.0%95.36%99.59%99.55%99.36%94.56%
Frac. Data Ret. (FDR)[10%]100.0%95.08%96.07%97.42%99.61%98.67%
FDR[20%]100.0%98.36%98.97%99.85%100.0%99.67%
Border Ctrl. (BC)[top k = 5]100.0%100.0%99.79%99.85%100.0%100.0%
BC[topk = 10]100.0%100.0%100.0%100.0%99.74%100.0%
FS + FDR[10%]100.0%95.36%99.79%98.48%99.61%98.89%
FS + FDR[20%]100.0%98.36%99.79%99.69%100.0%99.67%
FS+BC[topk = 5]100.0%100.0%100.0%100.0%99.74%100.0%
FS + BC[topk = 10]100.0%100.0%100.0%100.0%100.0%100.0%
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Method Sample accuracy on accumulated dataset after Batch1
#1#2#3#4#5#6
No Incremental Learning100.0%90.98%292.76%94.89%98.32%96.11%
Fisher Shut-off (FS)100.0%99.74%98.19%98.88%98.96%96.89%
Frac. Data Ret. (FDR)[10%]100.0%97.94%98.39%98.89%98.71%98.67%
FDR[20%]100.0%98.71%98.59%99.36%98.97%98.78%
Border Ctrl. (BC)[t opk = 5]100.0%100.0%100.0%99.84%100.0%100.0%
BC[topk = 10]100.0%100.0%100.0%99.84%100.0%100.0%
FS + FDR[10%]100.0%99.74%98.79%99.52%99.23%98.78%
FS + FDR[20%]100.0%100.0%99.19%99.52%99.48%99.11%
FS +BC[topk = 5]100.0%100.0%100.0%99.84%100.0%100.0%
FS + BC[topk = 10]100.0%100.0%100.0%100.0%100.0%100.0%
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MethodSample accuracy on accumulated dataset after Batch
#1#2#3#4#5#6
No Incremental Learning100.0%73.14%92.07%97.56%88.33%91.67%
Fisher Shut-off (FS)100.0%99.43%98.26%99.54%92.85%97.78%
Frac. Data Ret. (FDR)[10%]100.0%97.43%96.13%98.93%98.75%96.78%
FDR[20%]100.0%98.86%98.84%99.69%99.75%98.89%
Border Ctrl. (BC)[topk = 5]100.0%100.0%100.0%100.0%100.0%99.78%
BC[topk = 10]100.0%100.0%100.0%100.0%100.0%100.0%
FS + FDR[10%]100.0%99.71%98.84%99.54%98.75%98.33%
FS + FDR[20%]100.0%100.0%99.23%99.85%99.87%98.89%
FS +BC[topk = 5]100.0%100.0%100.0%100.0%100.0%100.0%
FS + BC[topk = 10]100.0%100.0%100.0%100.0%100.0%100.0%
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MethodSample accuracy on accumulated dataset after Batch
#1#2#3#4#5#6
No Incremental Learning100.0%89.34%93.80%95.60%97.81%84.78%
Fisher Shut-off (FS)100.0%95.36%99.59%99.55%99.36%94.56%
Frac. Data Ret. (FDR)[10%]100.0%95.08%96.07%97.42%99.61%98.67%
FDR[20%]100.0%98.36%98.97%99.85%100.0%99.67%
Border Ctrl. (BC)[top k = 5]100.0%100.0%99.79%99.85%100.0%100.0%
BC[topk = 10]100.0%100.0%100.0%100.0%99.74%100.0%
FS + FDR[10%]100.0%95.36%99.79%98.48%99.61%98.89%
FS + FDR[20%]100.0%98.36%99.79%99.69%100.0%99.67%
FS+BC[topk = 5]100.0%100.0%100.0%100.0%99.74%100.0%
FS + BC[topk = 10]100.0%100.0%100.0%100.0%100.0%100.0%
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SHARPNESS-AWARE TRAINING + +Juntang Zhuang1 ∗ j.zhuang@yale.edu + +Boqing Gong2, Liangzhe Yuan2, Yin $\mathbf { C } \mathbf { u } \mathbf { i } ^ { 2 }$ , Hartwig Adam2 {bgong, lzyuan, yincui, hadam}@google.com + +Nicha C. Dvornek1, Sekhar Tatikonda1, James S. Duncan1 {nicha.dvornek, sekhar.tatikonda, james.duncan}@yale.edu + +Ting Liu2liuti@google.com + +1 Yale University, 2 Google Research + +# ABSTRACT + +The recently proposed Sharpness-Aware Minimization (SAM) improves generalization by minimizing a perturbed loss defined as the maximum loss within a neighborhood in the parameter space. However, we show that both sharp and flat minima can have a low perturbed loss, implying that SAM does not always prefer flat minima. Instead, we define a surrogate gap, a measure equivalent to the dominant eigenvalue of Hessian at a local minimum when the radius of neighborhood (to derive the perturbed loss) is small. The surrogate gap is easy to compute and feasible for direct minimization during training. Based on the above observations, we propose Surrogate Gap Guided Sharpness-Aware Minimization (GSAM), a novel improvement over SAM with negligible computation overhead. Conceptually, GSAM consists of two steps: 1) a gradient descent like SAM to minimize the perturbed loss, and 2) an ascent step in the orthogonal direction (after gradient decomposition) to minimize the surrogate gap and yet not affect the perturbed loss. GSAM seeks a region with both small loss (by step 1) and low sharpness (by step 2), giving rise to a model with high generalization capabilities. Theoretically, we show the convergence of GSAM and provably better generalization than SAM. Empirically, GSAM consistently improves generalization (e.g., $+ 3 . 2 \%$ over SAM and $+ 5 . 4 \%$ over AdamW on ImageNet top-1 accuracy for ViT-B/32). Code is released at https://sites.google.com/view/gsam-iclr22/home. + +# 1 INTRODUCTION + +Modern neural networks are typically highly over-parameterized and easy to overfit to training data, yet the generalization performances on unseen data (test set) often suffer a gap from the training performance (Zhang et al., 2017a). Many studies try to understand the generalization of machine learning models, including the Bayesian perspective (McAllester, 1999; Neyshabur et al., 2017), the information perspective (Liang et al., 2019), the loss surface geometry perspective (Hochreiter & Schmidhuber, 1995; Jiang et al., 2019) and the kernel perspective (Jacot et al., 2018; Wei et al., 2019). Besides analyzing the properties of a model after training, some works study the influence of training and the optimization process, such as the implicit regularization of stochastic gradient descent (SGD) (Bottou, 2010; Zhou et al., 2020), the learning rate’s regularization effect (Li et al., 2019), and the influence of the batch size (Keskar et al., 2016). + +These studies have led to various modifications to the training process to improve generalization. Keskar & Socher (2017) proposed to use Adam in early training phases for fast convergence and then switch to SGD in late phases for better generalization. Izmailov et al. (2018) proposed to average weights to achieve a wider local minimum, which is expected to generalize better than sharp minima. A similar idea was later used in Lookahead (Zhang et al., 2019). Entropy-SGD (Chaudhari et al., 2019) derived the gradient of local entropy to avoid solutions in sharp valleys. Entropy-SGD has a nested Langevin iteration, inducing much higher computation costs than vanilla training. + +The recently proposed Sharpness-Aware Minimization (SAM) (Foret et al., 2020) is a generic training scheme that improves generalization and has been shown especially effective for Vision Transformers (Dosovitskiy et al., 2020) when large-scale pre-training is unavailable (Chen et al., 2021). Suppose vanilla training minimizes loss $f ( w )$ (e.g., the cross-entropy loss for classification), where $w$ is the parameter. SAM minimizes a perturbed loss defined as $f _ { p } ( w ) \triangleq \operatorname* { m a x } _ { | | \delta | | \leq \rho } f ( w + \delta )$ , which is the maximum loss within radius $\rho$ centered at the model parameter $w$ . Intuitively, vanilla training seeks a single point with a low loss, while SAM searches for a neighborhood within which the maximum loss is low. However, we show that a low perturbed loss $f _ { p }$ could appear in both flat and sharp minima, implying that only minimizing $f _ { p }$ is not always sharpness-aware. + +Although the perturbed loss $f _ { p } ( w )$ might disagree with sharpness, we find a surrogate gap defined as $h ( w ) \triangleq f _ { p } ( w ) - f ( w )$ agrees with sharpness — Lemma 3.3 shows that the surrogate gap $h$ is an equivalent measure of the dominant eigenvalue of Hessian at a local minimum. Inspired by this observation, we propose the Surrogate Gap Guided Sharpness Aware Minimization (GSAM) which jointly minimizes the perturbed loss $f _ { p }$ and the surrogate gap $h$ : a low perturbed loss $f _ { p }$ indicates a low training loss within the neighborhood, and a small surrogate gap $h$ avoids solutions in sharp valleys and hence narrows the generalization gap between training and test performances (Thm. 5.3). When both criteria are satisfied, we find a generalizable model with good performances. + +GSAM consists of two steps for each update: 1) descend gradient $\nabla f _ { p } ( w )$ to minimize the perturbed loss $f _ { p }$ (this step is exactly the same as SAM), and 2) decompose gradient $\nabla f ( w )$ of the original loss $\bar { f } ( w )$ into components that are parallel and orthogonal to $\nabla f _ { p } ( w )$ , i.e., $\nabla f ( w ) = \nabla _ { \| } f ( w ) +$ $\nabla _ { \perp } f ( w )$ , and perform an ascent step in $\nabla _ { \perp } f ( w )$ to minimize the surrogate gap $h ( w )$ . Note that this ascent step does not change the perturbed loss $f _ { p }$ because $\nabla f _ { \perp } ( w ) \perp \mathbf { \bar { \nabla } } \nabla f _ { p } ( \bar { w } )$ by construction. + +We summarize our contribution as follows: + +• We define surrogate gap, which measures the sharpness at local minima and is easy to compute. • We propose the GSAM method to improve the generalization of neural networks. GSAM is widely applicable and incurs negligible computation overhead compared to SAM. • We demonstrate the convergence of GSAM and its provably better generalization than SAM. • We empirically validate GSAM over image classification tasks with various neural architectures, including ResNets (He et al., 2016), Vision Transformers (Dosovitskiy et al., 2020), and MLP-Mixers (Tolstikhin et al., 2021). + +# 2 PRELIMINARIES + +# 2.1 NOTATIONS + +• $f ( w )$ : A loss function $f$ with parameter $w \in \mathbb { R } ^ { k }$ , where $k$ is the parameter dimension. + +• $\rho _ { t } \in \mathbb { R }$ : A scalar value controlling the amplitude of perturbation at step $t$ • $\epsilon \in \mathbb { R }$ : A small positive constant (to avoid division by $) , \epsilon = 1 0 ^ { - 1 2 }$ by default). + +• $\begin{array} { r } { w _ { t } ^ { a d v } \triangleq w _ { t } + \rho _ { t } \frac { \nabla f ( w _ { t } ) } { | | \nabla f ( w _ { t } ) | | + \epsilon } } \end{array}$ : The solution to $\operatorname* { m a x } _ { | | w ^ { \prime } - w _ { t } | | \leq \rho _ { t } } f ( w ^ { \prime } )$ when $\rho _ { t }$ is small. + +• $\begin{array} { r } { f _ { p } ( w _ { t } ) \triangleq \operatorname* { m a x } _ { | | \delta | | \leq \rho _ { t } } f ( w _ { t } + \delta ) \approx f ( w _ { t } ^ { a d v } ) } \end{array}$ : The perturbed loss induced by $f ( w _ { t } )$ . For each $w _ { t }$ , $f _ { p } ( w _ { t } )$ returns the worst possible loss $f$ within a ball of radius $\rho _ { t }$ centered at $w _ { t }$ . When $\rho _ { t }$ is small, by Taylor expansion, the solution to the maximization problem is equivalent to a gradient ascent from $w _ { t }$ to $w _ { t } ^ { a d v }$ . + +• $h ( w ) \triangleq f _ { p } ( w ) - f ( w )$ : The surrogate gap defined as the difference between $f _ { p } ( w )$ and $f ( w )$ + +• $\eta _ { t } \in \mathbb { R }$ : Learning rate at step $t$ . + +• $\alpha \in \mathbb { R }$ : A constant value that controls the scaled learning rate of the ascent step in GSAM. + +$g ^ { ( t ) } , g _ { p } ^ { ( t ) } \in \mathbb { R } ^ { k }$ : At the $t$ -th step, the noisy observation of the gradients $\nabla f ( w _ { t } )$ , $\nabla f _ { p } ( w _ { t } )$ of the original loss and perturbed loss, respectively. + +![](images/3396788a9126ef1da5043f16c4c9713656da2cf3b5a9e09a6b9971d5556bacff.jpg) + +Figure 1: Consider original loss $f$ (solid line), perturbed loss $\begin{array} { r } { f _ { p } \triangleq \operatorname* { m a x } _ { | | \delta | | \leq \rho } f ( w + \delta ) } \end{array}$ (dashed line), and surrogate gap $h ( w ) \triangleq f _ { p } ( w ) - f ( w )$ . Intuitively, $f _ { p }$ is approximately a max-pooled version of $f$ with a pooling kernel of width $2 \rho$ , and SAM minimizes $f _ { p }$ . From left to right are the local minima centered at $w _ { 1 } , w _ { 2 } , w _ { 3 }$ , and the valleys become flatter. Since $f _ { p } ( w _ { 1 } ) = f _ { p } ( w _ { 3 } ) < f _ { p } ( w _ { 2 } )$ , SAM prefers $w _ { 1 }$ and $w _ { 3 }$ to $w _ { 2 }$ . However, a low $f _ { p }$ could appear in both sharp $( w _ { 1 } )$ and flat $( w _ { 3 } )$ minima, so $f _ { p }$ might disagree with sharpness. On the contrary, a smaller surrogate gap $h$ indicates a flatter loss surface (Lemma 3.3). From $w _ { 1 }$ to $w _ { 3 }$ , the loss surface is flatter, and $h$ is smaller. + +• $\nabla f ( w _ { t } ) = \nabla f _ { \parallel } ( w _ { t } ) + \nabla f _ { \perp } ( w _ { t } )$ : Decompose $\nabla f ( w _ { t } )$ into parallel component $\nabla f _ { \parallel } ( \boldsymbol { w } _ { t } )$ and vertical component $\nabla f _ { \perp } ( w _ { t } )$ by projection $\nabla f ( w _ { t } )$ onto $\nabla f _ { p } ( w _ { t } )$ . + +# 2.2 SHARPNESS-AWARE MINIMIZATION + +Conventional optimization of neural networks typically minimizes the training loss $f ( w )$ by gradient descent w.r.t. $\nabla f ( w )$ and searches for a single point $w$ with a low loss. However, this vanilla training often falls into a sharp valley of the loss surface, resulting in inferior generalization performance (Chaudhari et al., 2019). Instead of searching for a single point solution, SAM seeks a region with low losses so that small perturbation to the model weights does not cause significant performance degradation. SAM formulates the problem as: + +$$ +\begin{array} { r } { \operatorname* { m i n } _ { w } f _ { p } ( w ) \mathrm { { w h e r e } } f _ { p } ( w ) \triangleq \operatorname* { m a x } _ { | | \delta | | \leq \rho } f ( w + \delta ) } \end{array} +$$ + +where $\rho$ is a predefined constant controlling the radius of a neighborhood. This perturbed loss $f _ { p }$ induced by $f ( w )$ is the maximum loss within the neighborhood. When the perturbed loss is minimized, the neighborhood corresponds to low losses (below the perturbed loss). For a small $\rho$ , using Taylor expansion around $w$ , the inner maximization in Eq. 1 turns into a linear constrained optimization with solution + +$$ +\arg \operatorname* { m a x } _ { | | \delta | | \leq \rho } f ( w + \delta ) = \arg \operatorname* { m a x } _ { | | \delta | | \leq \rho } f ( w ) + \delta ^ { \top } \nabla f ( w ) + O ( \rho ^ { 2 } ) = \rho \frac { \nabla f ( w ) } { | | \nabla f ( w ) | | } +$$ + +As a result, the optimization problem of SAM reduces to + +$$ +\operatorname* { m i n } _ { w } f _ { p } ( w ) \approx \operatorname* { m i n } _ { w } f ( w ^ { a d v } ) { \mathrm { ~ w h e r e ~ } } w ^ { a d v } \triangleq w + \rho { \frac { \nabla f ( w ) } { | | \nabla f ( w ) | | + \epsilon } } +$$ + +where $\epsilon$ is a scalar (default: 1e-12) to avoid division by 0, and $w ^ { a d v }$ is the “perturbed weight” with the highest loss within the neighborhood. Equivalently, SAM seeks a solution on the surface of the perturbed loss $f _ { p } ( w )$ rather than the original loss $f ( w )$ (Foret et al., 2020). + +# 3 THE SURROGATE GAP MEASURES THE SHARPNESS AT A LOCAL MINIMUM + +# 3.1 THE PERTURBED LOSS IS NOT ALWAYS SHARPNESS-AWARE + +Despite that SAM searches for a region of low losses, we show that a solution by SAM is not guaranteed to be flat. Throughout this paper we measure the sharpness at a local minimum of loss $f ( w )$ by the dominant eigenvalue $\sigma _ { m a x }$ (eigenvalue with the largest absolute value) of Hessian. For simplicity, we do not consider the influence of reparameterization on the geometry of loss surfaces, which is thoroughly discussed in (Laurent & Massart, 2000; Kwon et al., 2021). + +For $t = 1$ to $T$ 0) ρt schedule: ρt = ρmin + (ρmax−ρmin)(lr−lrmin)lr −lr ∇f (t) 1a) ∆wt = ρt ||∇f (t)||+ 1b) $w _ { t } ^ { a d v } = w _ { t } + \Delta w _ { t }$ 2) Get $\nabla f _ { p } ^ { ( t ) }$ by back-propagation at $w _ { t } ^ { a d v }$ . 3) $\begin{array} { r l r } { \mathrm { ~ } } & { { } } & { = \nabla f _ { \parallel } ^ { ( t ) } + \nabla f _ { \perp } ^ { ( t ) } } \end{array}$ Decompose $\nabla f ^ { ( t ) }$ into compo +nents that are parallel and orthogonal to $\nabla f _ { p } ^ { ( t ) }$ . + +![](images/872b59aec8a7a8aaa3e5f73fed10c4e224fe6b0af656dfe3a11efa43b6098e4f.jpg) +Figure 2: $\nabla f$ is decomposed into parallel and vertical $( \nabla f _ { \perp } )$ components by projection onto $\nabla f _ { p }$ . $\mathsf { \bar { V } } f ^ { G S A M } = \mathsf { \bar { V } } \bar { f } _ { p } - \alpha \nabla f _ { \perp }$ + +# Algorithm 1 GSAM Algorithm + +4) Update weights: + +Vanilla $\begin{array} { r l } & { w _ { t + 1 } = w _ { t } - \eta _ { t } } \\ & { w _ { t + 1 } = w _ { t } - \eta _ { t } \nabla f _ { p } ^ { ( t ) } } \\ & { w _ { t + 1 } = w _ { t } - \eta _ { t } \big ( \nabla f _ { p } ^ { ( t ) } - \alpha \nabla f _ { \bot } ^ { ( t ) } \big ) } \end{array}$ +SAM +GSAM + +Lemma 3.1. For some fixed $\rho _ { ; }$ , consider two local minima $w _ { 1 }$ and $w _ { 2 }$ , $f _ { p } ( w _ { 1 } ) \leq f _ { p } ( w _ { 2 } ) \neq$ $\sigma _ { m a x } ( w _ { 1 } ) \leq \sigma _ { m a x } ( w _ { 2 } )$ , where $\sigma _ { m a x }$ is the dominant eigenvalue of the Hessian. + +We leave the proof to Appendix. Fig. 1 illustrates Lemma 3.1 with an example. Consider three local minima denoted as $w _ { 1 }$ to $w _ { 3 }$ , and suppose the corresponding loss surfaces are flatter from $w _ { 1 }$ to $w _ { 3 }$ . For some fixed $\rho$ , we plot the perturbed loss $f _ { p }$ and surrogate gap $h \triangleq f _ { p } - f$ around each solution. Comparing $w _ { 2 }$ with $w _ { 3 }$ : Suppose their vanilla losses are equal, $f ( \tilde { w _ { 2 } } ) = f ( w _ { 3 } )$ , then $f _ { p } ( w _ { 2 } ) > f _ { p } ( \bar { w } _ { 3 } )$ because the loss surface is flatter around $w _ { 3 }$ , implying that SAM will prefer $w _ { 3 }$ to $w _ { 2 }$ . Comparing $w _ { 1 }$ and $w _ { 2 }$ : $f _ { p } ( w _ { 1 } ) < f _ { p } ( w _ { 2 } )$ , and SAM will favor $w _ { 1 }$ over $w _ { 2 }$ because it only cares about the perturbed loss $f _ { p }$ , even though the loss surface is sharper around $w _ { 1 }$ than $w _ { 2 }$ . + +# 3.2 THE SURROGATE GAP AGREES WITH SHARPNESS + +We introduce the surrogate gap that agrees with sharpness, defined as: + +$$ +\begin{array} { r } { h ( w ) \triangleq \operatorname* { m a x } _ { | | \delta | | \leq \rho } f ( w + \delta ) - f ( w ) \approx f ( w ^ { a d v } ) - f ( w ) } \end{array} +$$ + +Intuitively, the surrogate gap represents the difference between the maximum loss within the neighborhood and the loss at the center point. The surrogate gap has the following properties. + +Lemma 3.2. Suppose the perturbation amplitude $\rho$ is sufficiently small, then the approximation to the surrogate gap in Eq. 4 is always non-negative, $h ( w ) \approx f ( w ^ { a d v } ) - f ( w ) \geq 0 , \forall w$ . + +Lemma 3.3. For a local minimum $w ^ { * }$ , consider the dominate eigenvalue $\sigma _ { m a x }$ of the Hessian of loss $f$ as a measure of sharpness. Considering the neighborhood centered at $w ^ { * }$ with a small radius $\rho$ , the surrogate gap $h ( w ^ { * } )$ is an equivalent measure of the sharpness: $\sigma _ { m a x } \approx 2 h ( w ^ { * } ) / \rho ^ { 2 }$ . + +The proof is in Appendix. Lemma 3.2 tells that the surrogate gap is non-negative, and Lemma 3.3 shows that the loss surface is flatter as $h$ gets closer to 0. The two lemmas together indicate that we can find a region with a flat loss surface by minimizing the surrogate gap $h ( w )$ . + +# 4 SURROGATE GAP GUIDED SHARPNESS-AWARE MINIMIZATION + +4.1 GENERAL IDEA: SIMULTANEOUSLY MINIMIZE THE PERTURBED LOSS AND SURROGATE GAP + +Inspired by the analysis in Section 3, we propose Surrogate Gap Guided Sharpness-Aware Minimzation (GSAM) to simultaneously minimize two objectives, the perturbed loss $f _ { p }$ and the surrogate gap $h$ : + +$$ +\operatorname* { m i n } _ { w } \left( f _ { p } ( w ) , h ( w ) \right) +$$ + +Intuitively, by minimizng $f _ { p }$ we search for a region with a low perturbed loss similar to SAM, and by minimizing $h$ we search for a local minimum with a flat surface. A low perturbed loss implies low training losses within the neighborhood, and a flat loss surface reduces the generalization gap between training and test performances (Chaudhari et al., 2019). When both are minimized, the solution gives rise to high accuracy and good generalization. + +Potential caveat in optimization It is tempting and yet sub-optimal to combine the objectives in Eq. 5 to arrive at $\mathrm { m i n } _ { w }$ $f _ { p } ( w ) + \lambda h ( w )$ , where $\lambda$ is some positive scalar. One caveat when solving this weighted combination is the potential conflict between the gradients of the two terms, i.e., $\nabla f _ { p } ( w )$ and $\overline { { \nabla } h } ( \boldsymbol { w } )$ . We illustrate this conflict by Fig. 2, where $\bar { \nabla } h ( w ) = \nabla f _ { p } ( w ) - \nabla f ( w )$ (the grey dashed arrow) has a negative inner product with $\nabla f _ { p } ( w )$ and $\nabla f ( w )$ . Hence, the gradient descent for the surrogate gap could potentially increase the loss $f _ { p }$ , harming the model’s performance. We empirically validate this argument in Sec. 6.4. + +# 4.2 GRADIENT DECOMPOSITION AND ASCENT FOR THE MULTI-OBJECTIVE OPTIMIZATION + +Our primary goal is to minimize $f _ { p }$ because otherwise a flat solution of high loss is meaningless, and the minimization of $h$ should not increase $f _ { p }$ . We propose to decompose $\nabla f ( w _ { t } )$ and $\nabla h$ into components that are parallel and orthogonal to $\dot { \nabla } f _ { p } ( \boldsymbol { w } _ { t } )$ , respectively (see Fig. 2): + +$$ +\begin{array} { r l } & { \nabla f ( w _ { t } ) = \nabla f _ { \parallel } ( w _ { t } ) + \nabla f _ { \perp } ( w _ { t } ) } \\ & { \nabla h ( w _ { t } ) = \nabla h _ { \parallel } ( w _ { t } ) + \nabla h _ { \perp } ( w _ { t } ) } \\ & { \nabla h _ { \perp } ( w _ { t } ) = - \nabla f _ { \perp } ( w _ { t } ) } \end{array} +$$ + +The key is that updating in the direction of $\nabla h _ { \perp } ( w _ { t } )$ does not change the value of the perturbed loss $f _ { p } ( w _ { t } )$ because $\nabla h _ { \perp } \perp \nabla f _ { p }$ by construction. Therefore, we propose to perform a descent step in the $\nabla h _ { \perp } ( w _ { t } )$ direction, which is equivalent to an ascent step in the $\nabla f _ { \perp } ( w _ { t } )$ direction (because $\nabla h _ { \perp } = - \nabla f _ { \perp }$ by the definition of $h$ ), and achieve two goals simultaneously — it keeps the value of $f _ { p } ( w _ { t } )$ intact and meanwhile decreases the surrogate gap $h ( w _ { t } ) = f _ { p } ( w _ { t } ) - f ( w _ { t } )$ (by increasing $f ( w _ { t } )$ and not affect $f _ { p } ( w _ { t } ) )$ ). + +The full GSAM Algorithm is shown in Algo. 1 and Fig. 2, where $g ^ { ( t ) } , g _ { p } ^ { ( t ) }$ are noisy observations of $\nabla f ( w _ { t } )$ and $\nabla f _ { p } ( w _ { t } )$ , respectively, and $g _ { \parallel } ^ { ( t ) } , g _ { \perp } ^ { ( t ) }$ are noisy observations of $\nabla f _ { \parallel } ( w _ { t } )$ and $\nabla f _ { \perp } ( w _ { t } )$ , respectively, by projecting $g ^ { ( t ) }$ onto $g _ { p } ^ { ( t ) }$ . We introduce a constant $\alpha$ to scale the stepsize of the ascent step. Steps 1) to 2) are the same as SAM: At current point $w _ { t }$ , step 1) takes a gradient ascent to $w _ { t } ^ { a d v }$ followed by step 2) evaluating the gradient $g _ { p } ^ { ( t ) }$ at $w _ { t } ^ { a d v }$ . Step 3) projects $g ^ { ( t ) }$ onto $g _ { p } ^ { ( t ) }$ , which requires negligible computation compared to the forward and backward passes. In step 4), $- \eta _ { t } g _ { p } ^ { ( t ) }$ is the same as in SAM and minimizes the perturbed loss $f _ { p } ( w _ { t } )$ with gradient descent, and αηtg(t)⊥ performs an ascent step in the orthogonal direction of $g _ { p } ^ { ( t ) }$ to minimize the surrogate gap $h ( w _ { t } )$ ( equivalently increase $f ( w _ { t } )$ and keep $f _ { p } ( w _ { t } )$ intact). In coding, GSAM feeds the “surrogate gradient” ∇fGSAMt , g $\nabla f _ { t } ^ { G S A M } \triangleq g _ { p } ^ { ( t ) } - \alpha g _ { \perp } ^ { ( t ) }$ to first-order gradient optimizers such as SGD and Adam. + +The ascent step along $g _ { \perp } ^ { ( t ) }$ does not harm convergence SAM demonstrates that minimizing $f _ { p }$ makes the network generalize better than minimizing $f$ . Even though our ascent step along $g _ { \perp } ^ { ( t ) }$ increases $f ( w )$ , it does not affect $f _ { p } ( w )$ , so GSAM still decreases the perturbed loss $f _ { p }$ in a way similar to SAM. In Thm. 5.1, we formally prove the convergence of GSAM. In Sec. 6 and Appendix C, we empirically validate that the loss decreases and accuracy increases with training. + +Illustration with a toy example We demonstrate different algorithms by a numerical toy example shown in Fig. 3. The trajectory of GSAM is closer to the ridge and tends to find a flat minimum. Intuitively, since the loss surface is smoother along the ridge than in sharp local minima, the surrogate gap $h ( w )$ is small near the ridge, and the ascent step in GSAM minimizes $h$ to pushes the trajectory closer to the ridge. More concretely, $\nabla f ( w _ { t } )$ points to a sharp local solution and deviates from the ridge; in contrast, $w _ { t } ^ { a d v }$ is closer to the ridge and $\nabla f ( w _ { t } ^ { a d v } )$ is closer to the ridge descent direction than $\nabla f ( w _ { t } )$ . Note that $\nabla f _ { t } ^ { G S A M }$ and $\nabla f ( w _ { t } )$ always lie at different sides of $\nabla f _ { p } ( w _ { t } )$ by construction (see Fig. 2), hence $\nabla f _ { t } ^ { G S A M }$ pushes the trajectory closer to the ridge than $\nabla f _ { p } ( w _ { t } )$ does. The trajectory of GSAM is like descent along the ridge and tends to find flat minima. + +![](images/64c1996c768babc0d124bb80d0862cf1e6d992f4b1b70cc711ed289334144e3d.jpg) +Figure 3: Consider the loss surface with a few sharp local minima. Left: Overview of the procedures of SGD, SAM and GSAM. SGD takes a descent step at $w _ { t }$ using $\nabla f ( w _ { t } )$ (orange), which points to a sharp local minima. SAM first performs gradient ascent in the direction of $\nabla f ( w _ { t } )$ to reach $w _ { t } ^ { a d v }$ with a higher loss, followed by descent with gradient $\nabla f ( w _ { t } ^ { a d v } )$ (green) at the perturbed weight. Based on $\nabla f ( w _ { t } )$ and $\nabla f ( w _ { t } ^ { a d v } )$ , GSAM updates in a new direction (red) that points to a flatter region. Right: Trajectories by different methods. SGD and SAM fall into different sharp local minima, while GSAM reaches a flat region. A video is in the supplement for better visualization. + +# 5 THEORETICAL PROPERTIES OF GSAM + +# 5.1 CONVERGENCE DURING TRAINING + +Theorem 5.1. Consider a non-convex function $f ( w )$ with Lipschitz-smooth constant $L$ and lower bound $f _ { m i n }$ . Suppose we can access a noisy, bounded observation $g ^ { ( t ) } \ ( \vert \vert g ^ { ( t ) } \vert \vert _ { 2 } \leq G , \forall t )$ of the true gradient $\nabla f ( w _ { t } )$ at the $t$ -th step. For some constant $\alpha$ , with learning rate $\eta _ { t } = \eta _ { 0 } / \sqrt { t }$ , and perturbation amplitude $\rho _ { t }$ proportional to the learning rate, e.g., $\rho _ { t } = \rho _ { 0 } / \sqrt { t }$ , we have + +$$ +\frac { 1 } { T } \sum _ { t = 1 } ^ { T } \mathbb { E } \Big | \Big | \nabla f _ { p } ( w _ { t } ) \Big | \Big | _ { 2 } ^ { 2 } \leq \frac { C _ { 1 } + C _ { 2 } \log T } { \sqrt { T } } , \quad \frac { 1 } { T } \sum _ { t = 1 } ^ { T } \mathbb { E } \Big | \Big | \nabla f ( w _ { t } ) \Big | \Big | _ { 2 } ^ { 2 } \leq \frac { C _ { 3 } + C _ { 4 } \log T } { \sqrt { T } } +$$ + +where $C _ { 1 } , C _ { 2 } , C _ { 3 } , C _ { 4 }$ are some constants. + +Thm. 5.1 implies both $f _ { p }$ and $f$ converge in GSAM at rate $O ( \log T / \sqrt { T } )$ for non-convex stochastic optimization, matching the convergence rate of first-order gradient optimizers like Adam. + +# 5.2 GENERALIZATION OF GSAM + +In this section, we show the surrogate gap in GSAM is provably lower than SAM’s, so GSAM is expected to find a smoother minimum with better generalization. + +Theorem 5.2 (PAC-Bayesian Theorem (McAllester, 2003)). Suppose the training set has m elements drawn i.i.d. from the true distribution, and denote the loss on the training set as ${ \widehat { f } } ( w ) =$ Pmi=1 f (w, xi), where we use xi to denote the (input, target) pair of the i-th element. Let w be learned from the training set. Suppose $w$ is drawn from posterior distribution $\mathcal { Q } .$ Denote the prior distribution (independent of training) as $\mathcal { P }$ , then + +$$ +\mathbb { E } _ { w \sim Q } \mathbb { E } _ { x } f ( w , x ) \le \mathbb { E } _ { w \sim Q } \widehat { f } ( w ) + 4 \sqrt { \Big ( K L ( Q | | \mathcal { P } ) + \log \frac { 2 m } { a } \Big ) / m } w i t h \ p r o b a b i l i t y \ a t \ l e a s t 1 - a +$$ + +Corollary 5.2.1. Suppose perturbation $\delta$ is drawn from distribution $\delta \sim \mathcal { N } ( 0 , b ^ { 2 } I ^ { k } ) , \delta \in \mathbb { R } ^ { k }$ , $k$ is the dimension of $w$ , then with probability at least $\left( 1 - a \right) \left[ 1 - e ^ { - \left( \frac { \rho } { \sqrt { 2 } b } - \sqrt { k } \right) ^ { 2 } } \right]$ + +$$ +\begin{array} { r l } & { \mathbb { E } _ { w \sim Q } \mathbb { E } _ { x } f ( w , x ) \le \widehat { h } + C + 4 \sqrt { \Big ( K L ( Q | \mathcal { P } ) + \log \frac { 2 m } { a } \Big ) / m } } \\ & { \widehat { h } \triangleq \operatorname* { m a x } _ { | | \delta | | _ { 2 } \le \rho } \widehat { f } ( w + \delta ) - \widehat { f } ( w ) = \displaystyle \frac { 1 } { m } \sum _ { i = 1 } ^ { m } \Big [ \operatorname* { m a x } _ { | | \delta | | _ { 2 } \le \rho } f ( w + \delta , x _ { i } ) - f ( w , x _ { i } ) \Big ] } \end{array} +$$ + +where $C = { \widehat { f } } ( w )$ is the empirical training loss, and $\widehat { h }$ is the surrogate gap evaluated on the training set. + +Corollary 5.2.1 implies that minimizing $\widehat { h }$ (right hand side of Eq. 7) is expected to achieve a tighter upper bound of the generalization performance (left hand side of Eq. 7). The third term on the right of Eq. 7 is typically hard to analyze and often simplified to $L 2$ regularization (Foret et al., 2020). Note that $f _ { p } = C + \widehat { h }$ only holds when $\rho _ { t r a i n }$ (the perturbation amplitude specified by users during training) equals $\rho _ { t r u e }$ (the ground truth value determined by underlying data distribution); when $\rho _ { t r a i n } \neq \rho _ { t r u e }$ , $m i n ( f _ { p } , \widehat { h } )$ is more effective than $m i n ( f _ { p } )$ in terms of minimizing generalization loss. A detailed discussion is in Appendix A.7. + +Theorem 5.3 (Unlike SAM, GSAM decreases the surrogate gap). Under the assumption in Thm. 5.1, Thm. 5.2 and Corollary 5.2.1, we assume the Hessian has a lower-bound $| \sigma | _ { m i n }$ on the absolute value of eigenvalue, and the variance of noisy observation $g ^ { ( t ) }$ is lower-bounded by $c ^ { 2 }$ . The surrogate gap h can be minimized by the ascent step along the orthogonal direction $g _ { \perp } ^ { ( t ) }$ . During training we minimize the sample estimate of $h$ . We use $\Delta \widehat { h } _ { t }$ to denote the amount that the ascent step in GSAM decreases $\widehat { h }$ for the $t$ -th step. Compared to SAM, the proposed method generates a total decrease in surrogate gap $\Sigma _ { t = 1 } ^ { T } \Delta \widehat { h } _ { t }$ , which is bounded by + +$$ +\frac { \alpha c ^ { 2 } \rho _ { 0 } ^ { 2 } \eta _ { 0 } | \sigma | _ { m i n } ^ { 2 } } { G ^ { 2 } } \le \operatorname* { l i m } _ { T \to \infty } \sum _ { t = 1 } ^ { T } \Delta \widehat { h } _ { t } \le 2 . 7 \alpha L ^ { 2 } \eta _ { 0 } \rho _ { 0 } ^ { 2 } +$$ + +We provide proof in the appendix. The lower-bound of provably non-trivial decrease in the surrogate gap. Com $\Sigma _ { t = 1 } ^ { T } \Delta \widehat { h } _ { t }$ indicates that GSAM achieves aCorollary 5.2.1, GSAM provably improves the generalization performance over SAM. + +# 6 EXPERIMENTS + +# 6.1 GSAM IMPROVES TEST PERFORMANCE ON VARIOUS MODEL ARCHITECTURES + +We conduct experiments with ResNets (He et al., 2016), Vision Transformers (ViTs) (Dosovitskiy et al., 2020) and MLP-Mixers (Tolstikhin et al., 2021). Following the settings by Chen et al. (2021), we train on the ImageNet-1k (Deng et al., 2009) training set using the Inception-style (Szegedy et al., 2015) pre-processing without extra training data or strong augmentation. For all models, we search for the best learning rate and weight decay for vanilla training, and then use the same values for the experiments with SAM and GSAM. For ResNets, we search for $\rho$ from 0.01 to 0.05 with a stepsize 0.01. For ViTs and Mixers, we search for $\rho$ from 0.05 to 0.6 with a stepsize 0.05. In GSAM, we search for $\alpha$ in $\{ 0 . 0 1 , 0 . 0 2 , 0 . 0 3 \}$ for ResNets and $\alpha$ in $\{ 0 . 1 , 0 . 2 , 0 . 3 \}$ for ViTs and Mixers. Considering that each step in SAM and GSAM requires twice the computation of vanilla training, we experiment with the vanilla training for twice the epochs of SAM and GSAM, but we observe no significant improvements from the longer training (Table 5 in appendix). We summarize the best hyper-parameters for each model in Appendix B. + +We report the performances on ImageNet (Deng et al., 2009), ImageNet-v2 (Recht et al., 2019) and ImageNet-Real (Beyer et al., 2020) in Table 1. GSAM consistently improves over SAM and vanilla training (with SGD or AdamW): on ViT-B/32, GSAM achieves $+ 5 . 4 \%$ improvement over AdamW and $+ 3 . 2 \%$ over SAM in top-1 accuracy; on Mixer-B/32, GSAM achieves $+ 1 1 . 1 \%$ over AdamW and $+ 1 . 2 \%$ over SAM. We ignore the standard deviation since it is typically negligible $( < 0 . 1 \% )$ compared to the improvements. We also test the generalization performance on out-of-distribution data (ImageNet-R and ImageNet-C), and the observation is consistent with that on ImageNet, e.g., $+ 5 . 1 \%$ on ImageNet-R and $+ 5 . 9 \%$ on ImageNet-C for Mixer-B/32. + +# 6.2 GSAM FINDS A MINIMUM WHOSE HESSIAN HAS SMALL DOMINANT EIGENVALUES + +Lemma 3.3 indicates that the surrogate gap $h$ is an equivalent measure of the dominant eigenvalue of the Hessian, and minimizing $h$ equivalently searches for a flat minimum. We empirically validate this in Fig. 4. As shown in the left subfigure, for some fixed $\rho$ , increasing $\alpha$ decreases the dominant value and improves generalization (test accuracy). In the middle subfigure, we plot the dominant eigenvalues estimated by the surrogate gap, $\sigma _ { m a x } \approx 2 h / \rho ^ { 2 }$ (Lemma 3.3). In the right subfigure, we directly calculate the dominant eigenvalues using the power-iteration (Mises & Pollaczek-Geiringer, 1929). The estimated dominant eigenvalues (middle) match the real eigenvalues $\sigma _ { m a x }$ (right) in terms of the trend that $\sigma _ { m a x }$ decreases with $\alpha$ and $\rho$ . Note that the surrogate gap $h$ is derived over the whole training set, while the measured eigenvalues are over a subset to save computation. These results show that the ascent step in GSAM minimizes the dominant eigenvalue by minimizing the surrogate loss, validating $\mathrm { T h m } 5 . 3$ . + +Table 1: Top-1 Accuracy $( \% )$ on ImageNet datasets for ResNets, ViTs and MLP-Mixers trained with Vanilla SGD or AdamW, SAM, and GSAM optimizers. + +
ModelTrainingImageNet-v1ImageNet-RealImageNet-V2ImageNet-RImageNet-C
ResNet
ResNet50Vanilla (SGD)76.082.463.622.244.6
SAM76.983.364.423.846.5
GSAM77.283.964.623.647.6
ResNet101Vanilla (SGD)77.883.965.324.448.5
SAM78.684.866.725.951.3
GSAM78.985.267.326.351.8
ResNet152Vanilla (SGD)78.584.266.325.350.0
SAM79.384.967.325.752.2
GSAM80.085.968.627.354.1
ViT-S/32Vanilla (AdamW)68.4Vision Transformer 75.254.3
SAM70.577.556.919.0 21.443.3
GSAM73.880.460.422.546.2 48.2
ViT-S/16Vanilla (AdamW)74.480.461.720.046.5
SAM78.184.165.624.753.0
GSAM79.585.367.325.353.3
ViT-B/32Vanilla (AdamW)71.477.557.523.444.0
SAM73.680.360.024.050.7
GSAM76.882.763.025.151.7
ViT-B/16Vanilla (AdamW)74.679.861.320.146.6
SAM79.985.267.526.456.5
GSAM81.086.569.227.155.7
Mixer-S/32Vanilla (AdamW)63.9MLP-Mixer 70.349.516.935.2
SAM66.773.852.418.639.3
GSAM68.675.855.022.644.6
Mixer-S/16Vanilla (AdamW)68.875.154.815.935.6
SAM72.979.858.920.142.0
GSAM75.081.761.923.748.5
Mixer-S/8Vanilla (AdamW)70.276.256.115.434.6
SAM75.982.562.320.542.4
GSAM76.883.464.024.647.8
Mixer-B/32Vanilla (AdamW)62.568.147.614.633.8
SAM72.479.058.022.846.2
GSAM73.680.259.927.952.1
Mixer-B/16Vanilla (AdamW)66.472.150.814.533.8
SAM77.483.563.924.748.8
GSAM77.884.064.928.354.4
+ +![](images/0ab9a693376a9164b50af0714b51b883842522baa5f9217592cd485202ddf052.jpg) +Figure 4: Influence of $\rho$ (set as constant for ease of comparison, other experiments use decayed $\rho _ { t }$ schedule) and $\alpha$ on the training of ViT-B/32. Left: Top-1 accuracy on ImageNet. Middle: Estimation of the dominant eigenvalues from the surrogate gap, $\sigma _ { m a x } \approx 2 h / \rho ^ { 2 }$ . Right: Dominant eigenvalues of the Hessian calculated via the power iteration. Middle and right figures match in the trend of curves, validating that the surrogate gap can be viewed as a proxy of the dominant eigenvalue of Hessian. + +![](images/cd91a39c4fda1aa38347bb16bcf091af8803ccb572f7dae609b6c2bda9d3fc93.jpg) +Figure 5: Top-1 accuracy of Mixer-S/32 trained with different methods. “+ascent” represents applying the ascent step in Algo. 1 to an optimizer. Note that our GSAM is described as SAM+ascent $\mathbf { \Lambda } = \mathbf { G S A M } )$ for consistency. + +Table 2: Results $( \% )$ of GSAM and $\operatorname* { m i n } ( f _ { p } + \lambda h )$ on ViT-B/32 + +
Datasetmin(fp+Xh)GSAM
ImageNet75.476.8
ImageNet-Real81.182.7
ImageNet-v260.963.0
ImageNet-R23.925.1
+ +Table 3: Transfer learning results (top-1 accuracy, $\%$ ) + +
ViT-B/16ViT-S/16
VanillaSAMGSAMVanillaSAMGSAM
Cifar1098.198.698.897.698.298.4
Cifar10087.689.189.785.787.688.1
Flowers88.591.891.286.491.590.3
Pets91.993.194.490.492.993.5
mean91.593.293.590.092.692.6
+ +# 6.3 COMPARISON WITH METHODS IN THE LITERATURE + +Section 6.1 compares GSAM to SAM and vanilla training. In this subsection, we further compare GSAM against Entropy-SGD (Chaudhari et al., 2019) and Adaptive-SAM (ASAM) (Kwon et al., 2021), which are designed to improve generalization. Note that Entropy-SGD uses SGD in the inner Langevin iteration and can be combined with other base optimizers such as AdamW as the outer loop. For Entropy-SGD, we find the hyper-parameter “scope” from 0.0 and 0.9, and search for the inner-loop iteration number between 1 and 14. For ASAM, we search for $\rho$ between 1 and 7 $( 1 0 \times$ larger than in SAM) as recommended by the ASAM authors. Note that the only difference between ASAM and SAM is the derivation of the perturbation, so both can be combined with the proposed ascent step. As shown in Fig. 5, the proposed ascent step increases test accuracy when combined with both SAM and ASAM and outperforms Entropy-SGD and vanilla training. + +# 6.4 ADDITIONAL STUDIES + +GSAM outperforms a weighted combination of the perturbed loss and surrogate gap With an example in Fig. 2, we demonstrate that directly minimizing $f _ { p } ( w ) + \lambda h ( w )$ as discussed in Sec. 4.1 is sub-optimal because $\nabla h ( w )$ could conflict with $\nabla f _ { p } ( w )$ and $\nabla f ( w )$ . We empirically validate this argument on ViT-B/32. We search for $\lambda$ between 0.0 and 0.5 with a step 0.1 and search for $\rho$ in the same grid as SAM and GSAM. We report the best accuracy of each method. Top-1 accuracy in Table 2 show the superior performance of GSAM, validating our analysis. + +$\operatorname* { m i n } ( f _ { p } , h )$ vs. $\operatorname* { m i n } ( f , h )$ GSAM solves $\operatorname* { m i n } ( f _ { p } , h )$ by descent in $\nabla f _ { p }$ , decomposing $\nabla f$ onto $\nabla f _ { p }$ , and an ascent step in the orthogonal direction to increase $f$ while keep $f _ { p }$ intact. Alternatively, we can also optimize $\operatorname* { m i n } ( f , h )$ by descent in $\nabla f$ , decomposing $\nabla f _ { p }$ onto $\nabla f$ , and a descent step in the orthogonal direction to decrease $f _ { p }$ while keep $f$ intact. The two GSAM variations perform similarly (see Fig. 6, right). We choose $\operatorname* { m i n } ( f _ { p } , h )$ mainly to make the minimal change to SAM. + +GSAM benefits transfer learning Using weights trained on ImageNet-1k, we finetune models with SGD on downstream tasks including the CIFAR10/CIFAR100 (Krizhevsky et al., 2009), Oxfordflowers (Nilsback & Zisserman, 2008) and Oxford-IITPets (Parkhi et al., 2012). Results in Table 3 shows that GSAM leads to better transfer performance than vanilla training and SAM. + +![](images/819890a555e7fa3dd273a452709db6a6112e787f55b5798c88bf87f762261419.jpg) +Figure 6: Top-1 accuracy of ViT-B/32 for the additional studies (Section 6.4). Left: from left to right are performances under different data augmentations (details in Appendix B.3) , where the vanilla method is trained for $2 \times$ the epochs. Middle: performance with different base optimizers. Right: Comparison between $\operatorname* { m i n } ( f _ { p } , h )$ and $\operatorname* { m i n } ( f , h )$ . + +GSAM remains effective under various data augmentations We plot the top-1 accuracy of a ViT-B/32 model under various Mixup (Zhang et al., 2017b) augmentations in Fig. 6 (left subfigure). Under different augmentations, GSAM consistently outperforms SAM and vanilla training. + +GSAM is compatible with different base optimizers GSAM is generic and applicable to various base optimizers. We compare vanilla training, SAM and GSAM using AdamW (Loshchilov & Hutter, 2017) and AdaBelief (Zhuang et al., 2020) with default hyper-parameters. Fig. 6 (middle subfigure) shows that GSAM performs the best, and SAM improves over vanilla training. + +# 7 CONCLUSION + +We propose the surrogate gap as an equivalent measure of sharpness which is easy to compute and feasible to optimize. We propose the GSAM method, which improves the generalization over SAM at negligible computation cost. We show the convergence and provably better generalization of GSAM compared to SAM, and validate the superior performance of GSAM on various models. + +# ACKNOWLEDGEMENT + +We would like to thank Xiangning Chen (UCLA) and Hossein Mobahi (Google) for discussions, Yi Tay (Google) for help with datasets, and Yeqing Li, Xianzhi Du, and Shawn Wang (Google) for help with TensorFlow implementation. + +# ETHICS STATEMENT + +This paper focuses on the development of optimization methodologies and can be applied to the training of different deep neural networks for a wide range of applications. Therefore, the ethical impact of our work would primarily be determined by the specific models that are trained using our new optimization strategy. + +# REPRODUCIBILITY STATEMENT + +We provide the detailed proof of theoretical results in Appendix A and provide the data preprocessing and hyper-parameter settings in Appendix B. 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Adabelief optimizer: Adapting stepsizes by the belief in observed gradients. arXiv preprint arXiv:2010.07468, 2020. + +# A PROOFS + +A.1 PROOF OF LEMMA. 3.1 + +Suppose $\rho$ is small, perform Taylor expansion around the local minima $w$ , we have: + +$$ +\boldsymbol { f } ( \boldsymbol { w } + \delta ) = \boldsymbol { f } ( \boldsymbol { w } ) + \nabla \boldsymbol { f } ( \boldsymbol { w } ) ^ { \top } \delta + \frac { 1 } { 2 } \delta ^ { \top } \boldsymbol { H } \delta + O ( | | \delta | | ^ { 3 } ) +$$ + +where $H$ is the Hessian, and is positive semidefinite at a local minima. At a local minima, $\nabla f ( w ) =$ 0, hence we have + +$$ +f ( w + \delta ) = f ( w ) + \frac { 1 } { 2 } \delta ^ { \top } H \delta + O ( | | \delta | | ^ { 3 } ) +$$ + +and + +$$ +f _ { p } ( w ) = \operatorname* { m a x } _ { | | \delta | | \leq \rho } f ( w + \delta ) = f ( w ) + \frac { 1 } { 2 } \rho ^ { 2 } \sigma _ { m a x } ( H ) + O ( | | \delta | | ^ { 3 } ) +$$ + +where $\sigma _ { m a x }$ is the dominate eigenvalue (eigenvalue with the largest absolute value). Now consider two local minima $w _ { 1 }$ and $w _ { 2 }$ with dominate eigenvalue $\sigma _ { 1 }$ and $\sigma _ { 2 }$ respectively, we have + +$$ +f _ { p } ( w _ { 1 } ) \approx f ( w _ { 1 } ) + \frac { 1 } { 2 } \rho ^ { 2 } \sigma _ { 1 } \qquad f _ { p } ( w _ { 2 } ) \approx f ( w _ { 2 } ) + \frac { 1 } { 2 } \rho ^ { 2 } \sigma _ { 2 } +$$ + +We have $f _ { p } ( w _ { 1 } ) > f _ { p } ( w _ { 2 } ) \implies \sigma _ { 1 } > \sigma _ { 2 }$ and $\sigma _ { 1 } > \sigma _ { 2 } \implies f _ { p } ( w _ { 1 } ) > f _ { p } ( w _ { 2 } )$ because the relation between $f ( w _ { 1 } )$ and $f ( w _ { 2 } )$ is undetermined.  + +# A.2 PROOF OF LEMMA. 3.2 + +Since $\rho$ is small, we can perform Taylor expansion around $w$ + +$$ +\begin{array} { r l } & { h ( w ) = f ( w + \delta ) - f ( w ) } \\ & { \qquad = \delta ^ { \top } \nabla f ( w ) + O ( \rho ^ { 2 } ) } \\ & { \qquad = \rho | | \nabla f ( w ) | | _ { 2 } + O ( \rho ^ { 2 } ) > 0 } \end{array} +$$ + +where the last line is because δ is approximated as δ = ρ ∇f(w)||∇f(w)||2+ , h ence has the same direction as $\nabla f ( w )$ .  + +# A.3 PROOF OF LEMMA. 3.3 + +Since $\rho$ is small, we can approximate $f ( w )$ with a quadratic model around a local minima $w$ : + +$$ +f ( w + \delta ) = f ( w ) + \frac { 1 } { 2 } \delta ^ { \top } H \delta + O ( \rho ^ { 3 } ) +$$ + +where $H$ is the Hessian at $w$ , assumed to be positive semidefinite at local minima. Normalize $\delta$ such that $| | \delta | | _ { 2 } = \rho$ , Hence we have: + +$$ +h ( w ) = f _ { p } ( w ) - f ( w ) = \operatorname* { m a x } _ { | | \delta | | _ { 2 } \leq \rho } f ( w + \delta ) - f ( w ) = \frac { 1 } { 2 } \sigma _ { m a x } \rho ^ { 2 } + O ( \rho ^ { 3 } ) +$$ + +where $\sigma _ { m a x }$ is the dominate eigenvalue of the hessian $H$ , and first order term is 0 because the gradient is 0 at local minima. Therefore, we have $\sigma _ { m a x } \approx 2 h ( w ) / \rho ^ { 2 }$ .  + +# A.4 PROOF OF THM. 5.1 + +For simplicity we consider the base optimizer is SGD. For other optimizers such as Adam, we can derive similar results by applying standard proof techniques in the literature to our proof. + +STEP 1: CONVERGENCE W.R.T FUNCTION $f _ { p } ( w )$ + +For simplicity of notation, we denote the update at step $t$ as + +$$ +d _ { t } = - \eta _ { t } g _ { p } ^ { ( t ) } + \eta _ { t } \alpha g _ { \perp } ^ { ( t ) } +$$ + +By $L -$ smoothness of $f$ and the definition of $f _ { p } ( w _ { t } ) = f ( w _ { t } ^ { a d v } )$ , and definition of $d _ { t } = w _ { t + 1 } - w _ { t }$ and $w _ { t } ^ { a d v } = w _ { t } + \delta _ { t }$ we have + +$$ +\begin{array} { l } { \displaystyle f _ { p } ( w _ { t + 1 } ) = f ( w _ { t + 1 } ^ { a d v } ) \leq f ( w _ { t } ^ { a d v } ) + \langle \nabla f ( w _ { t } ^ { a d v } ) , w _ { t + 1 } ^ { a d v } - w _ { t } ^ { a d v } \rangle + \frac { L } { 2 } \Big \| | w _ { t + 1 } ^ { a d v } - w _ { t } ^ { a d v } \Big \| ^ { 2 } } \\ { \displaystyle \qquad = f ( w _ { t } ^ { a d v } ) + \langle \nabla f ( w _ { t } ^ { a d v } ) , w _ { t + 1 } + \delta _ { t + 1 } - w _ { t } - \delta _ { t } \rangle } \\ { \displaystyle \qquad + \frac { L } { 2 } \Big \| w _ { t + 1 } + \delta _ { t + 1 } - w _ { t } - \delta _ { t } \Big \| ^ { 2 } } \\ { \displaystyle \qquad \leq f ( w _ { t } ^ { a d v } ) + \langle \nabla f ( w _ { t } ^ { a d v } ) , d _ { t } \rangle + L \Big \| d _ { t } \Big \| ^ { 2 } } \\ { \displaystyle \qquad + \langle \nabla f ( w _ { t } ^ { a d v } ) , \delta _ { t + 1 } - \delta _ { t } \rangle + L \Big \| \delta _ { t + 1 } - \delta _ { t } \Big \| ^ { 2 } } \end{array} +$$ + +STEP 1.0: BOUND EQ. 18 + +We first bound Eq. 18. Take expectation conditioned on observation up to step $t$ (for simplicity of notation, we use $\mathbb { E }$ short for $\mathbb { E } _ { x }$ to denote expectation over all possible data points) conditioned on observations up to step $t$ , also by definition of $d _ { t }$ , we have + +$$ +\begin{array} { r l } & { \mathbb { E } f _ { p } ( w _ { t + 1 } ) - f _ { p } ( w _ { t } ) \leq - \eta _ { t } \langle \nabla f _ { p } ( w _ { t } ) , \mathbb { E } g _ { p } ^ { ( t ) } \rangle + \alpha \eta _ { t } \langle \nabla f _ { p } ( w _ { t } ) , \mathbb { E } g _ { \perp } ^ { ( t ) } \rangle } \\ & { \qquad + L \eta _ { t } ^ { 2 } \mathbb { E } \Big | \Big | - g _ { p } ^ { ( t ) } + \alpha g _ { \perp } ^ { ( t ) } \Big | \Big | _ { 2 } ^ { 2 } } \\ & { \qquad \leq - \eta _ { t } \mathbb { E } \Big | \Big | \nabla f _ { p } ( w _ { t } ) \Big | \Big | _ { 2 } ^ { 2 } + 0 + ( \alpha + 1 ) ^ { 2 } G ^ { 2 } \eta _ { t } ^ { 2 } } \end{array} +$$ + +Since $\mathbb { E } g _ { \perp } ^ { ( t ) }$ is orthogonal to $\nabla f _ { p } ( w _ { t } )$ by construction, + +$$ +\vert \vert g ^ { ( t ) } \vert \vert \le G \mathrm { b y } \mathrm { a s s u m p t i o n } \bigg ) +$$ + +STEP 1.1: BOUND EQ. 19 + +By definition of $\delta _ { t }$ , we have + +$$ +\begin{array} { c } { \displaystyle \delta _ { t } = \rho _ { t } \frac { g ^ { ( t ) } } { | | g ^ { ( t ) } | | + \epsilon } } \\ { \displaystyle \delta _ { t + 1 } = \rho _ { t + 1 } \frac { g ^ { ( t + 1 ) } } { | | g ^ { ( t + 1 ) } | | + \epsilon } } \end{array} +$$ + +where $g ^ { ( t ) }$ is the gradient of $f$ at $w _ { t }$ evaluated with a noisy data sample. When learning rate $\eta _ { t }$ is small, the update in weight $d _ { t }$ is small, and expected gradient is + +$$ +\nabla f ( w _ { t + 1 } ) = \nabla f ( w _ { t } + d _ { t } ) = \nabla f ( w _ { t } ) + H d _ { t } + O ( | | d _ { t } | | ^ { 2 } ) +$$ + +where $H$ is the Hessian at $w _ { t }$ . Therefore, we have + +$$ +\begin{array} { r l } & { \mathbb { E } \langle \nabla f ( w _ { t } ^ { a d v } ) , \delta _ { t + 1 } - \delta _ { t } \rangle = \langle \nabla f ( w _ { t } ^ { a d v } ) , \rho _ { t } \mathbb { E } \frac { g ^ { ( t ) } } { | | g ^ { ( t ) } | | + \epsilon } - \rho _ { t + 1 } \mathbb { E } \frac { g ^ { ( t + 1 ) } } { | | g ^ { ( t + 1 ) } | | + \epsilon } \rangle } \\ & { \qquad \leq | | \nabla f ( w _ { t } ^ { a d v } ) | | \rho _ { t } \Big | \Big | \mathbb { E } \frac { g ^ { ( t ) } } { | | g ^ { ( t ) } | | + \epsilon } - \mathbb { E } \frac { g ^ { ( t + 1 ) } } { | | g ^ { ( t + 1 ) } | | + \epsilon } \Big | \Big | } \\ & { \qquad \leq | | \nabla f ( w _ { t } ^ { a d v } ) | | \rho _ { t } \phi _ { t } } \end{array} +$$ + +where the first inequality is due to (1) $\rho _ { t }$ is monotonically decreasing with $t$ , and (2) triangle inequality that $\langle a , b \rangle \overset { \cdot } { \leq } | | \boldsymbol { \dot { a } } | | \cdot | | b | |$ . $\phi _ { t }$ is the angle between the unit vector in the direction of $\nabla f ( w _ { t } )$ + +and $\nabla f ( w _ { t + 1 } )$ . The second inequality comes from that (1) $\begin{array} { r } { \left\| \frac { g } { | | g | | + \epsilon } \right\| < 1 } \end{array}$ strictly, so we can replace $\delta _ { t }$ in Eq. 25 with a unit vector in corresponding directions multiplied by $\rho _ { t }$ and get the upper bound, (2) the norm of difference in unit vectors can be upper bounded by the arc length on a unit circle. + +When learning rate $\eta _ { t }$ and update stepsize $d _ { t }$ is small, $\phi _ { t }$ is also small. Using the limit that + +$$ +\tan x = x + O ( x ^ { 2 } ) , \quad \sin x = x + O ( x ^ { 2 } ) , \quad x \to 0 +$$ + +We have: + +$$ +\begin{array} { r l r } { { \tan \phi _ { t } = \frac { | | \nabla f ( w _ { t + 1 } ) - \nabla f ( w _ { t } ) | | } { | | \nabla f ( w _ { t } ) | | } + O ( \phi _ { t } ^ { 2 } ) } } \\ & { } & { = \frac { | | H d _ { t } + O ( | | d _ { t } | | ^ { 2 } ) | | } { | | \nabla f ( w _ { t } ) | | } + O ( \phi _ { t } ^ { 2 } ) } \\ & { } & { \leq \eta _ { t } L ( 1 + \alpha ) } \end{array} +$$ + +where the last inequality is due to (1) max eigenvalue of $H$ is upper bounded by $L$ because $f$ is $L -$ smooth, (2) $| | d _ { t } | | = | | \eta _ { t } ( g _ { \parallel } + \alpha g _ { \perp } ) | |$ and $\mathbb { E } g _ { t } = \nabla f ( w _ { t } )$ . + +Plug into Eq. 27, also note that the perturbation amplitude $\rho _ { t }$ is small so $w _ { t }$ is close to $w _ { t } ^ { a d v }$ , then we have + +$$ +\mathbb { E } \langle \nabla f ( w _ { t } ^ { a d v } ) , \delta _ { t + 1 } - \delta _ { t } \rangle \leq L ( 1 + \alpha ) G \rho _ { t } \eta _ { t } +$$ + +Similarly, we have + +$$ +\begin{array} { r l } & { \mathbb { E } \Big \lvert \Big \lvert \delta _ { t + 1 } - \delta _ { t } \Big \rvert \Big \rvert ^ { 2 } \leq \rho _ { t } ^ { 2 } \mathbb { E } \Big \lvert \Big \lvert \frac { g ^ { ( t ) } } { \lvert | g ^ { ( t ) } \rvert \rvert + \epsilon } - \frac { g ^ { ( t + 1 ) } } { \lvert | g ^ { ( t + 1 ) } \rvert \rvert + \epsilon } \Big \rvert \Big \rvert ^ { 2 } } \\ & { \qquad \leq \rho _ { t } ^ { 2 } \phi _ { t } ^ { 2 } } \\ & { \qquad \leq \rho _ { t } ^ { 2 } \eta _ { t } ^ { 2 } L ^ { 2 } ( 1 + \alpha ) ^ { 2 } } \end{array} +$$ + +STEP 1.2: TOTAL BOUND + +Reuse results from Eq. 21 (replace $L _ { p }$ with $2 L$ ) and plug into Eq. 18, and plug Eq. 31 and Eq. 34 into Eq. 19, we have + +$$ +\begin{array} { r l } & { \mathbb { E } f _ { p } ( w _ { t + 1 } ) - f _ { p } ( w _ { t } ) \leq - \eta _ { t } \mathbb { E } \Big | \Big | \nabla f _ { p } ( w _ { t } ) \Big | \Big | _ { 2 } ^ { 2 } + \frac { 2 L ( \alpha + 1 ) ^ { 2 } } { 2 } G ^ { 2 } \eta _ { t } ^ { 2 } } \\ & { \qquad + L ( 1 + \alpha ) G \rho _ { t } \eta _ { t } + \frac { 2 L ^ { 3 } ( 1 + \alpha ) ^ { 2 } } { 2 } \eta _ { t } ^ { 2 } \rho _ { t } ^ { 2 } } \end{array} +$$ + +Perform telescope sum, we have + +$$ +\begin{array} { l } { \displaystyle \mathbb { E } f _ { p } ( w _ { T } ) - f _ { p } ( w _ { 0 } ) \leq - \sum _ { t = 1 } ^ { T } \eta _ { t } \mathbb { E } | | \nabla f _ { p } ( w _ { t } ) | | ^ { 2 } + \left[ L ( 1 + \alpha ) ^ { 2 } G ^ { 2 } \eta _ { 0 } ^ { 2 } + L ( 1 + \alpha ) G \rho _ { 0 } \eta _ { 0 } \right] \sum _ { t = 1 } ^ { T } \frac { 1 } { t } } \\ { \displaystyle \qquad + L ^ { 3 } ( 1 + \alpha ) ^ { 2 } \eta _ { 0 } ^ { 2 } \rho _ { 0 } ^ { 2 } \sum _ { t = 1 } ^ { T } \frac { 1 } { t ^ { 2 } } } \end{array} +$$ + +Hence + +$$ +\eta _ { T } \sum _ { t = 1 } ^ { T } \mathbb { E } | | \nabla f _ { p } ( w _ { t } ) | | ^ { 2 } \leq \sum _ { t = 1 } ^ { T } \eta _ { t } \mathbb { E } | | \nabla f _ { p } ( w _ { t } ) | | ^ { 2 } \leq f _ { p } ( w _ { 0 } ) - \mathbb { E } f _ { p } ( w _ { T } ) + D \log T + \frac { \pi ^ { 2 } E } { 6 } +$$ + +where + +$$ +D = L ( 1 + \alpha ) ^ { 2 } G ^ { 2 } \eta _ { 0 } ^ { 2 } + L ( 1 + \alpha ) G \rho _ { 0 } \eta _ { 0 } , \quad E = L ^ { 3 } ( 1 + \alpha ) ^ { 2 } \eta _ { 0 } ^ { 2 } \rho _ { 0 } ^ { 2 } +$$ + +Note that $\begin{array} { r } { \eta _ { T } = \frac { \eta _ { 0 } } { \sqrt { T } } } \end{array}$ , we have + +$$ +\frac { 1 } { T } \sum _ { t = 1 } ^ { T } \mathbb { E } | | \nabla f _ { p } ( w _ { t } ) | | ^ { 2 } \leq \frac { f _ { p } ( w _ { 0 } ) - f _ { m i n } + \pi ^ { 2 } E / 6 } { \eta _ { 0 } } \frac { 1 } { \sqrt { T } } + \frac { D } { \eta _ { 0 } } \frac { \log T } { \sqrt { T } } +$$ + +which implies that GSAM enables $f _ { p }$ to converge at a rate of $O ( \log T / \sqrt { T } )$ , and all the constants here are well-bounded. + +STEP 2: CONVERGENCE W.R.T. FUNCTION $f ( w )$ + +We prove the risk for $f ( w )$ convergences for non-convex stochastic optimization case using SGD. Denote the update at step $t$ as + +$$ +d _ { t } = - \eta _ { t } g _ { p } ^ { ( t ) } + \alpha \eta _ { t } g _ { \perp } ^ { ( t ) } +$$ + +By smoothness of $f$ , we have + +$$ +\begin{array} { l } { f ( w _ { t + 1 } ) \leq f ( w _ { t } ) + \langle \nabla f ( w _ { t } ) , d _ { t } \rangle + \displaystyle \frac { L } { 2 } \Big \lvert \Big \lvert d _ { t } \Big \rvert \Big \rvert _ { 2 } ^ { 2 } } \\ { = f ( w _ { t } ) + \langle \nabla f ( w _ { t } ) , - \eta _ { t } g _ { p } ^ { ( t ) } + \alpha \eta _ { t } g _ { \perp } ^ { ( t ) } \rangle + \displaystyle \frac { L } { 2 } \Big \lvert \Big \lvert d _ { t } \Big \rvert \Big \rvert _ { 2 } ^ { 2 } } \end{array} +$$ + +For simplicity, we introduce a scalar $\beta _ { t }$ such that + +$$ +\nabla f _ { \parallel } ( w _ { t } ) = \beta _ { t } \nabla f _ { p } ( w _ { t } ) +$$ + +where $\nabla f _ { \parallel } ( w _ { t } )$ is the projection of $\nabla f ( w _ { t } )$ onto $\nabla f _ { p } ( w _ { t } )$ . When perturbation amplitude $\rho$ is small, we expect $\beta _ { t }$ to be very close to 1. + +Take expectation conditioned on observations up to step $t$ for both sides of Eq. 42, we have: + +$$ +\begin{array} { l } { \displaystyle \mathbb { E } f ( w _ { t + 1 } ) \leq f ( w _ { t } ) + \left. \nabla f ( w _ { t } ) , - \frac { \eta _ { t } } { \beta _ { t } } \Big ( \nabla f ( w _ { t } ) - \nabla f _ { \perp } ( w _ { t } ) \Big ) + \alpha \eta _ { t } \mathbb { E } g _ { \perp } ^ { ( t ) } \right. + \displaystyle \frac { L } { 2 } \mathbb { E } \Big \| \boldsymbol { d } _ { t } \Big \| _ { 2 } ^ { 2 } } \\ { = f ( w _ { t } ) - \displaystyle \frac { \eta _ { t } } { \beta _ { t } } \Big \| \nabla f ( w _ { t } ) \Big \| _ { 2 } ^ { 2 } + \Big ( \frac { 1 } { \beta _ { t } } + \alpha \Big ) \eta _ { t } \Big \langle \nabla f ( w _ { t } ) , \nabla f _ { \perp } ( w _ { t } ) \Big \rangle + \displaystyle \frac { L } { 2 } \mathbb { E } \Big \| \boldsymbol { d } _ { t } \Big \| _ { 2 } ^ { 2 } } \\ { = f ( w _ { t } ) - \displaystyle \frac { \eta _ { t } } { \beta _ { t } } \Big \| \nabla f ( w _ { t } ) \Big \| _ { 2 } ^ { 2 } + \Big ( \frac { 1 } { \beta _ { t } } + \alpha \Big ) \eta _ { t } \Big \langle \nabla f ( w _ { t } ) , \nabla f ( w _ { t } ) \sin \theta _ { t } \Big \rangle + \displaystyle \frac { L } { 2 } \mathbb { E } \Big \| \boldsymbol { d } _ { t } \Big \| _ { 2 } ^ { 2 } } \end{array} +$$ + +$$ += f ( w _ { t } ) - \frac { \eta _ { t } } { \beta _ { t } } \Big \lvert \Big | \nabla f ( w _ { t } ) \Big \rvert \Big | _ { 2 } ^ { 2 } + \Big ( \frac { 1 } { \beta _ { t } } + \alpha \Big ) \eta _ { t } \Big \lvert \Big | \nabla f ( w _ { t } ) \Big \rvert \Big | _ { 2 } ^ { 2 } ( | \tan \theta _ { t } | + O ( \theta _ { t } ^ { 2 } ) ) + \frac { L } { 2 } \mathbb { E } \Big \lvert \Big | d _ { t } \Big \rvert \Big | _ { 2 } ^ { 2 } +$$ + +$$ +\Big ( \sin x = x + O ( x ^ { 2 } ) , \tan x = x + O ( x ^ { 2 } ) \mathrm { w h e n } x 0 . \Big ) +$$ + +Also note when perturbation amplitude $\rho _ { t }$ is small, we have + +$$ +\nabla f _ { p } ( w _ { t } ) = \nabla f ( w _ { t } + \delta _ { t } ) = \nabla f ( w _ { t } ) + \frac { \rho _ { t } } { | | \nabla f ( w _ { t } ) | | _ { 2 } + \epsilon } H ( w _ { t } ) \nabla f ( w _ { t } ) + O ( \rho _ { t } ^ { 2 } ) +$$ + +where δt = ρt ∇f (wt)||∇f (wt)||2 by definition, $H ( w _ { t } )$ is the Hessian. Hence we have + +$$ +| \tan \theta _ { t } | \leq \frac { | | \nabla f _ { p } ( w _ { t } ) - \nabla f ( w _ { t } ) | | } { | | \nabla f ( w _ { t } ) | | } \leq \frac { \rho _ { t } L } { | | \nabla f ( w _ { t } ) | | } +$$ + +where $L$ is the Lipschitz constant of $f$ , and $L -$ smoothness of $f$ indicates the maximum absolute eigenvalue of $H$ is upper bounded by $L$ . Plug Eq. 49 into Eq. 47, we have + +$$ +\begin{array} { l } { \displaystyle \mathbb { E } f ( w _ { t + 1 } ) \leq f ( w _ { t } ) - \frac { \eta _ { t } } { \beta _ { t } } \Big \| \nabla f ( w _ { t } ) \Big \| _ { 2 } ^ { 2 } + \Big ( \frac { 1 } { \beta _ { t } } + \alpha \Big ) \eta _ { t } \Big \| \nabla f ( w _ { t } ) \Big \| _ { 2 } ^ { 2 } | \tan \theta _ { t } | + \frac { L } { 2 } \mathbb { E } \Big \| d t _ { t } \Big \| _ { 2 } ^ { 2 } } \\ { \displaystyle \leq f ( w _ { t } ) - \frac { \eta _ { t } } { \beta _ { t } } \Big \| \nabla f ( w _ { t } ) \Big \| _ { 2 } ^ { 2 } + \Big ( \frac { 1 } { \beta _ { t } } + \alpha \Big ) L \rho _ { t } \eta _ { t } \Big \| \nabla f ( w _ { t } ) \Big \| _ { 2 } + \frac { L } { 2 } \mathbb { E } \Big \| d t _ { t } \Big \| _ { 2 } ^ { 2 } } \\ { \displaystyle \leq f ( w _ { t } ) - \frac { \eta _ { t } } { \beta _ { t } } \Big \| \nabla f ( w _ { t } ) \Big \| _ { 2 } ^ { 2 } + \Big ( \frac { 1 } { \beta _ { t } } + \alpha \Big ) L \rho _ { t } \eta _ { t } G + \frac { L } { 2 } \mathbb { E } \Big \| d t _ { t } \Big \| _ { 2 } ^ { 2 } } \\ { \Big ( \mathrm { A s s u m e ~ g r a d i e n t ~ h a s ~ b o u n d e d ~ n o r m ~ } G _ { \cdot } \Big ) } \\ { \displaystyle \leq f ( w _ { t } ) - \frac { \eta _ { t } } { \beta _ { m a x } } \Big \| \nabla f ( w _ { t } ) \Big \| _ { 2 } ^ { 2 } + \Big ( \frac { 1 } { \beta _ { m i n } } + \alpha \Big ) L \rho _ { t } \eta _ { t } G + \frac { L } { 2 } \mathbb { E } ( \alpha + 1 ) ^ { 2 } G ^ { 2 } \eta _ { t } ^ { 2 } } \end{array} +$$ + + $\beta _ { t }$ is close to 1 assuming $\rho$ is small, + +Re-arranging above formula, we have + +$$ +\frac { \eta _ { t } } { \beta _ { m a x } } \Big | \Big | \nabla f ( w _ { t } ) \Big | \Big | _ { 2 } ^ { 2 } \le f ( w _ { t } ) - \mathbb { E } f ( w _ { t + 1 } ) + \Big ( \frac { 1 } { \beta _ { m i n } } + \alpha \Big ) L G \eta _ { t } \rho _ { t } + \frac { L } { 2 } ( \alpha + 1 ) ^ { 2 } G ^ { 2 } \eta _ { t } ^ { 2 } +$$ + +perform telescope sum and taking expectations on each step, we have + +$$ +\frac { 1 } { \beta _ { m a x } } \sum _ { t = 1 } ^ { T } \eta _ { t } \Big | \Big | \nabla f ( w _ { t } ) \Big | \Big | _ { 2 } ^ { 2 } \le f ( w _ { 0 } ) - \mathbb { E } f ( w _ { T } ) + \Big ( \frac { 1 } { \beta _ { m i n } } + \alpha \Big ) L G \sum _ { t = 1 } ^ { T } \eta _ { t } \rho _ { t } + \frac { L } { 2 } ( \alpha + 1 ) ^ { 2 } G ^ { 2 } \sum _ { t = 1 } ^ { T } \eta _ { t } ^ { 2 } +$$ + +Take the schedule to be $\begin{array} { r } { \eta _ { t } = \frac { \eta _ { 0 } } { \sqrt { t } } } \end{array}$ and $\begin{array} { r } { \rho _ { t } = \frac { \rho _ { 0 } } { \sqrt { t } } } \end{array}$ , then we have + +$$ +\begin{array} { r l r } { { \frac { \eta _ { 0 } } { 3 m \alpha \lambda } \frac { 1 } { \sqrt { T } } \sum _ { t = 1 } ^ { T } \| \nabla f ( w _ { t } ) \| _ { 2 } ^ { 2 } \le L H S } } \\ & { } & { \le R H S } \\ & { } & { \le { \cal J } ( w _ { 0 } ) - { \cal J } _ { m i n } + \Big ( \frac { 1 } { \beta _ { m i n } } + \alpha \Big ) L G \eta _ { 0 } \rho _ { 0 } \sum _ { t = 1 } ^ { T } \frac { 1 } { t } + \frac { L } { 2 } ( \alpha + 1 ) ^ { 2 } G ^ { 2 } \eta _ { 0 } ^ { 2 } \frac { T } { t - 1 } ; } \\ & { } & { \overset { ( \mathrm { 5 9 } ) } { \le } ~ } \\ & { } & { \le f ( w _ { 0 } ) - f _ { m i n } + \Big ( \frac { 1 } { \beta _ { m i n } } + \alpha \Big ) L G \eta _ { 0 } \rho _ { 0 } ( 1 + \log T ) } \\ & { } & { ~ + \frac { L } { 2 } ( \alpha + 1 ) ^ { 2 } G ^ { 2 } \eta _ { 0 } ^ { 2 } ( 1 + \log T ) ~ ( 6 0 ) } \end{array} +$$ + +Hence + +$$ +\frac { 1 } { T } \sum _ { t = 1 } ^ { T } \Big | \Big | \nabla f ( w _ { t } ) \Big | \Big | _ { 2 } ^ { 2 } \le \frac { C _ { 3 } } { \sqrt { T } } + C _ { 4 } \frac { \log T } { \sqrt { T } } +$$ + +where $C _ { 1 } , C _ { 4 }$ are some constants. This implies the convergence rate w.r.t $f ( w )$ is $O ( \log T / \sqrt { T } )$ + +STEP 3: CONVERGENCE W.R.T. SURROGATE GAP $h ( w )$ + +Note that we have proved convergence for $f _ { p } ( w )$ in step 1, and convergence for $f ( w )$ in step 3. Also note that + +$$ +\Big | \Big | \nabla h ( w _ { t } ) \Big | \Big | _ { 2 } ^ { 2 } = \Big | \Big | \nabla f _ { p } ( w _ { t } ) - \nabla f ( w _ { t } ) \Big | \Big | _ { 2 } ^ { 2 } \leq 2 \Big | \Big | \nabla f _ { p } ( w _ { t } ) \Big | \Big | _ { 2 } ^ { 2 } + 2 \Big | \Big | \nabla f ( w _ { t } ) \Big | \Big | _ { 2 } ^ { 2 } +$$ + +Hence + +$$ +\frac { 1 } { T } \sum _ { t = 1 } ^ { T } \Big | \Big | \nabla h ( w _ { t } ) \Big | \Big | _ { 2 } ^ { 2 } \le \frac { 2 } { T } \sum _ { t = 1 } ^ { T } \Big | \Big | \nabla f _ { p } ( w _ { t } ) \Big | \Big | _ { 2 } ^ { 2 } + \frac { 2 } { T } \sum _ { t = 1 } ^ { T } \Big | \Big | \nabla f ( w _ { t } ) \Big | \Big | _ { 2 } ^ { 2 } +$$ + +also converges at rate $O ( \log T / \sqrt { T } )$ because each item in the RHS converges at rate $O ( \log T \sqrt { T } )$ + +# A.5 PROOF OF COROLLARY. 5.2.1 + +Using the results from Thm. 5.2, with probability at least $1 - a$ , we have + +$$ +\mathbb { E } _ { w \sim \mathcal { Q } } \mathbb { E } _ { { x } } f ( w , { x } ) \le \mathbb { E } _ { w \sim \mathcal { Q } } \widehat { f } ( w ) + 4 \sqrt { \frac { K L ( \mathcal { Q } | | \mathcal { P } ) + \log \frac { 2 m } { a } } { m } } +$$ + +Assume $\delta \sim \mathcal { N } ( 0 , b ^ { 2 } I _ { k } )$ where $k$ is the dimension of model parameters, hence $\delta ^ { 2 }$ (element-wise square) follows a a Chi-square distribution. By Lemma.1 in Laurent $\&$ Massart (2000), we have + +$$ +\mathbb { P } \big ( | | \delta | | _ { 2 } ^ { 2 } - k b ^ { 2 } \geq 2 b ^ { 2 } \sqrt { k t } + 2 t b ^ { 2 } \big ) \leq e x p ( - t ) +$$ + +hence with probability at least $1 - 1 / \sqrt { n }$ , we have + +$$ +| | \delta | | _ { 2 } ^ { 2 } \leq b ^ { 2 } \Bigg ( 2 \log \sqrt { n } + k + 2 \sqrt { k \log \sqrt { n } } \Bigg ) \leq 2 b ^ { 2 } k \Bigg ( 1 + \sqrt { \frac { \log \sqrt { n } } { k } } \Bigg ) ^ { 2 } \leq \rho ^ { 2 } +$$ + +Therefore, with probability at least $\begin{array} { r } { 1 - 1 / \sqrt { n } = 1 - e x p \bigg ( - \big ( \frac { \rho } { \sqrt { 2 } b } - \sqrt { k } \big ) ^ { 2 } \bigg ) } \end{array}$ + +$$ +\begin{array} { r } { \mathbb { E } _ { \delta } \widehat { f } ( w + \delta ) \leq \operatorname* { m a x } _ { | | \delta | | _ { 2 } \leq \rho } \widehat { f } ( w + \delta ) } \end{array} +$$ + +Combine Eq. 65 and Eq. 67, subtract the same constant $C$ on both sides, and under the same assumption as in (Foret et al., 2020) that $\begin{array} { r } { \mathbb { E } _ { w \sim \mathcal { Q } } \mathbb { E } _ { x } f ( w , x ) \le \mathbb { E } _ { \delta \sim \mathcal { N } ( 0 , b ^ { 2 } I ^ { k } ) } \mathbb { E } _ { w \sim \mathcal { Q } } \mathbb { E } _ { x } f ( w + \delta , x ) \mathrm { w e } } \end{array}$ finish the proof.  + +# A.6 PROOF OF THM. 5.3 + +STEP 1: A SUFFICIENT CONDITION THAT THE LOSS GAP IS EXPECTED TO DECREASE FOR EACH STEP + +Take Taylor expansion, then the expected change of loss gap caused by descent step is + +$$ +\begin{array} { r l } & { \mathbb { E } \langle \nabla f _ { p } ( w _ { t } ) - \nabla f ( w _ { t } ) , - \eta _ { t } \nabla f _ { p } ( w _ { t } ) \rangle } \\ & { \Big ( w h e r e \mathbb { E } g _ { \perp } = \nabla f _ { \perp } ( w _ { t } ) \Big ) } \\ & { = \eta _ { t } \Bigg [ - | | \nabla f _ { p } ( w _ { t } ) | | _ { 2 } ^ { 2 } + | | \nabla f _ { p } ( w _ { t } ) | | _ { 2 } \big | | \nabla f ( w _ { t } ) | | _ { 2 } \cos \theta _ { t } \Bigg ] } \end{array} +$$ + +where $\theta _ { t }$ is the angle between vector $\nabla f _ { p } ( w _ { t } )$ and $\nabla f ( w _ { t } )$ . + +The expected change of loss gap caused by ascent step is + +$$ +\begin{array} { r } { \mathbb { E } \langle \nabla f _ { p } ( w _ { t } ) - \nabla f ( w _ { t } ) , \alpha \eta _ { t } \nabla f _ { \perp } ( w _ { t } ) \rangle = - \alpha \eta _ { t } \big \lvert \big \rvert \nabla f _ { \perp } ( w _ { t } ) \big \rvert \big \rvert _ { 2 } ^ { 2 } < 0 } \end{array} +$$ + +Above results demonstrate that ascent step decreases the loss gap, while descent step might increase the loss gap. A sufficient (but not necessary) condition for $\mathbb { E } \langle \nabla h ( w _ { t } ) , d t \rangle \leq 0$ requires $\alpha$ to be large or $| \big | \nabla f ( w _ { t } ) \big | \big | _ { 2 } \cos \theta _ { t } \leq \big | \big | \nabla f _ { p } ( w _ { t } ) \big | \big |$ . In practice, the perturbation amplitude $\rho$ is small and we can assume $\theta _ { t }$ is close to 0 and $\big | \big | \nabla f _ { p } ( w _ { t } ) \big | \big |$ is close to $\left| \left| \nabla f ( w _ { t } ) \right| \right|$ , we can also set the parameter $\alpha$ to be large in order to decrease the loss gap. + +STEP 2: UPPER AND LOWER BOUND OF DECREASE IN LOSS GAP (BY THE ASCENT STEP IN ORTHOGONAL GRADIENT DIRECTION) COMPARED TO SAM. + +Next we give an estimate of the decrease in $\widehat { h }$ caused by our ascent step. We refer to Eq. 69 and Eq. 70 to analyze the change in loss gap caused by the descent and ascent (orthogonally) respectively. It can be seen that gradient descent step might not decrease loss gap, in fact they often increase loss gap in practice; while the ascent step is guaranteed to decrease the loss gap. + +The decrease in loss gap is: + +$$ +\begin{array} { r l r } & { } & { \Delta \widehat { h } _ { t } = - \langle \nabla \widehat { f } _ { p } ( w _ { t } ) - \nabla \widehat { f } ( w _ { t } ) , \alpha \eta _ { t } \nabla \widehat { f } _ { \bot } ( w _ { t } ) \rangle = \alpha \eta _ { t } \big \lvert \big \lvert \nabla \widehat { f } _ { \bot } ( w _ { t } ) \big \rvert \big \rvert _ { 2 } ^ { 2 } } \\ & { } & { = \alpha \eta _ { t } \big \lvert \big \lvert \nabla \widehat { f } ( w _ { t } ) \big \rvert \big \rvert _ { 2 } ^ { 2 } \big \lvert \tan \theta _ { t } \big \rvert ^ { 2 } } \end{array} +$$ + +$$ +\begin{array} { r l r } { { \sum _ { t = 1 } ^ { T } \Delta \widehat { h } _ { t } \le \sum _ { t = 1 } ^ { T } \alpha L ^ { 2 } \eta _ { t } \rho _ { t } ^ { 2 } } } \\ & { } & { \Big ( \mathrm { B y ~ E q . ~ 4 9 } \Big ) } \\ & { } & { \le \displaystyle \sum _ { t = 1 } ^ { T } \alpha L ^ { 2 } \eta _ { 0 } \rho _ { 0 } ^ { 2 } \frac { 1 } { t ^ { 3 / 2 } } } \\ & { } & { \le 2 . 7 \alpha L ^ { 2 } \eta _ { 0 } \rho _ { 0 } ^ { 2 } } \end{array} +$$ + +Hence we derive an upper bound for $\Sigma _ { t = 1 } ^ { T } \Delta \widehat { h } _ { t }$ + +Next we derive a lower bound for $\Sigma _ { t = 1 } ^ { T } \Delta \widehat { h } _ { t }$ Note that when $\rho _ { t }$ is small, by Taylor expansion + +$$ +\nabla \widehat { f } _ { p } ( w _ { t } ) = \nabla \widehat { f } ( w _ { t } + \delta _ { t } ) = \nabla \widehat { f } ( w _ { t } ) + \frac { \rho _ { t } } { | | \nabla \widehat { f } ( w _ { t } ) | | } \widehat { H } ( w _ { t } ) \nabla \widehat { f } ( w _ { t } ) + O ( \rho _ { t } ^ { 2 } ) +$$ + +where $\widehat { H } ( w _ { t } )$ is the Hessian evaluated on training samples. Also when $\rho _ { t }$ is small, the angle $\theta _ { t }$ between $\nabla \widehat { f } _ { p } ( w _ { t } )$ and $\nabla \widehat { f } ( \boldsymbol { w } _ { t } )$ is small, by the limit that + +$$ +\begin{array} { l } { \tan x = x + O ( x ^ { 2 } ) , x \to 0 } \\ { \sin x = x + O ( x ^ { 2 } ) , x \to 0 } \end{array} +$$ + +We have + +$$ +\left| \tan \theta _ { t } \right| = \left| \sin \theta _ { t } \right| + O ( \theta _ { t } ^ { 2 } ) = | \theta _ { t } | + O ( \theta _ { t } ^ { 2 } ) +$$ + +Omitting high order term, we have + +$$ +| \tan \theta _ { t } | \approx | \theta _ { t } | = \frac { | | \nabla \widehat { f } _ { p } ( w _ { t } ) - \nabla \widehat { f } ( w _ { t } ) | | } { | | \widehat { f } ( w _ { t } ) | | } = \frac { | | \rho _ { t } \widehat { H } ( w _ { t } ) + O ( \rho _ { t } ^ { 2 } ) | | } { | | \nabla \widehat { f } ( w _ { t } ) | | } \geq \frac { \rho _ { t } | \sigma | _ { m i n } } { G } +$$ + +where $G$ is the upper-bound on norm of gradient, $| \sigma | _ { m i n }$ is the minimum absolute eigenvalue of the Hessian. The intuition is that as perturbation amplitude decreases, the angle $\theta _ { t }$ decreases at a similar rate, though the scale constant might be different. Hence we have + +$$ +\begin{array} { r l } { { \sum _ { t = 1 } ^ { T } \Delta \hat { h } _ { t } = \sum _ { t = 1 } ^ { T } \alpha \eta _ { t } \vert \big \vert \nabla \widehat { f } ( w _ { t } ) \big \vert \big \vert _ { 2 } ^ { 2 } \vert \tan \theta _ { t } \vert ^ { 2 } + O ( \theta _ { t } ^ { 4 } ) } \ ~ } \\ & { \geq \sum _ { t = 1 } ^ { T } \alpha \eta _ { t } c ^ { 2 } \Big ( \frac { \rho _ { t } \vert \sigma \vert _ { m i n } } { G } \Big ) ^ { 2 } } \\ & { = \frac { \alpha c ^ { 2 } \rho _ { 0 } ^ { 2 } \eta _ { 0 } \vert \sigma \vert _ { m i n } ^ { 2 } } { G ^ { 2 } } \sum _ { t = 1 } ^ { T } \frac { 1 } { t ^ { 3 / 2 } } } \\ & { \geq \frac { \alpha c ^ { 2 } \rho _ { 0 } ^ { 2 } \eta _ { 0 } \vert \sigma \vert _ { m i n } ^ { 2 } } { G ^ { 2 } } } \end{array} +$$ + +where $c ^ { 2 }$ is the lower bound of $| | \nabla \widehat { f } | | ^ { 2 }$ (e.g. due to noise in data and gradient observation). Results above indicate that the decrease in loss gap caused by the ascent step is non-trivial, hence our proposed method efficiently improves generalization compared with SAM.  + +# A.7 DISCUSSION ON COROLLARY 5.2.1 + +The comment “‘The corollary gives a bound on the risk in terms of the perturbed training loss if one removes $C$ from both sides”’ is correct. But there is a misunderstanding in the statement “‘the perturbed training loss is small then the model has a small risk”’: it’s only true when $\rho _ { t r a i n }$ for training equals its real value $\rho _ { t r u e }$ determined by the data distribution; in practice, we never know $\rho _ { t r u e }$ . In the following we show that the minimization of both $h$ and $f _ { p }$ is better than simply minimizing $f _ { p }$ when $\rho _ { t r u e } \neq \rho _ { t r a i n }$ . + +1. First, we re-write the conclusion of Corollary 5.2.1 as + +$$ +\begin{array} { r } { \mathbb { E } _ { w } \mathbb { E } _ { x } f ( w , x ) \le f _ { p } + R = C + \widehat { h } + R = C + \rho ^ { 2 } \sigma / 2 + R + O ( \rho ^ { 3 } ) } \\ { w i t h p r o b a b i l i t y \left( 1 - a \right) [ 1 - e ^ { - ( \frac { \rho } { \sqrt { 2 } b } - \sqrt { k } ) ^ { 2 } } ] } \end{array} +$$ + +where $R$ is the regularization term, $C$ is the training loss, $\sigma$ is the dominant eigenvalue of Hessian. As in lemma 3.3, we perform Taylor-expansion and can ignore the high-order term $O ( \rho ^ { 3 } )$ . We focus on + +$$ +f _ { p } = C + \widehat { h } = C + \rho ^ { 2 } \sigma / 2 +$$ + +2. When $\rho _ { t r u e } \neq \rho _ { t r a i n }$ , minimizing $h$ achieves a lower risk than only minimizing $f _ { p }$ . (1) Note that after training, $C$ (training loss) is fixed, but $h$ could vary with $\rho$ (e.g. when training on dataset A and testing on an unrelated dataset B, the training loss remains unchanged, but the risk would be huge and a large $\rho$ is required for a valid bound). (2) With an example, we show a low $f _ { p }$ is insufficient for generalization, and a low $\sigma$ is necessary: + +A Suppose we use $\rho _ { t r a i n }$ for training, and consider two solutions with $C _ { 1 } , \sigma _ { 1 }$ (SAM) and $C _ { 2 } , \sigma _ { 2 }$ (GSAM). Suppose they have the same $f _ { p }$ during training for some $\rho _ { t r a i n }$ , so $f _ { p 1 } = C _ { 1 } + \sigma _ { 1 } / 2 \times \rho _ { t r a i n } ^ { 2 } = C _ { 2 } + \sigma _ { 2 } / 2 \times \rho _ { t r a i n } ^ { 2 } = f _ { p 2 }$ Suppose $C _ { 1 } < C _ { 2 }$ so $\sigma _ { 1 } > \sigma _ { 2 }$ . +B When $\rho _ { t r u e } > \rho _ { t r a i n }$ , we have risk bound $\begin{array} { r } { - 1 = C _ { 1 } + \sigma _ { 1 } / 2 \times \rho _ { t r u e } ^ { 2 } + R > \mathrm { r i s k . b o u n d . } 2 = C _ { 2 } + \sigma _ { 2 } / 2 \times \rho _ { t r u e } ^ { 2 } + R } \end{array}$ This implies that a small $\sigma$ helps generalization, but only a low $f _ { p 1 }$ (caused by a low $C _ { 1 }$ and high $\sigma _ { 1 }$ ) is insufficient for a good generalization. +C Note that $\rho _ { t r a i n }$ is fixed during training, so minimizing $h _ { t r a i n }$ during training is equivalently minimizing $\sigma$ by Lemma 3.3 + +3. Why we are often unlucky to have $\rho _ { t r u e } ~ > ~ \rho _ { t r a i n }$ (1) First, the test sets are almost surely outside the convex hull of the training set because “‘interpolation almost surely never occurs in high-dimensional $( > ~ 1 0 0 )$ cases”’ Balestriero et al. (2021). As a result, the variability of (train $^ +$ test) sets is almost surely larger than the variability of (train) set. Since $\rho$ increases with data variability (see point 4 below), we have $\rho _ { t r u e } > \rho _ { t r a i n - s e t }$ almost surely. (2) Second, we don’t know the value of $\rho _ { t r u e }$ and can only guess it. In practice, we often guess a small value because training often diverges with large $\rho$ (as observed in Foret et al. (2020); Chen et al. (2021)). + +4. Why $\rho$ increases with data variability. In Corollary 5.2.1, we assume weight perturbation $\delta \sim \mathcal { N } ( 0 , b ^ { 2 } I ^ { k } )$ . The meaning of $b$ is the following. If we can randomly sample a fixed number of samples from the underlying distribution, then training the model from scratch (with a fixed seed for random initialization) gives rise to a set of weights. Repeating this process, we get many sets of weights, and their standard deviation is $b$ . Since the number of training samples is limited and fixed, the more variability in data, the more variability in weights, and the larger $b$ . Note that Corollary stated that the bound holds with probability proportional to $[ 1 - e ^ { - ( \frac { \rho } { \sqrt { 2 } b } - \sqrt { k } ) ^ { 2 } } ]$ . In order for the result to hold with a fixed probability, $\rho$ must stay proportional to $b$ , hence $\rho$ also increases with the variability of data. + +Table 4: Hyper-parameters to reproduce experimental results + +
ModelPmaxPminαlrmaxlrminWeight DecayBase OptimizerEpochsWarmup StepsLR schedule
ResNet500.040.020.011.61.6e-20.3SGD905kLinear
ResNet1010.040.020.011.61.6e-20.3SGD905kLinear
ResNet5120.040.020.0051.61.6e-20.3SGD905kLinear
ViT-S/320.60.00.43e-33e-50.3AdamW30010kLinear
ViT-S/160.60.01.03e-33e-50.3AdamW30010kLinear
ViT-B/320.60.10.63e-33e-50.3AdamW30010kLinear
ViT-B/160.60.20.43e-33e-50.3AdamW30010kLinear
Mixer-S/320.50.00.23e-33e-50.3AdamW30010kLinear
Mixer-S/160.50.00.63e-33e-50.3AdamW30010kLinear
Mixer-S/80.50.10.13e-33e-50.3AdamW30010kLinear
Mixer-B/320.70.20.053e-33e-50.3AdamW30010kLinear
Mixer-B/160.50.20.013e-33e-50.3AdamW30010kLinear
+ +# B EXPERIMENTAL DETAILS + +# B.1 TRAINING DETAILS + +For ViT and Mixer, we search the learning rate in $\{ 1 \mathrm { e } { - } 3 , 3 \mathrm { e } { - } 3 , 1 \mathrm { e } { - } 2 , 3 \mathrm { e } { - } 3 \}$ , and search weight decay in $\{ 0 . 0 0 3 , 0 . 0 3 , 0 . 3 \}$ . For ResNet, we search the learning rate in $\left. 1 . 6 , 0 . 1 6 , 0 . 0 1 6 \right.$ , and search the weight decay in $\{ 0 . 0 0 1 , 0 . 0 1 , 0 . 1 \}$ . For ViT and Mixer, we use the AdamW optimizer with $\beta _ { 1 } = 0 . 9 , \beta _ { 2 } = 0 . 9 9 $ ; for ResNet we use SGD with momentum $\ l = 0 . 9$ . We train ResNets for 90 epochs, and train ViTs and Mixers for 300 epochs following the settings in (Chen et al., 2021) and (Dosovitskiy et al., 2020). Considering that SAM and GSAM uses twice the computation of vanilla training for each step, for vanilla training we try $2 \times$ longer training, and does not find significant improvement as in Table. 5. + +We first search the optimal learning rate and weight decay for vanilla training, and keep these two hyper-parameters fixed for SAM and GSAM. For ViT and Mixer, we search $\rho$ in $\{ 0 . 1 , \bar { 0 . 2 } , 0 . 3 , 0 . 4 ,$ $0 . 5 , 0 . 6 \}$ for SAM and GSAM; for ResNet, we search $\rho$ from 0.01 to 0.05 with a stepsize 0.01. For ASAM, we amplify $\rho$ by $1 0 \times$ compared to SAM, as recommended by Kwon et al. (2021). For GSAM, we search $\alpha$ in $\{ 0 . 1 , 0 . 2 , 0 . 3 \}$ throughout the paper. We report the best configuration of each individual model in Table. 4. + +# B.2 TRANSFER LEARNING EXPERIMENTS + +Using weights trained on ImageNet-1k, we finetune models with SGD on downstream tasks including the CIFAR10/CIFAR100 (Krizhevsky et al., 2009), Oxford-flowers (Nilsback & Zisserman, 2008) and Oxford-IITPets (Parkhi et al., 2012). For all experiments, we use the SGD optimizer with no weight decay under a linear learning rate schedule and gradient clipping with global norm 1. We search the maximum learning rate in $\left. 0 . 0 0 1 , 0 . 0 0 3 , 0 . 0 1 , 0 . 0 3 \right.$ . On Cifar datasets, we train models for 10k steps with a warmup step of 500; on Oxford datasets, we train models for 500 steps with a wamup step of 100. + +# B.3 EXPERIMENTAL SETUP WITH ABLATION STUDIES ON DATA AUGMENTATION + +We follow the settings in (Tolstikhin et al., 2021) to perform ablation studies on data augmentation. In the left subfigure of Fig. 6, “Light” refers to Inception-style data augmentation with random flip and crop of images, “Medium” refers to the mixup augmentation with probability 0.2 and RandAug magnitude 10; “Strong” refers to the mixup augmentation with probability 0.2 and RandAug magnitude 15. + +# C ABLATION STUDIES AND DISCUSSIONS + +# C.1 INFLUENCE OF $\rho$ AND $\alpha$ + +We plot the performance of a ViT-B/32 model varying with $\rho$ (Fig. 7a) and $\alpha$ (Fig. 7b). We empirically validate that fine-tuning $\rho$ in SAM can not achieve comparable performance with GSAM, as + +![](images/24eba187fef4c92575a570adcf3b277bd800c56050a22796582502f9883ba719.jpg) +Figure 7: Performance of GSAM varying with $\rho$ and $\alpha$ + +Table 5: Top-1 accuracy of ViT-B/32 on ImageNet with Inception-style data augmentation. For vanilla training we report results for training 300 epochs and 600 epochs, for GSAM we report the results for 300 epochs. + +
MethodEpochsImageNetImageNet-RealImageNet-v2ImageNet-R
Vanilla30071.477.557.523.4
60072.078.257.923.6
GSAM30076.882.763.025.1
+ +shown in Fig. 7a. Considering that GSAM has one more parameter $\alpha$ , we plot the accuracy varying with $\alpha$ in Fig. 7b, and show that GSAM consistently outperforms SAM and vanilla training. + +# C.2 CONSTANT $\rho$ V.S. DECAYED $\rho _ { t }$ SCHEDULE + +Note that Thm. 5.1 assumes $\rho _ { t }$ to decay with $t$ in order to prove the convergence, while SAM uses a constant $\rho$ during training. To eliminate the influence of $\rho _ { t }$ schedule, we conduct ablation study as in Table. 6. The ascent step in GSAM can be applied to both constant $\rho$ or a decayed $\rho _ { t }$ schedule, and improves accuracy for both cases. Without ascent step, constant $\rho$ and decayed $\rho _ { t }$ achieve similar performance. Results in Table. 6 implies that the ascent step in GSAM is the main reason for improvement of generalization performance. + +![](images/9b390274fa28b5ba47c294dfef588e29c76201cc8670a2da5a44ef3d22931fcd.jpg) +Figure 8: The value of $\cos \theta _ { t }$ varying with training steps, where $\theta _ { t }$ is the angle between $\nabla f ( w _ { t } )$ and $\nabla f _ { p } ( w _ { t } )$ as in Fig. 2. + +![](images/c33dd62d4567c39ae55b11964e44b74a04f56f0f5e2ffdc6a47b45abc7f9be5f.jpg) +Figure 9: Surrogate gap curve under different $\alpha$ values. + +Table 6: Top-1 Accuracy on ViT-B/32 on ImageNet. Ablation studies on constant $\rho$ or a decayed $\rho _ { t }$ + +
VanillaConstant p (SAM)Constant ρ+ascentDecayed ptIDecayed pt+ascent
72.075.876.275.876.8
+ +# C.3 VISUALIZE THE TRAINING PROCESS + +In the proof of Thm. 5.3, our analysis relies on assumption that $\theta _ { t }$ is small. We empirically validated this assumption by plotting $\cos \theta _ { t }$ in Fig. 8, where $\theta _ { t }$ is the angle between $\nabla f ( \bar { w } _ { t } )$ and $\nabla f _ { p } ( w _ { t } )$ . Note that the cosine value is calculated in the parameter space of dimension $8 . 8 \times 1 0 ^ { 7 }$ , and in high-dimensional space two random vectors are highly likely to be perpendicular. In Fig. 8 the cosine value is always above 0.9, indicating that $\nabla f ( w _ { t } )$ and $\nabla f _ { p } ( w _ { t } )$ point to very close directions considering the high dimension of parameters. This empirically validates our assumption that $\theta _ { t }$ is small during training. + +We also plot the surrogate gap during training in Fig. 9. As $\alpha$ increases, the surrogate gap decreases, validating that the ascent step in GSAM efficiently minimizes the surrogate gap. Furthermore, the surrogate gap increases with training steps for any fixed $\alpha$ , indicating that the training process gradually falls into local minimum in order to minimize the training loss. + +# D RELATED WORKS + +Besides SAM and ASAM, other methods were proposed in the literature to improve generalization: Lin et al. (2020) proposed extrapolation of gradient, Xie et al. (2021) proposed to manipulate the noise in gradient, and Damian et al. (2021) proved label noise improves generalization, Yue et al. (2020) proposed to adjust learning rate according to sharpness, and Zheng et al. (2021) proposed model perturbation with similar idea to SAM. Izmailov et al. (2018) proposed averaging weights to improve generalization, and Heo et al. (2020) restricted the norm of updated weights to improve generalization. Many of aforementioned methods can be combined with GSAM to further improve generalization. + +Besides modified training schemes, there are other two types of techniques to improve generalization: data augmentation and model regularization. Data augmentation typically generates new data from training samples; besides standard data augmentation such as flipping or rotation of images, recent data augmentations include label smoothing (Muller et al., 2019) and mixup (M ¨ uller et al., ¨ 2019) which trains on convex combinations of both inputs and labels, automatically learned augmentation (Cubuk et al., 2018), and cutout (DeVries & Taylor, 2017) which randomly masks out parts of an image. Model regularization typically applies auxiliary losses besides the training loss such as weight decay (Loshchilov & Hutter, 2017), other methods randomly modify the model architecture during training, such as dropout (Srivastava et al., 2014) and shake-shake regularization (Gastaldi, 2017). Note that the data augmentation and model regularization literature mentioned here typically train with the standard back-propagation (Rumelhart et al., 1985) and first-order gradient optimizers, and both techniques can be combined with GSAM. + +Besides SGD, Adam and AdaBelief, GSAM can be combined with other first-order gradient optimizers, such as AdaBound (Luo et al., 2019), RAdam (Liu et al., 2019), Yogi (Zaheer et al., 2018), AdaGrad (Duchi et al., 2011), AMSGrad (Reddi et al., 2019) and AdaDelta (Zeiler, 2012). \ No newline at end of file diff --git a/parse/dev/edONMAnhLu-/edONMAnhLu-_content_list.json b/parse/dev/edONMAnhLu-/edONMAnhLu-_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..561ffb71a3fda01aa411e0d04f566aaf835d769b --- /dev/null +++ b/parse/dev/edONMAnhLu-/edONMAnhLu-_content_list.json @@ -0,0 +1,3995 @@ +[ + { + "type": "text", + "text": "SURROGATE GAP MINIMIZATIONIMPROVES SHARPNESS-AWARE TRAINING", + "text_level": 1, + "bbox": [ + 174, + 99, + 679, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Juntang Zhuang1 ∗ j.zhuang@yale.edu ", + "bbox": [ + 184, + 170, + 351, + 199 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Boqing Gong2, Liangzhe Yuan2, Yin $\\mathbf { C } \\mathbf { u } \\mathbf { i } ^ { 2 }$ , Hartwig Adam2 {bgong, lzyuan, yincui, hadam}@google.com ", + "bbox": [ + 401, + 169, + 812, + 199 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Nicha C. Dvornek1, Sekhar Tatikonda1, James S. Duncan1 {nicha.dvornek, sekhar.tatikonda, james.duncan}@yale.edu ", + "bbox": [ + 183, + 218, + 725, + 248 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Ting Liu2liuti@google.com", + "bbox": [ + 184, + 267, + 343, + 297 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 Yale University, 2 Google Research ", + "bbox": [ + 441, + 284, + 790, + 297 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 334, + 544, + 349 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "The recently proposed Sharpness-Aware Minimization (SAM) improves generalization by minimizing a perturbed loss defined as the maximum loss within a neighborhood in the parameter space. However, we show that both sharp and flat minima can have a low perturbed loss, implying that SAM does not always prefer flat minima. Instead, we define a surrogate gap, a measure equivalent to the dominant eigenvalue of Hessian at a local minimum when the radius of neighborhood (to derive the perturbed loss) is small. The surrogate gap is easy to compute and feasible for direct minimization during training. Based on the above observations, we propose Surrogate Gap Guided Sharpness-Aware Minimization (GSAM), a novel improvement over SAM with negligible computation overhead. Conceptually, GSAM consists of two steps: 1) a gradient descent like SAM to minimize the perturbed loss, and 2) an ascent step in the orthogonal direction (after gradient decomposition) to minimize the surrogate gap and yet not affect the perturbed loss. GSAM seeks a region with both small loss (by step 1) and low sharpness (by step 2), giving rise to a model with high generalization capabilities. Theoretically, we show the convergence of GSAM and provably better generalization than SAM. Empirically, GSAM consistently improves generalization (e.g., $+ 3 . 2 \\%$ over SAM and $+ 5 . 4 \\%$ over AdamW on ImageNet top-1 accuracy for ViT-B/32). Code is released at https://sites.google.com/view/gsam-iclr22/home. ", + "bbox": [ + 233, + 364, + 764, + 628 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 655, + 336, + 670 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Modern neural networks are typically highly over-parameterized and easy to overfit to training data, yet the generalization performances on unseen data (test set) often suffer a gap from the training performance (Zhang et al., 2017a). Many studies try to understand the generalization of machine learning models, including the Bayesian perspective (McAllester, 1999; Neyshabur et al., 2017), the information perspective (Liang et al., 2019), the loss surface geometry perspective (Hochreiter & Schmidhuber, 1995; Jiang et al., 2019) and the kernel perspective (Jacot et al., 2018; Wei et al., 2019). Besides analyzing the properties of a model after training, some works study the influence of training and the optimization process, such as the implicit regularization of stochastic gradient descent (SGD) (Bottou, 2010; Zhou et al., 2020), the learning rate’s regularization effect (Li et al., 2019), and the influence of the batch size (Keskar et al., 2016). ", + "bbox": [ + 174, + 685, + 825, + 824 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "These studies have led to various modifications to the training process to improve generalization. Keskar & Socher (2017) proposed to use Adam in early training phases for fast convergence and then switch to SGD in late phases for better generalization. Izmailov et al. (2018) proposed to average weights to achieve a wider local minimum, which is expected to generalize better than sharp minima. A similar idea was later used in Lookahead (Zhang et al., 2019). Entropy-SGD (Chaudhari et al., 2019) derived the gradient of local entropy to avoid solutions in sharp valleys. Entropy-SGD has a nested Langevin iteration, inducing much higher computation costs than vanilla training. ", + "bbox": [ + 176, + 832, + 823, + 901 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 103, + 823, + 132 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The recently proposed Sharpness-Aware Minimization (SAM) (Foret et al., 2020) is a generic training scheme that improves generalization and has been shown especially effective for Vision Transformers (Dosovitskiy et al., 2020) when large-scale pre-training is unavailable (Chen et al., 2021). Suppose vanilla training minimizes loss $f ( w )$ (e.g., the cross-entropy loss for classification), where $w$ is the parameter. SAM minimizes a perturbed loss defined as $f _ { p } ( w ) \\triangleq \\operatorname* { m a x } _ { | | \\delta | | \\leq \\rho } f ( w + \\delta )$ , which is the maximum loss within radius $\\rho$ centered at the model parameter $w$ . Intuitively, vanilla training seeks a single point with a low loss, while SAM searches for a neighborhood within which the maximum loss is low. However, we show that a low perturbed loss $f _ { p }$ could appear in both flat and sharp minima, implying that only minimizing $f _ { p }$ is not always sharpness-aware. ", + "bbox": [ + 173, + 138, + 825, + 267 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Although the perturbed loss $f _ { p } ( w )$ might disagree with sharpness, we find a surrogate gap defined as $h ( w ) \\triangleq f _ { p } ( w ) - f ( w )$ agrees with sharpness — Lemma 3.3 shows that the surrogate gap $h$ is an equivalent measure of the dominant eigenvalue of Hessian at a local minimum. Inspired by this observation, we propose the Surrogate Gap Guided Sharpness Aware Minimization (GSAM) which jointly minimizes the perturbed loss $f _ { p }$ and the surrogate gap $h$ : a low perturbed loss $f _ { p }$ indicates a low training loss within the neighborhood, and a small surrogate gap $h$ avoids solutions in sharp valleys and hence narrows the generalization gap between training and test performances (Thm. 5.3). When both criteria are satisfied, we find a generalizable model with good performances. ", + "bbox": [ + 173, + 272, + 825, + 387 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "GSAM consists of two steps for each update: 1) descend gradient $\\nabla f _ { p } ( w )$ to minimize the perturbed loss $f _ { p }$ (this step is exactly the same as SAM), and 2) decompose gradient $\\nabla f ( w )$ of the original loss $\\bar { f } ( w )$ into components that are parallel and orthogonal to $\\nabla f _ { p } ( w )$ , i.e., $\\nabla f ( w ) = \\nabla _ { \\| } f ( w ) +$ $\\nabla _ { \\perp } f ( w )$ , and perform an ascent step in $\\nabla _ { \\perp } f ( w )$ to minimize the surrogate gap $h ( w )$ . Note that this ascent step does not change the perturbed loss $f _ { p }$ because $\\nabla f _ { \\perp } ( w ) \\perp \\mathbf { \\bar { \\nabla } } \\nabla f _ { p } ( \\bar { w } )$ by construction. ", + "bbox": [ + 174, + 393, + 825, + 465 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We summarize our contribution as follows: ", + "bbox": [ + 178, + 472, + 455, + 486 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "• We define surrogate gap, which measures the sharpness at local minima and is easy to compute. • We propose the GSAM method to improve the generalization of neural networks. GSAM is widely applicable and incurs negligible computation overhead compared to SAM. • We demonstrate the convergence of GSAM and its provably better generalization than SAM. • We empirically validate GSAM over image classification tasks with various neural architectures, including ResNets (He et al., 2016), Vision Transformers (Dosovitskiy et al., 2020), and MLP-Mixers (Tolstikhin et al., 2021). ", + "bbox": [ + 191, + 497, + 826, + 612 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 PRELIMINARIES ", + "text_level": 1, + "bbox": [ + 174, + 631, + 339, + 647 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.1 NOTATIONS ", + "text_level": 1, + "bbox": [ + 174, + 661, + 297, + 676 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "• $f ( w )$ : A loss function $f$ with parameter $w \\in \\mathbb { R } ^ { k }$ , where $k$ is the parameter dimension. ", + "bbox": [ + 187, + 684, + 767, + 699 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "• $\\rho _ { t } \\in \\mathbb { R }$ : A scalar value controlling the amplitude of perturbation at step $t$ • $\\epsilon \\in \\mathbb { R }$ : A small positive constant (to avoid division by $) , \\epsilon = 1 0 ^ { - 1 2 }$ by default). ", + "bbox": [ + 191, + 703, + 679, + 718 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 191, + 723, + 728, + 738 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "• $\\begin{array} { r } { w _ { t } ^ { a d v } \\triangleq w _ { t } + \\rho _ { t } \\frac { \\nabla f ( w _ { t } ) } { | | \\nabla f ( w _ { t } ) | | + \\epsilon } } \\end{array}$ : The solution to $\\operatorname* { m a x } _ { | | w ^ { \\prime } - w _ { t } | | \\leq \\rho _ { t } } f ( w ^ { \\prime } )$ when $\\rho _ { t }$ is small. ", + "bbox": [ + 186, + 743, + 771, + 763 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "• $\\begin{array} { r } { f _ { p } ( w _ { t } ) \\triangleq \\operatorname* { m a x } _ { | | \\delta | | \\leq \\rho _ { t } } f ( w _ { t } + \\delta ) \\approx f ( w _ { t } ^ { a d v } ) } \\end{array}$ : The perturbed loss induced by $f ( w _ { t } )$ . For each $w _ { t }$ , $f _ { p } ( w _ { t } )$ returns the worst possible loss $f$ within a ball of radius $\\rho _ { t }$ centered at $w _ { t }$ . When $\\rho _ { t }$ is small, by Taylor expansion, the solution to the maximization problem is equivalent to a gradient ascent from $w _ { t }$ to $w _ { t } ^ { a d v }$ . ", + "bbox": [ + 189, + 770, + 826, + 828 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "• $h ( w ) \\triangleq f _ { p } ( w ) - f ( w )$ : The surrogate gap defined as the difference between $f _ { p } ( w )$ and $f ( w )$ ", + "bbox": [ + 186, + 833, + 821, + 849 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "• $\\eta _ { t } \\in \\mathbb { R }$ : Learning rate at step $t$ . ", + "bbox": [ + 191, + 853, + 411, + 868 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "• $\\alpha \\in \\mathbb { R }$ : A constant value that controls the scaled learning rate of the ascent step in GSAM. ", + "bbox": [ + 191, + 872, + 800, + 887 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "$g ^ { ( t ) } , g _ { p } ^ { ( t ) } \\in \\mathbb { R } ^ { k }$ : At the $t$ -th step, the noisy observation of the gradients $\\nabla f ( w _ { t } )$ , $\\nabla f _ { p } ( w _ { t } )$ of the original loss and perturbed loss, respectively. ", + "bbox": [ + 189, + 895, + 825, + 924 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/3396788a9126ef1da5043f16c4c9713656da2cf3b5a9e09a6b9971d5556bacff.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 202, + 95, + 792, + 226 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Figure 1: Consider original loss $f$ (solid line), perturbed loss $\\begin{array} { r } { f _ { p } \\triangleq \\operatorname* { m a x } _ { | | \\delta | | \\leq \\rho } f ( w + \\delta ) } \\end{array}$ (dashed line), and surrogate gap $h ( w ) \\triangleq f _ { p } ( w ) - f ( w )$ . Intuitively, $f _ { p }$ is approximately a max-pooled version of $f$ with a pooling kernel of width $2 \\rho$ , and SAM minimizes $f _ { p }$ . From left to right are the local minima centered at $w _ { 1 } , w _ { 2 } , w _ { 3 }$ , and the valleys become flatter. Since $f _ { p } ( w _ { 1 } ) = f _ { p } ( w _ { 3 } ) < f _ { p } ( w _ { 2 } )$ , SAM prefers $w _ { 1 }$ and $w _ { 3 }$ to $w _ { 2 }$ . However, a low $f _ { p }$ could appear in both sharp $( w _ { 1 } )$ and flat $( w _ { 3 } )$ minima, so $f _ { p }$ might disagree with sharpness. On the contrary, a smaller surrogate gap $h$ indicates a flatter loss surface (Lemma 3.3). From $w _ { 1 }$ to $w _ { 3 }$ , the loss surface is flatter, and $h$ is smaller. ", + "bbox": [ + 173, + 234, + 826, + 338 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "• $\\nabla f ( w _ { t } ) = \\nabla f _ { \\parallel } ( w _ { t } ) + \\nabla f _ { \\perp } ( w _ { t } )$ : Decompose $\\nabla f ( w _ { t } )$ into parallel component $\\nabla f _ { \\parallel } ( \\boldsymbol { w } _ { t } )$ and vertical component $\\nabla f _ { \\perp } ( w _ { t } )$ by projection $\\nabla f ( w _ { t } )$ onto $\\nabla f _ { p } ( w _ { t } )$ . ", + "bbox": [ + 187, + 362, + 823, + 395 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.2 SHARPNESS-AWARE MINIMIZATION ", + "text_level": 1, + "bbox": [ + 176, + 410, + 464, + 426 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Conventional optimization of neural networks typically minimizes the training loss $f ( w )$ by gradient descent w.r.t. $\\nabla f ( w )$ and searches for a single point $w$ with a low loss. However, this vanilla training often falls into a sharp valley of the loss surface, resulting in inferior generalization performance (Chaudhari et al., 2019). Instead of searching for a single point solution, SAM seeks a region with low losses so that small perturbation to the model weights does not cause significant performance degradation. SAM formulates the problem as: ", + "bbox": [ + 173, + 436, + 825, + 521 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/8c47f85f7358270d9da62d6bbd51249f252f6991006f8cf9bf682305602bbccb.jpg", + "text": "$$\n\\begin{array} { r } { \\operatorname* { m i n } _ { w } f _ { p } ( w ) \\mathrm { { w h e r e } } f _ { p } ( w ) \\triangleq \\operatorname* { m a x } _ { | | \\delta | | \\leq \\rho } f ( w + \\delta ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 334, + 529, + 665, + 547 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $\\rho$ is a predefined constant controlling the radius of a neighborhood. This perturbed loss $f _ { p }$ induced by $f ( w )$ is the maximum loss within the neighborhood. When the perturbed loss is minimized, the neighborhood corresponds to low losses (below the perturbed loss). For a small $\\rho$ , using Taylor expansion around $w$ , the inner maximization in Eq. 1 turns into a linear constrained optimization with solution ", + "bbox": [ + 173, + 553, + 825, + 623 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/d68fc447aa136e919b90d96aa2968756b30c0333441d53be974cf52f8d5bd76d.jpg", + "text": "$$\n\\arg \\operatorname* { m a x } _ { | | \\delta | | \\leq \\rho } f ( w + \\delta ) = \\arg \\operatorname* { m a x } _ { | | \\delta | | \\leq \\rho } f ( w ) + \\delta ^ { \\top } \\nabla f ( w ) + O ( \\rho ^ { 2 } ) = \\rho \\frac { \\nabla f ( w ) } { | | \\nabla f ( w ) | | }\n$$", + "text_format": "latex", + "bbox": [ + 214, + 630, + 784, + 664 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "As a result, the optimization problem of SAM reduces to ", + "bbox": [ + 174, + 670, + 547, + 685 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/cdd3efe72c87369ed3b1d091ea0854469a79ec0f5fc254ac61161845cd521bf1.jpg", + "text": "$$\n\\operatorname* { m i n } _ { w } f _ { p } ( w ) \\approx \\operatorname* { m i n } _ { w } f ( w ^ { a d v } ) { \\mathrm { ~ w h e r e ~ } } w ^ { a d v } \\triangleq w + \\rho { \\frac { \\nabla f ( w ) } { | | \\nabla f ( w ) | | + \\epsilon } }\n$$", + "text_format": "latex", + "bbox": [ + 277, + 691, + 723, + 726 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $\\epsilon$ is a scalar (default: 1e-12) to avoid division by 0, and $w ^ { a d v }$ is the “perturbed weight” with the highest loss within the neighborhood. Equivalently, SAM seeks a solution on the surface of the perturbed loss $f _ { p } ( w )$ rather than the original loss $f ( w )$ (Foret et al., 2020). ", + "bbox": [ + 174, + 733, + 825, + 776 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 THE SURROGATE GAP MEASURES THE SHARPNESS AT A LOCAL MINIMUM ", + "text_level": 1, + "bbox": [ + 171, + 796, + 813, + 813 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.1 THE PERTURBED LOSS IS NOT ALWAYS SHARPNESS-AWARE ", + "text_level": 1, + "bbox": [ + 173, + 828, + 622, + 843 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Despite that SAM searches for a region of low losses, we show that a solution by SAM is not guaranteed to be flat. Throughout this paper we measure the sharpness at a local minimum of loss $f ( w )$ by the dominant eigenvalue $\\sigma _ { m a x }$ (eigenvalue with the largest absolute value) of Hessian. For simplicity, we do not consider the influence of reparameterization on the geometry of loss surfaces, which is thoroughly discussed in (Laurent & Massart, 2000; Kwon et al., 2021). ", + "bbox": [ + 174, + 853, + 825, + 924 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "For $t = 1$ to $T$ 0) ρt schedule: ρt = ρmin + (ρmax−ρmin)(lr−lrmin)lr −lr ∇f (t) 1a) ∆wt = ρt ||∇f (t)||+\u000f 1b) $w _ { t } ^ { a d v } = w _ { t } + \\Delta w _ { t }$ 2) Get $\\nabla f _ { p } ^ { ( t ) }$ by back-propagation at $w _ { t } ^ { a d v }$ . 3) $\\begin{array} { r l r } { \\mathrm { ~ } } & { { } } & { = \\nabla f _ { \\parallel } ^ { ( t ) } + \\nabla f _ { \\perp } ^ { ( t ) } } \\end{array}$ Decompose $\\nabla f ^ { ( t ) }$ into compo \nnents that are parallel and orthogonal to $\\nabla f _ { p } ^ { ( t ) }$ . ", + "bbox": [ + 419, + 141, + 826, + 265 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/872b59aec8a7a8aaa3e5f73fed10c4e224fe6b0af656dfe3a11efa43b6098e4f.jpg", + "image_caption": [ + "Figure 2: $\\nabla f$ is decomposed into parallel and vertical $( \\nabla f _ { \\perp } )$ components by projection onto $\\nabla f _ { p }$ . $\\mathsf { \\bar { V } } f ^ { G S A M } = \\mathsf { \\bar { V } } \\bar { f } _ { p } - \\alpha \\nabla f _ { \\perp }$ " + ], + "image_footnote": [], + "bbox": [ + 184, + 113, + 403, + 270 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Algorithm 1 GSAM Algorithm ", + "text_level": 1, + "bbox": [ + 405, + 122, + 616, + 137 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4) Update weights: ", + "bbox": [ + 433, + 265, + 558, + 277 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Vanilla $\\begin{array} { r l } & { w _ { t + 1 } = w _ { t } - \\eta _ { t } } \\\\ & { w _ { t + 1 } = w _ { t } - \\eta _ { t } \\nabla f _ { p } ^ { ( t ) } } \\\\ & { w _ { t + 1 } = w _ { t } - \\eta _ { t } \\big ( \\nabla f _ { p } ^ { ( t ) } - \\alpha \\nabla f _ { \\bot } ^ { ( t ) } \\big ) } \\end{array}$ \nSAM \nGSAM ", + "bbox": [ + 467, + 276, + 761, + 332 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Lemma 3.1. For some fixed $\\rho _ { ; }$ , consider two local minima $w _ { 1 }$ and $w _ { 2 }$ , $f _ { p } ( w _ { 1 } ) \\leq f _ { p } ( w _ { 2 } ) \\neq$ $\\sigma _ { m a x } ( w _ { 1 } ) \\leq \\sigma _ { m a x } ( w _ { 2 } )$ , where $\\sigma _ { m a x }$ is the dominant eigenvalue of the Hessian. ", + "bbox": [ + 173, + 362, + 820, + 392 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We leave the proof to Appendix. Fig. 1 illustrates Lemma 3.1 with an example. Consider three local minima denoted as $w _ { 1 }$ to $w _ { 3 }$ , and suppose the corresponding loss surfaces are flatter from $w _ { 1 }$ to $w _ { 3 }$ . For some fixed $\\rho$ , we plot the perturbed loss $f _ { p }$ and surrogate gap $h \\triangleq f _ { p } - f$ around each solution. Comparing $w _ { 2 }$ with $w _ { 3 }$ : Suppose their vanilla losses are equal, $f ( \\tilde { w _ { 2 } } ) = f ( w _ { 3 } )$ , then $f _ { p } ( w _ { 2 } ) > f _ { p } ( \\bar { w } _ { 3 } )$ because the loss surface is flatter around $w _ { 3 }$ , implying that SAM will prefer $w _ { 3 }$ to $w _ { 2 }$ . Comparing $w _ { 1 }$ and $w _ { 2 }$ : $f _ { p } ( w _ { 1 } ) < f _ { p } ( w _ { 2 } )$ , and SAM will favor $w _ { 1 }$ over $w _ { 2 }$ because it only cares about the perturbed loss $f _ { p }$ , even though the loss surface is sharper around $w _ { 1 }$ than $w _ { 2 }$ . ", + "bbox": [ + 173, + 401, + 825, + 502 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.2 THE SURROGATE GAP AGREES WITH SHARPNESS ", + "text_level": 1, + "bbox": [ + 173, + 517, + 552, + 532 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We introduce the surrogate gap that agrees with sharpness, defined as: ", + "bbox": [ + 171, + 544, + 632, + 559 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/d0dbbd89e8de3b43b04b536ddabb4b177de49bb9d5a399b105fb3fea337bdabb.jpg", + "text": "$$\n\\begin{array} { r } { h ( w ) \\triangleq \\operatorname* { m a x } _ { | | \\delta | | \\leq \\rho } f ( w + \\delta ) - f ( w ) \\approx f ( w ^ { a d v } ) - f ( w ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 308, + 563, + 691, + 583 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Intuitively, the surrogate gap represents the difference between the maximum loss within the neighborhood and the loss at the center point. The surrogate gap has the following properties. ", + "bbox": [ + 174, + 587, + 821, + 616 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Lemma 3.2. Suppose the perturbation amplitude $\\rho$ is sufficiently small, then the approximation to the surrogate gap in Eq. 4 is always non-negative, $h ( w ) \\approx f ( w ^ { a d v } ) - f ( w ) \\geq 0 , \\forall w$ . ", + "bbox": [ + 173, + 618, + 825, + 647 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Lemma 3.3. For a local minimum $w ^ { * }$ , consider the dominate eigenvalue $\\sigma _ { m a x }$ of the Hessian of loss $f$ as a measure of sharpness. Considering the neighborhood centered at $w ^ { * }$ with a small radius $\\rho$ , the surrogate gap $h ( w ^ { * } )$ is an equivalent measure of the sharpness: $\\sigma _ { m a x } \\approx 2 h ( w ^ { * } ) / \\rho ^ { 2 }$ . ", + "bbox": [ + 174, + 650, + 823, + 693 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The proof is in Appendix. Lemma 3.2 tells that the surrogate gap is non-negative, and Lemma 3.3 shows that the loss surface is flatter as $h$ gets closer to 0. The two lemmas together indicate that we can find a region with a flat loss surface by minimizing the surrogate gap $h ( w )$ . ", + "bbox": [ + 174, + 703, + 825, + 746 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4 SURROGATE GAP GUIDED SHARPNESS-AWARE MINIMIZATION ", + "text_level": 1, + "bbox": [ + 174, + 765, + 728, + 782 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4.1 GENERAL IDEA: SIMULTANEOUSLY MINIMIZE THE PERTURBED LOSS AND SURROGATE GAP ", + "bbox": [ + 174, + 796, + 812, + 824 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Inspired by the analysis in Section 3, we propose Surrogate Gap Guided Sharpness-Aware Minimzation (GSAM) to simultaneously minimize two objectives, the perturbed loss $f _ { p }$ and the surrogate gap $h$ : ", + "bbox": [ + 173, + 835, + 823, + 877 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/9792aae0717fc879fbc94f0fb316680ebdfdaa6f830266912be569c38ffd7482.jpg", + "text": "$$\n\\operatorname* { m i n } _ { w } \\left( f _ { p } ( w ) , h ( w ) \\right)\n$$", + "text_format": "latex", + "bbox": [ + 429, + 876, + 570, + 895 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Intuitively, by minimizng $f _ { p }$ we search for a region with a low perturbed loss similar to SAM, and by minimizing $h$ we search for a local minimum with a flat surface. A low perturbed loss implies low training losses within the neighborhood, and a flat loss surface reduces the generalization gap between training and test performances (Chaudhari et al., 2019). When both are minimized, the solution gives rise to high accuracy and good generalization. ", + "bbox": [ + 171, + 895, + 828, + 924 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 823, + 146 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Potential caveat in optimization It is tempting and yet sub-optimal to combine the objectives in Eq. 5 to arrive at $\\mathrm { m i n } _ { w }$ $f _ { p } ( w ) + \\lambda h ( w )$ , where $\\lambda$ is some positive scalar. One caveat when solving this weighted combination is the potential conflict between the gradients of the two terms, i.e., $\\nabla f _ { p } ( w )$ and $\\overline { { \\nabla } h } ( \\boldsymbol { w } )$ . We illustrate this conflict by Fig. 2, where $\\bar { \\nabla } h ( w ) = \\nabla f _ { p } ( w ) - \\nabla f ( w )$ (the grey dashed arrow) has a negative inner product with $\\nabla f _ { p } ( w )$ and $\\nabla f ( w )$ . Hence, the gradient descent for the surrogate gap could potentially increase the loss $f _ { p }$ , harming the model’s performance. We empirically validate this argument in Sec. 6.4. ", + "bbox": [ + 173, + 152, + 825, + 251 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4.2 GRADIENT DECOMPOSITION AND ASCENT FOR THE MULTI-OBJECTIVE OPTIMIZATION ", + "text_level": 1, + "bbox": [ + 174, + 280, + 805, + 295 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Our primary goal is to minimize $f _ { p }$ because otherwise a flat solution of high loss is meaningless, and the minimization of $h$ should not increase $f _ { p }$ . We propose to decompose $\\nabla f ( w _ { t } )$ and $\\nabla h$ into components that are parallel and orthogonal to $\\dot { \\nabla } f _ { p } ( \\boldsymbol { w } _ { t } )$ , respectively (see Fig. 2): ", + "bbox": [ + 174, + 310, + 825, + 354 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/bfe642ad9dc93e7e579b1ee21ac9a7dda338ecd01d5984fbaa9ec0612a2953b5.jpg", + "text": "$$\n\\begin{array} { r l } & { \\nabla f ( w _ { t } ) = \\nabla f _ { \\parallel } ( w _ { t } ) + \\nabla f _ { \\perp } ( w _ { t } ) } \\\\ & { \\nabla h ( w _ { t } ) = \\nabla h _ { \\parallel } ( w _ { t } ) + \\nabla h _ { \\perp } ( w _ { t } ) } \\\\ & { \\nabla h _ { \\perp } ( w _ { t } ) = - \\nabla f _ { \\perp } ( w _ { t } ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 382, + 369, + 616, + 430 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The key is that updating in the direction of $\\nabla h _ { \\perp } ( w _ { t } )$ does not change the value of the perturbed loss $f _ { p } ( w _ { t } )$ because $\\nabla h _ { \\perp } \\perp \\nabla f _ { p }$ by construction. Therefore, we propose to perform a descent step in the $\\nabla h _ { \\perp } ( w _ { t } )$ direction, which is equivalent to an ascent step in the $\\nabla f _ { \\perp } ( w _ { t } )$ direction (because $\\nabla h _ { \\perp } = - \\nabla f _ { \\perp }$ by the definition of $h$ ), and achieve two goals simultaneously — it keeps the value of $f _ { p } ( w _ { t } )$ intact and meanwhile decreases the surrogate gap $h ( w _ { t } ) = f _ { p } ( w _ { t } ) - f ( w _ { t } )$ (by increasing $f ( w _ { t } )$ and not affect $f _ { p } ( w _ { t } ) )$ ). ", + "bbox": [ + 173, + 444, + 825, + 530 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The full GSAM Algorithm is shown in Algo. 1 and Fig. 2, where $g ^ { ( t ) } , g _ { p } ^ { ( t ) }$ are noisy observations of $\\nabla f ( w _ { t } )$ and $\\nabla f _ { p } ( w _ { t } )$ , respectively, and $g _ { \\parallel } ^ { ( t ) } , g _ { \\perp } ^ { ( t ) }$ are noisy observations of $\\nabla f _ { \\parallel } ( w _ { t } )$ and $\\nabla f _ { \\perp } ( w _ { t } )$ , respectively, by projecting $g ^ { ( t ) }$ onto $g _ { p } ^ { ( t ) }$ . We introduce a constant $\\alpha$ to scale the stepsize of the ascent step. Steps 1) to 2) are the same as SAM: At current point $w _ { t }$ , step 1) takes a gradient ascent to $w _ { t } ^ { a d v }$ followed by step 2) evaluating the gradient $g _ { p } ^ { ( t ) }$ at $w _ { t } ^ { a d v }$ . Step 3) projects $g ^ { ( t ) }$ onto $g _ { p } ^ { ( t ) }$ , which requires negligible computation compared to the forward and backward passes. In step 4), $- \\eta _ { t } g _ { p } ^ { ( t ) }$ is the same as in SAM and minimizes the perturbed loss $f _ { p } ( w _ { t } )$ with gradient descent, and αηtg(t)⊥ performs an ascent step in the orthogonal direction of $g _ { p } ^ { ( t ) }$ to minimize the surrogate gap $h ( w _ { t } )$ ( equivalently increase $f ( w _ { t } )$ and keep $f _ { p } ( w _ { t } )$ intact). In coding, GSAM feeds the “surrogate gradient” ∇fGSAMt , g $\\nabla f _ { t } ^ { G S A M } \\triangleq g _ { p } ^ { ( t ) } - \\alpha g _ { \\perp } ^ { ( t ) }$ to first-order gradient optimizers such as SGD and Adam. ", + "bbox": [ + 173, + 536, + 826, + 705 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The ascent step along $g _ { \\perp } ^ { ( t ) }$ does not harm convergence SAM demonstrates that minimizing $f _ { p }$ makes the network generalize better than minimizing $f$ . Even though our ascent step along $g _ { \\perp } ^ { ( t ) }$ increases $f ( w )$ , it does not affect $f _ { p } ( w )$ , so GSAM still decreases the perturbed loss $f _ { p }$ in a way similar to SAM. In Thm. 5.1, we formally prove the convergence of GSAM. In Sec. 6 and Appendix C, we empirically validate that the loss decreases and accuracy increases with training. ", + "bbox": [ + 174, + 713, + 825, + 790 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Illustration with a toy example We demonstrate different algorithms by a numerical toy example shown in Fig. 3. The trajectory of GSAM is closer to the ridge and tends to find a flat minimum. Intuitively, since the loss surface is smoother along the ridge than in sharp local minima, the surrogate gap $h ( w )$ is small near the ridge, and the ascent step in GSAM minimizes $h$ to pushes the trajectory closer to the ridge. More concretely, $\\nabla f ( w _ { t } )$ points to a sharp local solution and deviates from the ridge; in contrast, $w _ { t } ^ { a d v }$ is closer to the ridge and $\\nabla f ( w _ { t } ^ { a d v } )$ is closer to the ridge descent direction than $\\nabla f ( w _ { t } )$ . Note that $\\nabla f _ { t } ^ { G S A M }$ and $\\nabla f ( w _ { t } )$ always lie at different sides of $\\nabla f _ { p } ( w _ { t } )$ by construction (see Fig. 2), hence $\\nabla f _ { t } ^ { G S A M }$ pushes the trajectory closer to the ridge than $\\nabla f _ { p } ( w _ { t } )$ does. The trajectory of GSAM is like descent along the ridge and tends to find flat minima. ", + "bbox": [ + 173, + 796, + 825, + 924 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/64c1996c768babc0d124bb80d0862cf1e6d992f4b1b70cc711ed289334144e3d.jpg", + "image_caption": [ + "Figure 3: Consider the loss surface with a few sharp local minima. Left: Overview of the procedures of SGD, SAM and GSAM. SGD takes a descent step at $w _ { t }$ using $\\nabla f ( w _ { t } )$ (orange), which points to a sharp local minima. SAM first performs gradient ascent in the direction of $\\nabla f ( w _ { t } )$ to reach $w _ { t } ^ { a d v }$ with a higher loss, followed by descent with gradient $\\nabla f ( w _ { t } ^ { a d v } )$ (green) at the perturbed weight. Based on $\\nabla f ( w _ { t } )$ and $\\nabla f ( w _ { t } ^ { a d v } )$ , GSAM updates in a new direction (red) that points to a flatter region. Right: Trajectories by different methods. SGD and SAM fall into different sharp local minima, while GSAM reaches a flat region. A video is in the supplement for better visualization. " + ], + "image_footnote": [], + "bbox": [ + 174, + 102, + 821, + 241 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5 THEORETICAL PROPERTIES OF GSAM", + "text_level": 1, + "bbox": [ + 174, + 375, + 524, + 391 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5.1 CONVERGENCE DURING TRAINING ", + "text_level": 1, + "bbox": [ + 176, + 405, + 455, + 420 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Theorem 5.1. Consider a non-convex function $f ( w )$ with Lipschitz-smooth constant $L$ and lower bound $f _ { m i n }$ . Suppose we can access a noisy, bounded observation $g ^ { ( t ) } \\ ( \\vert \\vert g ^ { ( t ) } \\vert \\vert _ { 2 } \\leq G , \\forall t )$ of the true gradient $\\nabla f ( w _ { t } )$ at the $t$ -th step. For some constant $\\alpha$ , with learning rate $\\eta _ { t } = \\eta _ { 0 } / \\sqrt { t }$ , and perturbation amplitude $\\rho _ { t }$ proportional to the learning rate, e.g., $\\rho _ { t } = \\rho _ { 0 } / \\sqrt { t }$ , we have ", + "bbox": [ + 173, + 430, + 826, + 494 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/e60ff970a4b27ce0576e009a2164943517791f8b464b999f6ecdd471230e8f1a.jpg", + "text": "$$\n\\frac { 1 } { T } \\sum _ { t = 1 } ^ { T } \\mathbb { E } \\Big | \\Big | \\nabla f _ { p } ( w _ { t } ) \\Big | \\Big | _ { 2 } ^ { 2 } \\leq \\frac { C _ { 1 } + C _ { 2 } \\log T } { \\sqrt { T } } , \\quad \\frac { 1 } { T } \\sum _ { t = 1 } ^ { T } \\mathbb { E } \\Big | \\Big | \\nabla f ( w _ { t } ) \\Big | \\Big | _ { 2 } ^ { 2 } \\leq \\frac { C _ { 3 } + C _ { 4 } \\log T } { \\sqrt { T } }\n$$", + "text_format": "latex", + "bbox": [ + 222, + 500, + 777, + 542 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "where $C _ { 1 } , C _ { 2 } , C _ { 3 } , C _ { 4 }$ are some constants. ", + "bbox": [ + 174, + 546, + 449, + 563 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Thm. 5.1 implies both $f _ { p }$ and $f$ converge in GSAM at rate $O ( \\log T / \\sqrt { T } )$ for non-convex stochastic optimization, matching the convergence rate of first-order gradient optimizers like Adam. ", + "bbox": [ + 173, + 573, + 825, + 603 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5.2 GENERALIZATION OF GSAM ", + "text_level": 1, + "bbox": [ + 176, + 619, + 418, + 633 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In this section, we show the surrogate gap in GSAM is provably lower than SAM’s, so GSAM is expected to find a smoother minimum with better generalization. ", + "bbox": [ + 176, + 645, + 823, + 674 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Theorem 5.2 (PAC-Bayesian Theorem (McAllester, 2003)). Suppose the training set has m elements drawn i.i.d. from the true distribution, and denote the loss on the training set as ${ \\widehat { f } } ( w ) =$ Pmi=1 f (w, xi), where we use xi to denote the (input, target) pair of the i-th element. Let w be learned from the training set. Suppose $w$ is drawn from posterior distribution $\\mathcal { Q } .$ Denote the prior distribution (independent of training) as $\\mathcal { P }$ , then ", + "bbox": [ + 173, + 676, + 825, + 751 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/40e4313f846d41c34d8f7d3b9fbf975b19216b7aae4dd612155f8d60d593487b.jpg", + "text": "$$\n\\mathbb { E } _ { w \\sim Q } \\mathbb { E } _ { x } f ( w , x ) \\le \\mathbb { E } _ { w \\sim Q } \\widehat { f } ( w ) + 4 \\sqrt { \\Big ( K L ( Q | | \\mathcal { P } ) + \\log \\frac { 2 m } { a } \\Big ) / m } w i t h \\ p r o b a b i l i t y \\ a t \\ l e a s t 1 - a\n$$", + "text_format": "latex", + "bbox": [ + 181, + 756, + 836, + 790 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Corollary 5.2.1. Suppose perturbation $\\delta$ is drawn from distribution $\\delta \\sim \\mathcal { N } ( 0 , b ^ { 2 } I ^ { k } ) , \\delta \\in \\mathbb { R } ^ { k }$ , $k$ is the dimension of $w$ , then with probability at least $\\left( 1 - a \\right) \\left[ 1 - e ^ { - \\left( \\frac { \\rho } { \\sqrt { 2 } b } - \\sqrt { k } \\right) ^ { 2 } } \\right]$ ", + "bbox": [ + 173, + 796, + 825, + 840 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/989ae25385843a025e7352d261ebb78d803542672e08d784cb8edd340dadf643.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathbb { E } _ { w \\sim Q } \\mathbb { E } _ { x } f ( w , x ) \\le \\widehat { h } + C + 4 \\sqrt { \\Big ( K L ( Q | \\mathcal { P } ) + \\log \\frac { 2 m } { a } \\Big ) / m } } \\\\ & { \\widehat { h } \\triangleq \\operatorname* { m a x } _ { | | \\delta | | _ { 2 } \\le \\rho } \\widehat { f } ( w + \\delta ) - \\widehat { f } ( w ) = \\displaystyle \\frac { 1 } { m } \\sum _ { i = 1 } ^ { m } \\Big [ \\operatorname* { m a x } _ { | | \\delta | | _ { 2 } \\le \\rho } f ( w + \\delta , x _ { i } ) - f ( w , x _ { i } ) \\Big ] } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 218, + 845, + 777, + 924 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "where $C = { \\widehat { f } } ( w )$ is the empirical training loss, and $\\widehat { h }$ is the surrogate gap evaluated on the training set. ", + "bbox": [ + 173, + 102, + 823, + 132 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Corollary 5.2.1 implies that minimizing $\\widehat { h }$ (right hand side of Eq. 7) is expected to achieve a tighter upper bound of the generalization performance (left hand side of Eq. 7). The third term on the right of Eq. 7 is typically hard to analyze and often simplified to $L 2$ regularization (Foret et al., 2020). Note that $f _ { p } = C + \\widehat { h }$ only holds when $\\rho _ { t r a i n }$ (the perturbation amplitude specified by users during training) equals $\\rho _ { t r u e }$ (the ground truth value determined by underlying data distribution); when $\\rho _ { t r a i n } \\neq \\rho _ { t r u e }$ , $m i n ( f _ { p } , \\widehat { h } )$ is more effective than $m i n ( f _ { p } )$ in terms of minimizing generalization loss. A detailed discussion is in Appendix A.7. ", + "bbox": [ + 173, + 141, + 825, + 247 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Theorem 5.3 (Unlike SAM, GSAM decreases the surrogate gap). Under the assumption in Thm. 5.1, Thm. 5.2 and Corollary 5.2.1, we assume the Hessian has a lower-bound $| \\sigma | _ { m i n }$ on the absolute value of eigenvalue, and the variance of noisy observation $g ^ { ( t ) }$ is lower-bounded by $c ^ { 2 }$ . The surrogate gap h can be minimized by the ascent step along the orthogonal direction $g _ { \\perp } ^ { ( t ) }$ . During training we minimize the sample estimate of $h$ . We use $\\Delta \\widehat { h } _ { t }$ to denote the amount that the ascent step in GSAM decreases $\\widehat { h }$ for the $t$ -th step. Compared to SAM, the proposed method generates a total decrease in surrogate gap $\\Sigma _ { t = 1 } ^ { T } \\Delta \\widehat { h } _ { t }$ , which is bounded by ", + "bbox": [ + 173, + 250, + 825, + 363 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/3f670afed15392088428627fda19523afa663ca466b07b6112b86954f1322bf2.jpg", + "text": "$$\n\\frac { \\alpha c ^ { 2 } \\rho _ { 0 } ^ { 2 } \\eta _ { 0 } | \\sigma | _ { m i n } ^ { 2 } } { G ^ { 2 } } \\le \\operatorname* { l i m } _ { T \\to \\infty } \\sum _ { t = 1 } ^ { T } \\Delta \\widehat { h } _ { t } \\le 2 . 7 \\alpha L ^ { 2 } \\eta _ { 0 } \\rho _ { 0 } ^ { 2 }\n$$", + "text_format": "latex", + "bbox": [ + 336, + 367, + 661, + 411 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We provide proof in the appendix. The lower-bound of provably non-trivial decrease in the surrogate gap. Com $\\Sigma _ { t = 1 } ^ { T } \\Delta \\widehat { h } _ { t }$ indicates that GSAM achieves aCorollary 5.2.1, GSAM provably improves the generalization performance over SAM. ", + "bbox": [ + 173, + 424, + 825, + 469 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "6 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 489, + 326, + 506 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "6.1 GSAM IMPROVES TEST PERFORMANCE ON VARIOUS MODEL ARCHITECTURES ", + "text_level": 1, + "bbox": [ + 173, + 515, + 753, + 529 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We conduct experiments with ResNets (He et al., 2016), Vision Transformers (ViTs) (Dosovitskiy et al., 2020) and MLP-Mixers (Tolstikhin et al., 2021). Following the settings by Chen et al. (2021), we train on the ImageNet-1k (Deng et al., 2009) training set using the Inception-style (Szegedy et al., 2015) pre-processing without extra training data or strong augmentation. For all models, we search for the best learning rate and weight decay for vanilla training, and then use the same values for the experiments with SAM and GSAM. For ResNets, we search for $\\rho$ from 0.01 to 0.05 with a stepsize 0.01. For ViTs and Mixers, we search for $\\rho$ from 0.05 to 0.6 with a stepsize 0.05. In GSAM, we search for $\\alpha$ in $\\{ 0 . 0 1 , 0 . 0 2 , 0 . 0 3 \\}$ for ResNets and $\\alpha$ in $\\{ 0 . 1 , 0 . 2 , 0 . 3 \\}$ for ViTs and Mixers. Considering that each step in SAM and GSAM requires twice the computation of vanilla training, we experiment with the vanilla training for twice the epochs of SAM and GSAM, but we observe no significant improvements from the longer training (Table 5 in appendix). We summarize the best hyper-parameters for each model in Appendix B. ", + "bbox": [ + 173, + 539, + 825, + 707 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We report the performances on ImageNet (Deng et al., 2009), ImageNet-v2 (Recht et al., 2019) and ImageNet-Real (Beyer et al., 2020) in Table 1. GSAM consistently improves over SAM and vanilla training (with SGD or AdamW): on ViT-B/32, GSAM achieves $+ 5 . 4 \\%$ improvement over AdamW and $+ 3 . 2 \\%$ over SAM in top-1 accuracy; on Mixer-B/32, GSAM achieves $+ 1 1 . 1 \\%$ over AdamW and $+ 1 . 2 \\%$ over SAM. We ignore the standard deviation since it is typically negligible $( < 0 . 1 \\% )$ compared to the improvements. We also test the generalization performance on out-of-distribution data (ImageNet-R and ImageNet-C), and the observation is consistent with that on ImageNet, e.g., $+ 5 . 1 \\%$ on ImageNet-R and $+ 5 . 9 \\%$ on ImageNet-C for Mixer-B/32. ", + "bbox": [ + 174, + 713, + 825, + 825 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "6.2 GSAM FINDS A MINIMUM WHOSE HESSIAN HAS SMALL DOMINANT EIGENVALUES ", + "text_level": 1, + "bbox": [ + 174, + 842, + 787, + 856 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Lemma 3.3 indicates that the surrogate gap $h$ is an equivalent measure of the dominant eigenvalue of the Hessian, and minimizing $h$ equivalently searches for a flat minimum. We empirically validate this in Fig. 4. As shown in the left subfigure, for some fixed $\\rho$ , increasing $\\alpha$ decreases the dominant value and improves generalization (test accuracy). In the middle subfigure, we plot the dominant eigenvalues estimated by the surrogate gap, $\\sigma _ { m a x } \\approx 2 h / \\rho ^ { 2 }$ (Lemma 3.3). In the right subfigure, we directly calculate the dominant eigenvalues using the power-iteration (Mises & Pollaczek-Geiringer, 1929). The estimated dominant eigenvalues (middle) match the real eigenvalues $\\sigma _ { m a x }$ (right) in terms of the trend that $\\sigma _ { m a x }$ decreases with $\\alpha$ and $\\rho$ . Note that the surrogate gap $h$ is derived over the whole training set, while the measured eigenvalues are over a subset to save computation. These results show that the ascent step in GSAM minimizes the dominant eigenvalue by minimizing the surrogate loss, validating $\\mathrm { T h m } 5 . 3$ . ", + "bbox": [ + 174, + 867, + 823, + 924 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/5c4f69bdaf0137628bc66cbc0edd7bc00a340cab255c733de095876fcfdf42f6.jpg", + "table_caption": [ + "Table 1: Top-1 Accuracy $( \\% )$ on ImageNet datasets for ResNets, ViTs and MLP-Mixers trained with Vanilla SGD or AdamW, SAM, and GSAM optimizers. " + ], + "table_footnote": [], + "table_body": "
ModelTrainingImageNet-v1ImageNet-RealImageNet-V2ImageNet-RImageNet-C
ResNet
ResNet50Vanilla (SGD)76.082.463.622.244.6
SAM76.983.364.423.846.5
GSAM77.283.964.623.647.6
ResNet101Vanilla (SGD)77.883.965.324.448.5
SAM78.684.866.725.951.3
GSAM78.985.267.326.351.8
ResNet152Vanilla (SGD)78.584.266.325.350.0
SAM79.384.967.325.752.2
GSAM80.085.968.627.354.1
ViT-S/32Vanilla (AdamW)68.4Vision Transformer 75.254.3
SAM70.577.556.919.0 21.443.3
GSAM73.880.460.422.546.2 48.2
ViT-S/16Vanilla (AdamW)74.480.461.720.046.5
SAM78.184.165.624.753.0
GSAM79.585.367.325.353.3
ViT-B/32Vanilla (AdamW)71.477.557.523.444.0
SAM73.680.360.024.050.7
GSAM76.882.763.025.151.7
ViT-B/16Vanilla (AdamW)74.679.861.320.146.6
SAM79.985.267.526.456.5
GSAM81.086.569.227.155.7
Mixer-S/32Vanilla (AdamW)63.9MLP-Mixer 70.349.516.935.2
SAM66.773.852.418.639.3
GSAM68.675.855.022.644.6
Mixer-S/16Vanilla (AdamW)68.875.154.815.935.6
SAM72.979.858.920.142.0
GSAM75.081.761.923.748.5
Mixer-S/8Vanilla (AdamW)70.276.256.115.434.6
SAM75.982.562.320.542.4
GSAM76.883.464.024.647.8
Mixer-B/32Vanilla (AdamW)62.568.147.614.633.8
SAM72.479.058.022.846.2
GSAM73.680.259.927.952.1
Mixer-B/16Vanilla (AdamW)66.472.150.814.533.8
SAM77.483.563.924.748.8
GSAM77.884.064.928.354.4
", + "bbox": [ + 181, + 156, + 815, + 627 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/0ab9a693376a9164b50af0714b51b883842522baa5f9217592cd485202ddf052.jpg", + "image_caption": [ + "Figure 4: Influence of $\\rho$ (set as constant for ease of comparison, other experiments use decayed $\\rho _ { t }$ schedule) and $\\alpha$ on the training of ViT-B/32. Left: Top-1 accuracy on ImageNet. Middle: Estimation of the dominant eigenvalues from the surrogate gap, $\\sigma _ { m a x } \\approx 2 h / \\rho ^ { 2 }$ . Right: Dominant eigenvalues of the Hessian calculated via the power iteration. Middle and right figures match in the trend of curves, validating that the surrogate gap can be viewed as a proxy of the dominant eigenvalue of Hessian. " + ], + "image_footnote": [], + "bbox": [ + 176, + 676, + 820, + 803 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/cd91a39c4fda1aa38347bb16bcf091af8803ccb572f7dae609b6c2bda9d3fc93.jpg", + "image_caption": [ + "Figure 5: Top-1 accuracy of Mixer-S/32 trained with different methods. “+ascent” represents applying the ascent step in Algo. 1 to an optimizer. Note that our GSAM is described as SAM+ascent $\\mathbf { \\Lambda } = \\mathbf { G S A M } )$ for consistency. " + ], + "image_footnote": [], + "bbox": [ + 176, + 102, + 823, + 212 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/c064c031e0083de17096b8b38cc645290dd2916ffc029f3e63637a25e3288675.jpg", + "table_caption": [ + "Table 2: Results $( \\% )$ of GSAM and $\\operatorname* { m i n } ( f _ { p } + \\lambda h )$ on ViT-B/32 " + ], + "table_footnote": [], + "table_body": "
Datasetmin(fp+Xh)GSAM
ImageNet75.476.8
ImageNet-Real81.182.7
ImageNet-v260.963.0
ImageNet-R23.925.1
", + "bbox": [ + 176, + 323, + 434, + 387 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/7f251f330cbeb8ac646a9d0a83e0ad1d7d98ac8fc1a340c6106add1f96c7c19f.jpg", + "table_caption": [ + "Table 3: Transfer learning results (top-1 accuracy, $\\%$ ) " + ], + "table_footnote": [], + "table_body": "
ViT-B/16ViT-S/16
VanillaSAMGSAMVanillaSAMGSAM
Cifar1098.198.698.897.698.298.4
Cifar10087.689.189.785.787.688.1
Flowers88.591.891.286.491.590.3
Pets91.993.194.490.492.993.5
mean91.593.293.590.092.692.6
", + "bbox": [ + 475, + 303, + 839, + 386 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 411, + 825, + 510 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "6.3 COMPARISON WITH METHODS IN THE LITERATURE ", + "text_level": 1, + "bbox": [ + 174, + 527, + 566, + 541 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Section 6.1 compares GSAM to SAM and vanilla training. In this subsection, we further compare GSAM against Entropy-SGD (Chaudhari et al., 2019) and Adaptive-SAM (ASAM) (Kwon et al., 2021), which are designed to improve generalization. Note that Entropy-SGD uses SGD in the inner Langevin iteration and can be combined with other base optimizers such as AdamW as the outer loop. For Entropy-SGD, we find the hyper-parameter “scope” from 0.0 and 0.9, and search for the inner-loop iteration number between 1 and 14. For ASAM, we search for $\\rho$ between 1 and 7 $( 1 0 \\times$ larger than in SAM) as recommended by the ASAM authors. Note that the only difference between ASAM and SAM is the derivation of the perturbation, so both can be combined with the proposed ascent step. As shown in Fig. 5, the proposed ascent step increases test accuracy when combined with both SAM and ASAM and outperforms Entropy-SGD and vanilla training. ", + "bbox": [ + 173, + 553, + 825, + 691 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "6.4 ADDITIONAL STUDIES ", + "text_level": 1, + "bbox": [ + 176, + 703, + 369, + 717 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "GSAM outperforms a weighted combination of the perturbed loss and surrogate gap With an example in Fig. 2, we demonstrate that directly minimizing $f _ { p } ( w ) + \\lambda h ( w )$ as discussed in Sec. 4.1 is sub-optimal because $\\nabla h ( w )$ could conflict with $\\nabla f _ { p } ( w )$ and $\\nabla f ( w )$ . We empirically validate this argument on ViT-B/32. We search for $\\lambda$ between 0.0 and 0.5 with a step 0.1 and search for $\\rho$ in the same grid as SAM and GSAM. We report the best accuracy of each method. Top-1 accuracy in Table 2 show the superior performance of GSAM, validating our analysis. ", + "bbox": [ + 173, + 728, + 825, + 811 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "$\\operatorname* { m i n } ( f _ { p } , h )$ vs. $\\operatorname* { m i n } ( f , h )$ GSAM solves $\\operatorname* { m i n } ( f _ { p } , h )$ by descent in $\\nabla f _ { p }$ , decomposing $\\nabla f$ onto $\\nabla f _ { p }$ , and an ascent step in the orthogonal direction to increase $f$ while keep $f _ { p }$ intact. Alternatively, we can also optimize $\\operatorname* { m i n } ( f , h )$ by descent in $\\nabla f$ , decomposing $\\nabla f _ { p }$ onto $\\nabla f$ , and a descent step in the orthogonal direction to decrease $f _ { p }$ while keep $f$ intact. The two GSAM variations perform similarly (see Fig. 6, right). We choose $\\operatorname* { m i n } ( f _ { p } , h )$ mainly to make the minimal change to SAM. ", + "bbox": [ + 174, + 818, + 825, + 888 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "GSAM benefits transfer learning Using weights trained on ImageNet-1k, we finetune models with SGD on downstream tasks including the CIFAR10/CIFAR100 (Krizhevsky et al., 2009), Oxfordflowers (Nilsback & Zisserman, 2008) and Oxford-IITPets (Parkhi et al., 2012). Results in Table 3 shows that GSAM leads to better transfer performance than vanilla training and SAM. ", + "bbox": [ + 173, + 895, + 823, + 924 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/819890a555e7fa3dd273a452709db6a6112e787f55b5798c88bf87f762261419.jpg", + "image_caption": [ + "Figure 6: Top-1 accuracy of ViT-B/32 for the additional studies (Section 6.4). Left: from left to right are performances under different data augmentations (details in Appendix B.3) , where the vanilla method is trained for $2 \\times$ the epochs. Middle: performance with different base optimizers. Right: Comparison between $\\operatorname* { m i n } ( f _ { p } , h )$ and $\\operatorname* { m i n } ( f , h )$ . " + ], + "image_footnote": [], + "bbox": [ + 176, + 103, + 820, + 222 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "", + "bbox": [ + 176, + 296, + 821, + 324 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "GSAM remains effective under various data augmentations We plot the top-1 accuracy of a ViT-B/32 model under various Mixup (Zhang et al., 2017b) augmentations in Fig. 6 (left subfigure). Under different augmentations, GSAM consistently outperforms SAM and vanilla training. ", + "bbox": [ + 174, + 332, + 825, + 373 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "GSAM is compatible with different base optimizers GSAM is generic and applicable to various base optimizers. We compare vanilla training, SAM and GSAM using AdamW (Loshchilov & Hutter, 2017) and AdaBelief (Zhuang et al., 2020) with default hyper-parameters. Fig. 6 (middle subfigure) shows that GSAM performs the best, and SAM improves over vanilla training. ", + "bbox": [ + 174, + 380, + 825, + 436 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "7 CONCLUSION ", + "text_level": 1, + "bbox": [ + 176, + 457, + 318, + 473 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "We propose the surrogate gap as an equivalent measure of sharpness which is easy to compute and feasible to optimize. We propose the GSAM method, which improves the generalization over SAM at negligible computation cost. We show the convergence and provably better generalization of GSAM compared to SAM, and validate the superior performance of GSAM on various models. ", + "bbox": [ + 176, + 482, + 825, + 537 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "ACKNOWLEDGEMENT ", + "text_level": 1, + "bbox": [ + 176, + 560, + 357, + 574 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "We would like to thank Xiangning Chen (UCLA) and Hossein Mobahi (Google) for discussions, Yi Tay (Google) for help with datasets, and Yeqing Li, Xianzhi Du, and Shawn Wang (Google) for help with TensorFlow implementation. ", + "bbox": [ + 174, + 590, + 823, + 632 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "ETHICS STATEMENT ", + "text_level": 1, + "bbox": [ + 176, + 654, + 343, + 670 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "This paper focuses on the development of optimization methodologies and can be applied to the training of different deep neural networks for a wide range of applications. Therefore, the ethical impact of our work would primarily be determined by the specific models that are trained using our new optimization strategy. ", + "bbox": [ + 174, + 685, + 825, + 742 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "REPRODUCIBILITY STATEMENT ", + "text_level": 1, + "bbox": [ + 176, + 762, + 436, + 779 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "We provide the detailed proof of theoretical results in Appendix A and provide the data preprocessing and hyper-parameter settings in Appendix B. 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", + "bbox": [ + 174, + 756, + 825, + 799 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A PROOFS ", + "text_level": 1, + "bbox": [ + 176, + 102, + 277, + 118 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "A.1 PROOF OF LEMMA. 3.1 ", + "bbox": [ + 176, + 135, + 379, + 150 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Suppose $\\rho$ is small, perform Taylor expansion around the local minima $w$ , we have: ", + "bbox": [ + 173, + 162, + 720, + 179 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/28b35d584b6083c1a1ea5cc348276b221e3c2af029722fc4d947dbba9784eac6.jpg", + "text": "$$\n\\boldsymbol { f } ( \\boldsymbol { w } + \\delta ) = \\boldsymbol { f } ( \\boldsymbol { w } ) + \\nabla \\boldsymbol { f } ( \\boldsymbol { w } ) ^ { \\top } \\delta + \\frac { 1 } { 2 } \\delta ^ { \\top } \\boldsymbol { H } \\delta + O ( | | \\delta | | ^ { 3 } )\n$$", + "text_format": "latex", + "bbox": [ + 318, + 186, + 679, + 218 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "where $H$ is the Hessian, and is positive semidefinite at a local minima. At a local minima, $\\nabla f ( w ) =$ 0, hence we have ", + "bbox": [ + 171, + 227, + 823, + 256 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/4921ed1eb3ff577736addf39aee8a7852e63341de6c1327a62673cd80a691852.jpg", + "text": "$$\nf ( w + \\delta ) = f ( w ) + \\frac { 1 } { 2 } \\delta ^ { \\top } H \\delta + O ( | | \\delta | | ^ { 3 } )\n$$", + "text_format": "latex", + "bbox": [ + 361, + 257, + 637, + 287 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "and ", + "bbox": [ + 173, + 295, + 200, + 309 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/7207d7734e91bf561bbebf49dcaaf4695bce0a892fe8c783aa3d10e28b3115bf.jpg", + "text": "$$\nf _ { p } ( w ) = \\operatorname* { m a x } _ { | | \\delta | | \\leq \\rho } f ( w + \\delta ) = f ( w ) + \\frac { 1 } { 2 } \\rho ^ { 2 } \\sigma _ { m a x } ( H ) + O ( | | \\delta | | ^ { 3 } )\n$$", + "text_format": "latex", + "bbox": [ + 274, + 309, + 723, + 339 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "where $\\sigma _ { m a x }$ is the dominate eigenvalue (eigenvalue with the largest absolute value). Now consider two local minima $w _ { 1 }$ and $w _ { 2 }$ with dominate eigenvalue $\\sigma _ { 1 }$ and $\\sigma _ { 2 }$ respectively, we have ", + "bbox": [ + 173, + 345, + 823, + 375 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/743dc716e5819b1d0b9541154b6de6da85499afd14817fac169264e7b126ec4e.jpg", + "text": "$$\nf _ { p } ( w _ { 1 } ) \\approx f ( w _ { 1 } ) + \\frac { 1 } { 2 } \\rho ^ { 2 } \\sigma _ { 1 } \\qquad f _ { p } ( w _ { 2 } ) \\approx f ( w _ { 2 } ) + \\frac { 1 } { 2 } \\rho ^ { 2 } \\sigma _ { 2 }\n$$", + "text_format": "latex", + "bbox": [ + 271, + 383, + 723, + 415 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "We have $f _ { p } ( w _ { 1 } ) > f _ { p } ( w _ { 2 } ) \\implies \\sigma _ { 1 } > \\sigma _ { 2 }$ and $\\sigma _ { 1 } > \\sigma _ { 2 } \\implies f _ { p } ( w _ { 1 } ) > f _ { p } ( w _ { 2 } )$ because the relation between $f ( w _ { 1 } )$ and $f ( w _ { 2 } )$ is undetermined. \u0003 ", + "bbox": [ + 173, + 424, + 825, + 453 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "A.2 PROOF OF LEMMA. 3.2 ", + "text_level": 1, + "bbox": [ + 176, + 473, + 379, + 487 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Since $\\rho$ is small, we can perform Taylor expansion around $w$ ", + "bbox": [ + 173, + 500, + 576, + 515 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/ef9460eba714e155cb184bb9fc72c162245191261d6214f1c0c62b653d65b971.jpg", + "text": "$$\n\\begin{array} { r l } & { h ( w ) = f ( w + \\delta ) - f ( w ) } \\\\ & { \\qquad = \\delta ^ { \\top } \\nabla f ( w ) + O ( \\rho ^ { 2 } ) } \\\\ & { \\qquad = \\rho | | \\nabla f ( w ) | | _ { 2 } + O ( \\rho ^ { 2 } ) > 0 } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 382, + 523, + 614, + 585 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "where the last line is because δ is approximated as δ = ρ ∇f(w)||∇f(w)||2+\u000f , h ence has the same direction as $\\nabla f ( w )$ . \u0003 ", + "bbox": [ + 173, + 594, + 826, + 630 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "A.3 PROOF OF LEMMA. 3.3 ", + "text_level": 1, + "bbox": [ + 176, + 648, + 379, + 665 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Since $\\rho$ is small, we can approximate $f ( w )$ with a quadratic model around a local minima $w$ : ", + "bbox": [ + 169, + 676, + 782, + 693 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/e193c38e5c8efd8a485b1d1917e0e741cdea9b300865106c923ff07c1d95d8ba.jpg", + "text": "$$\nf ( w + \\delta ) = f ( w ) + \\frac { 1 } { 2 } \\delta ^ { \\top } H \\delta + O ( \\rho ^ { 3 } )\n$$", + "text_format": "latex", + "bbox": [ + 369, + 702, + 627, + 732 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "where $H$ is the Hessian at $w$ , assumed to be positive semidefinite at local minima. Normalize $\\delta$ such that $| | \\delta | | _ { 2 } = \\rho$ , Hence we have: ", + "bbox": [ + 171, + 741, + 825, + 770 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/266770658db8ee578fb42a9532ffea10b591945e46e2429298703fe2af9dc6a3.jpg", + "text": "$$\nh ( w ) = f _ { p } ( w ) - f ( w ) = \\operatorname* { m a x } _ { | | \\delta | | _ { 2 } \\leq \\rho } f ( w + \\delta ) - f ( w ) = \\frac { 1 } { 2 } \\sigma _ { m a x } \\rho ^ { 2 } + O ( \\rho ^ { 3 } )\n$$", + "text_format": "latex", + "bbox": [ + 238, + 779, + 759, + 810 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "where $\\sigma _ { m a x }$ is the dominate eigenvalue of the hessian $H$ , and first order term is 0 because the gradient is 0 at local minima. Therefore, we have $\\sigma _ { m a x } \\approx 2 h ( w ) / \\rho ^ { 2 }$ . \u0003 ", + "bbox": [ + 173, + 818, + 826, + 848 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "A.4 PROOF OF THM. 5.1 ", + "text_level": 1, + "bbox": [ + 174, + 867, + 356, + 882 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "For simplicity we consider the base optimizer is SGD. For other optimizers such as Adam, we can derive similar results by applying standard proof techniques in the literature to our proof. ", + "bbox": [ + 173, + 895, + 825, + 925 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "STEP 1: CONVERGENCE W.R.T FUNCTION $f _ { p } ( w )$ ", + "bbox": [ + 176, + 103, + 508, + 118 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "For simplicity of notation, we denote the update at step $t$ as ", + "bbox": [ + 173, + 127, + 563, + 142 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/cd63ddc604c5092cc5ab7263081c6610f34bdf9a65fa7f0129e5cd7272b6fe24.jpg", + "text": "$$\nd _ { t } = - \\eta _ { t } g _ { p } ^ { ( t ) } + \\eta _ { t } \\alpha g _ { \\perp } ^ { ( t ) }\n$$", + "text_format": "latex", + "bbox": [ + 419, + 147, + 578, + 170 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "By $L -$ smoothness of $f$ and the definition of $f _ { p } ( w _ { t } ) = f ( w _ { t } ^ { a d v } )$ , and definition of $d _ { t } = w _ { t + 1 } - w _ { t }$ and $w _ { t } ^ { a d v } = w _ { t } + \\delta _ { t }$ we have ", + "bbox": [ + 173, + 184, + 823, + 215 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/a19fc06e8f31b3a8054ac99a0e2cf8399e61a14ff8b3ff2c1ad492f93f5f5c63.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle f _ { p } ( w _ { t + 1 } ) = f ( w _ { t + 1 } ^ { a d v } ) \\leq f ( w _ { t } ^ { a d v } ) + \\langle \\nabla f ( w _ { t } ^ { a d v } ) , w _ { t + 1 } ^ { a d v } - w _ { t } ^ { a d v } \\rangle + \\frac { L } { 2 } \\Big \\| | w _ { t + 1 } ^ { a d v } - w _ { t } ^ { a d v } \\Big \\| ^ { 2 } } \\\\ { \\displaystyle \\qquad = f ( w _ { t } ^ { a d v } ) + \\langle \\nabla f ( w _ { t } ^ { a d v } ) , w _ { t + 1 } + \\delta _ { t + 1 } - w _ { t } - \\delta _ { t } \\rangle } \\\\ { \\displaystyle \\qquad + \\frac { L } { 2 } \\Big \\| w _ { t + 1 } + \\delta _ { t + 1 } - w _ { t } - \\delta _ { t } \\Big \\| ^ { 2 } } \\\\ { \\displaystyle \\qquad \\leq f ( w _ { t } ^ { a d v } ) + \\langle \\nabla f ( w _ { t } ^ { a d v } ) , d _ { t } \\rangle + L \\Big \\| d _ { t } \\Big \\| ^ { 2 } } \\\\ { \\displaystyle \\qquad + \\langle \\nabla f ( w _ { t } ^ { a d v } ) , \\delta _ { t + 1 } - \\delta _ { t } \\rangle + L \\Big \\| \\delta _ { t + 1 } - \\delta _ { t } \\Big \\| ^ { 2 } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 197, + 219, + 767, + 364 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "STEP 1.0: BOUND EQ. 18 ", + "bbox": [ + 174, + 375, + 354, + 388 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "We first bound Eq. 18. Take expectation conditioned on observation up to step $t$ (for simplicity of notation, we use $\\mathbb { E }$ short for $\\mathbb { E } _ { x }$ to denote expectation over all possible data points) conditioned on observations up to step $t$ , also by definition of $d _ { t }$ , we have ", + "bbox": [ + 176, + 398, + 823, + 440 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/3dfdb64a9cfe1173301d4a70dc535f0d1d7b3f83557bb30d9768a9f1c2642d6a.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathbb { E } f _ { p } ( w _ { t + 1 } ) - f _ { p } ( w _ { t } ) \\leq - \\eta _ { t } \\langle \\nabla f _ { p } ( w _ { t } ) , \\mathbb { E } g _ { p } ^ { ( t ) } \\rangle + \\alpha \\eta _ { t } \\langle \\nabla f _ { p } ( w _ { t } ) , \\mathbb { E } g _ { \\perp } ^ { ( t ) } \\rangle } \\\\ & { \\qquad + L \\eta _ { t } ^ { 2 } \\mathbb { E } \\Big | \\Big | - g _ { p } ^ { ( t ) } + \\alpha g _ { \\perp } ^ { ( t ) } \\Big | \\Big | _ { 2 } ^ { 2 } } \\\\ & { \\qquad \\leq - \\eta _ { t } \\mathbb { E } \\Big | \\Big | \\nabla f _ { p } ( w _ { t } ) \\Big | \\Big | _ { 2 } ^ { 2 } + 0 + ( \\alpha + 1 ) ^ { 2 } G ^ { 2 } \\eta _ { t } ^ { 2 } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 243, + 446, + 709, + 531 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "\u0010Since $\\mathbb { E } g _ { \\perp } ^ { ( t ) }$ is orthogonal to $\\nabla f _ { p } ( w _ { t } )$ by construction, ", + "bbox": [ + 385, + 535, + 751, + 554 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/9ae2c2b3352ce584ab4182f560bdde8808bbf9d983ed4d02a121d5487a0ee18d.jpg", + "text": "$$\n\\vert \\vert g ^ { ( t ) } \\vert \\vert \\le G \\mathrm { b y } \\mathrm { a s s u m p t i o n } \\bigg )\n$$", + "text_format": "latex", + "bbox": [ + 383, + 559, + 573, + 585 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "STEP 1.1: BOUND EQ. 19 ", + "bbox": [ + 174, + 597, + 354, + 612 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "By definition of $\\delta _ { t }$ , we have ", + "bbox": [ + 174, + 621, + 357, + 636 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/ea9c73a05c1cea2a2e924ae06615925b147cdf993cb89f0a16f572da8dcb485d.jpg", + "text": "$$\n\\begin{array} { c } { \\displaystyle \\delta _ { t } = \\rho _ { t } \\frac { g ^ { ( t ) } } { | | g ^ { ( t ) } | | + \\epsilon } } \\\\ { \\displaystyle \\delta _ { t + 1 } = \\rho _ { t + 1 } \\frac { g ^ { ( t + 1 ) } } { | | g ^ { ( t + 1 ) } | | + \\epsilon } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 411, + 641, + 588, + 717 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "where $g ^ { ( t ) }$ is the gradient of $f$ at $w _ { t }$ evaluated with a noisy data sample. When learning rate $\\eta _ { t }$ is small, the update in weight $d _ { t }$ is small, and expected gradient is ", + "bbox": [ + 174, + 722, + 823, + 751 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/66d5587c943510d804f97ad5d872bdff19987196998589c07f6d7da24cab05e7.jpg", + "text": "$$\n\\nabla f ( w _ { t + 1 } ) = \\nabla f ( w _ { t } + d _ { t } ) = \\nabla f ( w _ { t } ) + H d _ { t } + O ( | | d _ { t } | | ^ { 2 } )\n$$", + "text_format": "latex", + "bbox": [ + 302, + 755, + 696, + 773 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "where $H$ is the Hessian at $w _ { t }$ . Therefore, we have ", + "bbox": [ + 174, + 779, + 503, + 794 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/cbce51e846918014e8bd6f901dfc5f6cb50a82e104b54a04130da60f2f10d12d.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathbb { E } \\langle \\nabla f ( w _ { t } ^ { a d v } ) , \\delta _ { t + 1 } - \\delta _ { t } \\rangle = \\langle \\nabla f ( w _ { t } ^ { a d v } ) , \\rho _ { t } \\mathbb { E } \\frac { g ^ { ( t ) } } { | | g ^ { ( t ) } | | + \\epsilon } - \\rho _ { t + 1 } \\mathbb { E } \\frac { g ^ { ( t + 1 ) } } { | | g ^ { ( t + 1 ) } | | + \\epsilon } \\rangle } \\\\ & { \\qquad \\leq | | \\nabla f ( w _ { t } ^ { a d v } ) | | \\rho _ { t } \\Big | \\Big | \\mathbb { E } \\frac { g ^ { ( t ) } } { | | g ^ { ( t ) } | | + \\epsilon } - \\mathbb { E } \\frac { g ^ { ( t + 1 ) } } { | | g ^ { ( t + 1 ) } | | + \\epsilon } \\Big | \\Big | } \\\\ & { \\qquad \\leq | | \\nabla f ( w _ { t } ^ { a d v } ) | | \\rho _ { t } \\phi _ { t } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 235, + 797, + 764, + 892 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "where the first inequality is due to (1) $\\rho _ { t }$ is monotonically decreasing with $t$ , and (2) triangle inequality that $\\langle a , b \\rangle \\overset { \\cdot } { \\leq } | | \\boldsymbol { \\dot { a } } | | \\cdot | | b | |$ . $\\phi _ { t }$ is the angle between the unit vector in the direction of $\\nabla f ( w _ { t } )$ ", + "bbox": [ + 173, + 895, + 826, + 925 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "and $\\nabla f ( w _ { t + 1 } )$ . The second inequality comes from that (1) $\\begin{array} { r } { \\left\\| \\frac { g } { | | g | | + \\epsilon } \\right\\| < 1 } \\end{array}$ strictly, so we can replace $\\delta _ { t }$ in Eq. 25 with a unit vector in corresponding directions multiplied by $\\rho _ { t }$ and get the upper bound, (2) the norm of difference in unit vectors can be upper bounded by the arc length on a unit circle. ", + "bbox": [ + 173, + 99, + 826, + 154 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "When learning rate $\\eta _ { t }$ and update stepsize $d _ { t }$ is small, $\\phi _ { t }$ is also small. Using the limit that ", + "bbox": [ + 169, + 159, + 766, + 174 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/d2741b641fbd5d39ad4e3a0e7ad4692c73ddf101eafd3789adf5b2fecfd358a1.jpg", + "text": "$$\n\\tan x = x + O ( x ^ { 2 } ) , \\quad \\sin x = x + O ( x ^ { 2 } ) , \\quad x \\to 0\n$$", + "text_format": "latex", + "bbox": [ + 325, + 176, + 674, + 195 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "We have: ", + "bbox": [ + 173, + 199, + 236, + 213 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/6b3e9bc56ca4195a5a795ca50251ddea8b8ba95ec6e33927914298df06833550.jpg", + "text": "$$\n\\begin{array} { r l r } { { \\tan \\phi _ { t } = \\frac { | | \\nabla f ( w _ { t + 1 } ) - \\nabla f ( w _ { t } ) | | } { | | \\nabla f ( w _ { t } ) | | } + O ( \\phi _ { t } ^ { 2 } ) } } \\\\ & { } & { = \\frac { | | H d _ { t } + O ( | | d _ { t } | | ^ { 2 } ) | | } { | | \\nabla f ( w _ { t } ) | | } + O ( \\phi _ { t } ^ { 2 } ) } \\\\ & { } & { \\leq \\eta _ { t } L ( 1 + \\alpha ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 349, + 214, + 648, + 304 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "where the last inequality is due to (1) max eigenvalue of $H$ is upper bounded by $L$ because $f$ is $L -$ smooth, (2) $| | d _ { t } | | = | | \\eta _ { t } ( g _ { \\parallel } + \\alpha g _ { \\perp } ) | |$ and $\\mathbb { E } g _ { t } = \\nabla f ( w _ { t } )$ . ", + "bbox": [ + 173, + 306, + 823, + 338 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Plug into Eq. 27, also note that the perturbation amplitude $\\rho _ { t }$ is small so $w _ { t }$ is close to $w _ { t } ^ { a d v }$ , then we have ", + "bbox": [ + 171, + 344, + 821, + 371 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/1537a5c50c1dbc3ef55e80028d160facf36f2e371f8f6300f550c2d55ddfa66b.jpg", + "text": "$$\n\\mathbb { E } \\langle \\nabla f ( w _ { t } ^ { a d v } ) , \\delta _ { t + 1 } - \\delta _ { t } \\rangle \\leq L ( 1 + \\alpha ) G \\rho _ { t } \\eta _ { t }\n$$", + "text_format": "latex", + "bbox": [ + 352, + 369, + 645, + 388 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Similarly, we have ", + "bbox": [ + 174, + 390, + 297, + 405 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/04f0adcb4a94b0d329507155496b97c769a2ff1c2ebac217cfbd7bcf008d9ae6.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathbb { E } \\Big \\lvert \\Big \\lvert \\delta _ { t + 1 } - \\delta _ { t } \\Big \\rvert \\Big \\rvert ^ { 2 } \\leq \\rho _ { t } ^ { 2 } \\mathbb { E } \\Big \\lvert \\Big \\lvert \\frac { g ^ { ( t ) } } { \\lvert | g ^ { ( t ) } \\rvert \\rvert + \\epsilon } - \\frac { g ^ { ( t + 1 ) } } { \\lvert | g ^ { ( t + 1 ) } \\rvert \\rvert + \\epsilon } \\Big \\rvert \\Big \\rvert ^ { 2 } } \\\\ & { \\qquad \\leq \\rho _ { t } ^ { 2 } \\phi _ { t } ^ { 2 } } \\\\ & { \\qquad \\leq \\rho _ { t } ^ { 2 } \\eta _ { t } ^ { 2 } L ^ { 2 } ( 1 + \\alpha ) ^ { 2 } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 313, + 407, + 679, + 487 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "STEP 1.2: TOTAL BOUND ", + "bbox": [ + 174, + 496, + 349, + 510 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Reuse results from Eq. 21 (replace $L _ { p }$ with $2 L$ ) and plug into Eq. 18, and plug Eq. 31 and Eq. 34 into Eq. 19, we have ", + "bbox": [ + 173, + 518, + 821, + 547 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/8d7edc3744a10c9176ce6c029eb0ed82d22fc7b04ef3cfd85d31ed89b4647da0.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathbb { E } f _ { p } ( w _ { t + 1 } ) - f _ { p } ( w _ { t } ) \\leq - \\eta _ { t } \\mathbb { E } \\Big | \\Big | \\nabla f _ { p } ( w _ { t } ) \\Big | \\Big | _ { 2 } ^ { 2 } + \\frac { 2 L ( \\alpha + 1 ) ^ { 2 } } { 2 } G ^ { 2 } \\eta _ { t } ^ { 2 } } \\\\ & { \\qquad + L ( 1 + \\alpha ) G \\rho _ { t } \\eta _ { t } + \\frac { 2 L ^ { 3 } ( 1 + \\alpha ) ^ { 2 } } { 2 } \\eta _ { t } ^ { 2 } \\rho _ { t } ^ { 2 } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 284, + 551, + 714, + 618 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Perform telescope sum, we have ", + "bbox": [ + 173, + 619, + 388, + 635 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/2c3b2095d759f2e1ba95b32aff23dd37fa708a61fe1b8370db9103759f247c79.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle \\mathbb { E } f _ { p } ( w _ { T } ) - f _ { p } ( w _ { 0 } ) \\leq - \\sum _ { t = 1 } ^ { T } \\eta _ { t } \\mathbb { E } | | \\nabla f _ { p } ( w _ { t } ) | | ^ { 2 } + \\left[ L ( 1 + \\alpha ) ^ { 2 } G ^ { 2 } \\eta _ { 0 } ^ { 2 } + L ( 1 + \\alpha ) G \\rho _ { 0 } \\eta _ { 0 } \\right] \\sum _ { t = 1 } ^ { T } \\frac { 1 } { t } } \\\\ { \\displaystyle \\qquad + L ^ { 3 } ( 1 + \\alpha ) ^ { 2 } \\eta _ { 0 } ^ { 2 } \\rho _ { 0 } ^ { 2 } \\sum _ { t = 1 } ^ { T } \\frac { 1 } { t ^ { 2 } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 191, + 638, + 805, + 727 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Hence ", + "bbox": [ + 173, + 729, + 218, + 743 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/8033fd7307ba95c395370951681da1b0e5808711c1565776885d1ab177a6866f.jpg", + "text": "$$\n\\eta _ { T } \\sum _ { t = 1 } ^ { T } \\mathbb { E } | | \\nabla f _ { p } ( w _ { t } ) | | ^ { 2 } \\leq \\sum _ { t = 1 } ^ { T } \\eta _ { t } \\mathbb { E } | | \\nabla f _ { p } ( w _ { t } ) | | ^ { 2 } \\leq f _ { p } ( w _ { 0 } ) - \\mathbb { E } f _ { p } ( w _ { T } ) + D \\log T + \\frac { \\pi ^ { 2 } E } { 6 }\n$$", + "text_format": "latex", + "bbox": [ + 194, + 746, + 776, + 789 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "where ", + "bbox": [ + 174, + 792, + 217, + 805 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/a699fb33123efdc7742ede04f1ed39fcbe04c8b29549e58fddfc87f9d2c7b225.jpg", + "text": "$$\nD = L ( 1 + \\alpha ) ^ { 2 } G ^ { 2 } \\eta _ { 0 } ^ { 2 } + L ( 1 + \\alpha ) G \\rho _ { 0 } \\eta _ { 0 } , \\quad E = L ^ { 3 } ( 1 + \\alpha ) ^ { 2 } \\eta _ { 0 } ^ { 2 } \\rho _ { 0 } ^ { 2 }\n$$", + "text_format": "latex", + "bbox": [ + 277, + 803, + 718, + 821 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Note that $\\begin{array} { r } { \\eta _ { T } = \\frac { \\eta _ { 0 } } { \\sqrt { T } } } \\end{array}$ , we have ", + "bbox": [ + 174, + 821, + 367, + 840 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/ceaf518ddfe428cdbaee2eff819b2a044eee3988bb99ff66c9335ba1babb5d55.jpg", + "text": "$$\n\\frac { 1 } { T } \\sum _ { t = 1 } ^ { T } \\mathbb { E } | | \\nabla f _ { p } ( w _ { t } ) | | ^ { 2 } \\leq \\frac { f _ { p } ( w _ { 0 } ) - f _ { m i n } + \\pi ^ { 2 } E / 6 } { \\eta _ { 0 } } \\frac { 1 } { \\sqrt { T } } + \\frac { D } { \\eta _ { 0 } } \\frac { \\log T } { \\sqrt { T } }\n$$", + "text_format": "latex", + "bbox": [ + 276, + 845, + 722, + 888 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "which implies that GSAM enables $f _ { p }$ to converge at a rate of $O ( \\log T / \\sqrt { T } )$ , and all the constants here are well-bounded. ", + "bbox": [ + 174, + 895, + 823, + 924 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "STEP 2: CONVERGENCE W.R.T. FUNCTION $f ( w )$ ", + "bbox": [ + 174, + 103, + 506, + 118 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "We prove the risk for $f ( w )$ convergences for non-convex stochastic optimization case using SGD. Denote the update at step $t$ as ", + "bbox": [ + 173, + 127, + 823, + 156 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/1d7d827fd6faeec2f42b6122add3ec93b21a508350edc2e75519691905278f90.jpg", + "text": "$$\nd _ { t } = - \\eta _ { t } g _ { p } ^ { ( t ) } + \\alpha \\eta _ { t } g _ { \\perp } ^ { ( t ) }\n$$", + "text_format": "latex", + "bbox": [ + 418, + 154, + 578, + 176 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "By smoothness of $f$ , we have ", + "bbox": [ + 174, + 174, + 369, + 189 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/80408dea0ba6777038c88511debc506ed6a9785b24fc61fa401aa68f3080fdb0.jpg", + "text": "$$\n\\begin{array} { l } { f ( w _ { t + 1 } ) \\leq f ( w _ { t } ) + \\langle \\nabla f ( w _ { t } ) , d _ { t } \\rangle + \\displaystyle \\frac { L } { 2 } \\Big \\lvert \\Big \\lvert d _ { t } \\Big \\rvert \\Big \\rvert _ { 2 } ^ { 2 } } \\\\ { = f ( w _ { t } ) + \\langle \\nabla f ( w _ { t } ) , - \\eta _ { t } g _ { p } ^ { ( t ) } + \\alpha \\eta _ { t } g _ { \\perp } ^ { ( t ) } \\rangle + \\displaystyle \\frac { L } { 2 } \\Big \\lvert \\Big \\lvert d _ { t } \\Big \\rvert \\Big \\rvert _ { 2 } ^ { 2 } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 290, + 190, + 705, + 253 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "For simplicity, we introduce a scalar $\\beta _ { t }$ such that ", + "bbox": [ + 174, + 253, + 496, + 268 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/2bec24f3d1faa63344eb49ddccb1cc488afc4c1f504f6d27c42cb7a4a50bcc0d.jpg", + "text": "$$\n\\nabla f _ { \\parallel } ( w _ { t } ) = \\beta _ { t } \\nabla f _ { p } ( w _ { t } )\n$$", + "text_format": "latex", + "bbox": [ + 419, + 270, + 578, + 287 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "where $\\nabla f _ { \\parallel } ( w _ { t } )$ is the projection of $\\nabla f ( w _ { t } )$ onto $\\nabla f _ { p } ( w _ { t } )$ . When perturbation amplitude $\\rho$ is small, we expect $\\beta _ { t }$ to be very close to 1. ", + "bbox": [ + 173, + 290, + 823, + 319 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Take expectation conditioned on observations up to step $t$ for both sides of Eq. 42, we have: ", + "bbox": [ + 173, + 324, + 774, + 340 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/6a11115ba457eb26736107642e6570fac8fafa4999b1f67fd0ebcad8012d83b5.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle \\mathbb { E } f ( w _ { t + 1 } ) \\leq f ( w _ { t } ) + \\left. \\nabla f ( w _ { t } ) , - \\frac { \\eta _ { t } } { \\beta _ { t } } \\Big ( \\nabla f ( w _ { t } ) - \\nabla f _ { \\perp } ( w _ { t } ) \\Big ) + \\alpha \\eta _ { t } \\mathbb { E } g _ { \\perp } ^ { ( t ) } \\right. + \\displaystyle \\frac { L } { 2 } \\mathbb { E } \\Big \\| \\boldsymbol { d } _ { t } \\Big \\| _ { 2 } ^ { 2 } } \\\\ { = f ( w _ { t } ) - \\displaystyle \\frac { \\eta _ { t } } { \\beta _ { t } } \\Big \\| \\nabla f ( w _ { t } ) \\Big \\| _ { 2 } ^ { 2 } + \\Big ( \\frac { 1 } { \\beta _ { t } } + \\alpha \\Big ) \\eta _ { t } \\Big \\langle \\nabla f ( w _ { t } ) , \\nabla f _ { \\perp } ( w _ { t } ) \\Big \\rangle + \\displaystyle \\frac { L } { 2 } \\mathbb { E } \\Big \\| \\boldsymbol { d } _ { t } \\Big \\| _ { 2 } ^ { 2 } } \\\\ { = f ( w _ { t } ) - \\displaystyle \\frac { \\eta _ { t } } { \\beta _ { t } } \\Big \\| \\nabla f ( w _ { t } ) \\Big \\| _ { 2 } ^ { 2 } + \\Big ( \\frac { 1 } { \\beta _ { t } } + \\alpha \\Big ) \\eta _ { t } \\Big \\langle \\nabla f ( w _ { t } ) , \\nabla f ( w _ { t } ) \\sin \\theta _ { t } \\Big \\rangle + \\displaystyle \\frac { L } { 2 } \\mathbb { E } \\Big \\| \\boldsymbol { d } _ { t } \\Big \\| _ { 2 } ^ { 2 } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 171, + 340, + 790, + 452 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/d95c83b261fe0a209af3aa8b22f6c2df6d45eba22070a53365eb1612f98a5f80.jpg", + "text": "$$\n= f ( w _ { t } ) - \\frac { \\eta _ { t } } { \\beta _ { t } } \\Big \\lvert \\Big | \\nabla f ( w _ { t } ) \\Big \\rvert \\Big | _ { 2 } ^ { 2 } + \\Big ( \\frac { 1 } { \\beta _ { t } } + \\alpha \\Big ) \\eta _ { t } \\Big \\lvert \\Big | \\nabla f ( w _ { t } ) \\Big \\rvert \\Big | _ { 2 } ^ { 2 } ( | \\tan \\theta _ { t } | + O ( \\theta _ { t } ^ { 2 } ) ) + \\frac { L } { 2 } \\mathbb { E } \\Big \\lvert \\Big | d _ { t } \\Big \\rvert \\Big | _ { 2 } ^ { 2 }\n$$", + "text_format": "latex", + "bbox": [ + 240, + 493, + 826, + 526 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/c27b0b6dca24e2fd2ba850832ae4914ce6cb08dcb9fc4c7c0529fdbc76b05883.jpg", + "text": "$$\n\\Big ( \\sin x = x + O ( x ^ { 2 } ) , \\tan x = x + O ( x ^ { 2 } ) \\mathrm { w h e n } x 0 . \\Big )\n$$", + "text_format": "latex", + "bbox": [ + 243, + 542, + 619, + 570 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Also note when perturbation amplitude $\\rho _ { t }$ is small, we have ", + "bbox": [ + 176, + 570, + 568, + 585 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/08cd5188fba0b18a24baa081c3bfcbed324e16e0e8c99d9c5c1962579a43a7a7.jpg", + "text": "$$\n\\nabla f _ { p } ( w _ { t } ) = \\nabla f ( w _ { t } + \\delta _ { t } ) = \\nabla f ( w _ { t } ) + \\frac { \\rho _ { t } } { | | \\nabla f ( w _ { t } ) | | _ { 2 } + \\epsilon } H ( w _ { t } ) \\nabla f ( w _ { t } ) + O ( \\rho _ { t } ^ { 2 } )\n$$", + "text_format": "latex", + "bbox": [ + 230, + 585, + 766, + 617 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "where δt = ρt ∇f (wt)||∇f (wt)||2 by definition, $H ( w _ { t } )$ is the Hessian. Hence we have ", + "bbox": [ + 178, + 618, + 681, + 640 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/0b2df70ef33b914732e7b0f3cf38a9f3e8c16286d7c8496ff0d2d2c65088392b.jpg", + "text": "$$\n| \\tan \\theta _ { t } | \\leq \\frac { | | \\nabla f _ { p } ( w _ { t } ) - \\nabla f ( w _ { t } ) | | } { | | \\nabla f ( w _ { t } ) | | } \\leq \\frac { \\rho _ { t } L } { | | \\nabla f ( w _ { t } ) | | }\n$$", + "text_format": "latex", + "bbox": [ + 333, + 641, + 663, + 675 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "where $L$ is the Lipschitz constant of $f$ , and $L -$ smoothness of $f$ indicates the maximum absolute eigenvalue of $H$ is upper bounded by $L$ . Plug Eq. 49 into Eq. 47, we have ", + "bbox": [ + 181, + 678, + 820, + 705 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/d63d187e4cbdeb884cf173569bd4b8d2390260b9532a0b5978d07917539114b1.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle \\mathbb { E } f ( w _ { t + 1 } ) \\leq f ( w _ { t } ) - \\frac { \\eta _ { t } } { \\beta _ { t } } \\Big \\| \\nabla f ( w _ { t } ) \\Big \\| _ { 2 } ^ { 2 } + \\Big ( \\frac { 1 } { \\beta _ { t } } + \\alpha \\Big ) \\eta _ { t } \\Big \\| \\nabla f ( w _ { t } ) \\Big \\| _ { 2 } ^ { 2 } | \\tan \\theta _ { t } | + \\frac { L } { 2 } \\mathbb { E } \\Big \\| d t _ { t } \\Big \\| _ { 2 } ^ { 2 } } \\\\ { \\displaystyle \\leq f ( w _ { t } ) - \\frac { \\eta _ { t } } { \\beta _ { t } } \\Big \\| \\nabla f ( w _ { t } ) \\Big \\| _ { 2 } ^ { 2 } + \\Big ( \\frac { 1 } { \\beta _ { t } } + \\alpha \\Big ) L \\rho _ { t } \\eta _ { t } \\Big \\| \\nabla f ( w _ { t } ) \\Big \\| _ { 2 } + \\frac { L } { 2 } \\mathbb { E } \\Big \\| d t _ { t } \\Big \\| _ { 2 } ^ { 2 } } \\\\ { \\displaystyle \\leq f ( w _ { t } ) - \\frac { \\eta _ { t } } { \\beta _ { t } } \\Big \\| \\nabla f ( w _ { t } ) \\Big \\| _ { 2 } ^ { 2 } + \\Big ( \\frac { 1 } { \\beta _ { t } } + \\alpha \\Big ) L \\rho _ { t } \\eta _ { t } G + \\frac { L } { 2 } \\mathbb { E } \\Big \\| d t _ { t } \\Big \\| _ { 2 } ^ { 2 } } \\\\ { \\Big ( \\mathrm { A s s u m e ~ g r a d i e n t ~ h a s ~ b o u n d e d ~ n o r m ~ } G _ { \\cdot } \\Big ) } \\\\ { \\displaystyle \\leq f ( w _ { t } ) - \\frac { \\eta _ { t } } { \\beta _ { m a x } } \\Big \\| \\nabla f ( w _ { t } ) \\Big \\| _ { 2 } ^ { 2 } + \\Big ( \\frac { 1 } { \\beta _ { m i n } } + \\alpha \\Big ) L \\rho _ { t } \\eta _ { t } G + \\frac { L } { 2 } \\mathbb { E } ( \\alpha + 1 ) ^ { 2 } G ^ { 2 } \\eta _ { t } ^ { 2 } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 194, + 707, + 777, + 867 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "\u0010 $\\beta _ { t }$ is close to 1 assuming $\\rho$ is small, ", + "bbox": [ + 266, + 869, + 509, + 891 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Re-arranging above formula, we have ", + "bbox": [ + 174, + 103, + 423, + 118 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/fc53b6907c0309f7854755775d8cabd7005aa049d6da3dc1273139c3a188312d.jpg", + "text": "$$\n\\frac { \\eta _ { t } } { \\beta _ { m a x } } \\Big | \\Big | \\nabla f ( w _ { t } ) \\Big | \\Big | _ { 2 } ^ { 2 } \\le f ( w _ { t } ) - \\mathbb { E } f ( w _ { t + 1 } ) + \\Big ( \\frac { 1 } { \\beta _ { m i n } } + \\alpha \\Big ) L G \\eta _ { t } \\rho _ { t } + \\frac { L } { 2 } ( \\alpha + 1 ) ^ { 2 } G ^ { 2 } \\eta _ { t } ^ { 2 }\n$$", + "text_format": "latex", + "bbox": [ + 202, + 122, + 767, + 155 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "perform telescope sum and taking expectations on each step, we have ", + "bbox": [ + 173, + 157, + 630, + 172 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/a17189880b90ed64605b201439c689b66005dcf701a00d38cce341ee4fab9378.jpg", + "text": "$$\n\\frac { 1 } { \\beta _ { m a x } } \\sum _ { t = 1 } ^ { T } \\eta _ { t } \\Big | \\Big | \\nabla f ( w _ { t } ) \\Big | \\Big | _ { 2 } ^ { 2 } \\le f ( w _ { 0 } ) - \\mathbb { E } f ( w _ { T } ) + \\Big ( \\frac { 1 } { \\beta _ { m i n } } + \\alpha \\Big ) L G \\sum _ { t = 1 } ^ { T } \\eta _ { t } \\rho _ { t } + \\frac { L } { 2 } ( \\alpha + 1 ) ^ { 2 } G ^ { 2 } \\sum _ { t = 1 } ^ { T } \\eta _ { t } ^ { 2 }\n$$", + "text_format": "latex", + "bbox": [ + 169, + 175, + 816, + 219 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Take the schedule to be $\\begin{array} { r } { \\eta _ { t } = \\frac { \\eta _ { 0 } } { \\sqrt { t } } } \\end{array}$ and $\\begin{array} { r } { \\rho _ { t } = \\frac { \\rho _ { 0 } } { \\sqrt { t } } } \\end{array}$ , then we have ", + "bbox": [ + 173, + 229, + 570, + 248 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/0d84c9a9a4b06e53ff0d50912f26830b89db510d3133780418f7b606c9b20c6e.jpg", + "text": "$$\n\\begin{array} { r l r } { { \\frac { \\eta _ { 0 } } { 3 m \\alpha \\lambda } \\frac { 1 } { \\sqrt { T } } \\sum _ { t = 1 } ^ { T } \\| \\nabla f ( w _ { t } ) \\| _ { 2 } ^ { 2 } \\le L H S } } \\\\ & { } & { \\le R H S } \\\\ & { } & { \\le { \\cal J } ( w _ { 0 } ) - { \\cal J } _ { m i n } + \\Big ( \\frac { 1 } { \\beta _ { m i n } } + \\alpha \\Big ) L G \\eta _ { 0 } \\rho _ { 0 } \\sum _ { t = 1 } ^ { T } \\frac { 1 } { t } + \\frac { L } { 2 } ( \\alpha + 1 ) ^ { 2 } G ^ { 2 } \\eta _ { 0 } ^ { 2 } \\frac { T } { t - 1 } ; } \\\\ & { } & { \\overset { ( \\mathrm { 5 9 } ) } { \\le } ~ } \\\\ & { } & { \\le f ( w _ { 0 } ) - f _ { m i n } + \\Big ( \\frac { 1 } { \\beta _ { m i n } } + \\alpha \\Big ) L G \\eta _ { 0 } \\rho _ { 0 } ( 1 + \\log T ) } \\\\ & { } & { ~ + \\frac { L } { 2 } ( \\alpha + 1 ) ^ { 2 } G ^ { 2 } \\eta _ { 0 } ^ { 2 } ( 1 + \\log T ) ~ ( 6 0 ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 253, + 839, + 440 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Hence ", + "bbox": [ + 173, + 440, + 218, + 455 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/53ebef6edee69527480c48c303324f981a7f5a73fea79ed10a94d132718d19ab.jpg", + "text": "$$\n\\frac { 1 } { T } \\sum _ { t = 1 } ^ { T } \\Big | \\Big | \\nabla f ( w _ { t } ) \\Big | \\Big | _ { 2 } ^ { 2 } \\le \\frac { C _ { 3 } } { \\sqrt { T } } + C _ { 4 } \\frac { \\log T } { \\sqrt { T } }\n$$", + "text_format": "latex", + "bbox": [ + 369, + 450, + 629, + 493 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "where $C _ { 1 } , C _ { 4 }$ are some constants. This implies the convergence rate w.r.t $f ( w )$ is $O ( \\log T / \\sqrt { T } )$ ", + "bbox": [ + 169, + 497, + 805, + 512 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "STEP 3: CONVERGENCE W.R.T. SURROGATE GAP $h ( w )$ ", + "bbox": [ + 173, + 526, + 550, + 541 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Note that we have proved convergence for $f _ { p } ( w )$ in step 1, and convergence for $f ( w )$ in step 3. Also note that ", + "bbox": [ + 173, + 550, + 826, + 579 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/0a22cc0714d6196261bc016efd3b077b5031c520fc1d4a80d4bb90fe275387fd.jpg", + "text": "$$\n\\Big | \\Big | \\nabla h ( w _ { t } ) \\Big | \\Big | _ { 2 } ^ { 2 } = \\Big | \\Big | \\nabla f _ { p } ( w _ { t } ) - \\nabla f ( w _ { t } ) \\Big | \\Big | _ { 2 } ^ { 2 } \\leq 2 \\Big | \\Big | \\nabla f _ { p } ( w _ { t } ) \\Big | \\Big | _ { 2 } ^ { 2 } + 2 \\Big | \\Big | \\nabla f ( w _ { t } ) \\Big | \\Big | _ { 2 } ^ { 2 }\n$$", + "text_format": "latex", + "bbox": [ + 248, + 580, + 748, + 611 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Hence ", + "bbox": [ + 173, + 613, + 218, + 627 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/d1f4ceb0c5e6fdf40b58051b79f9dadc4f2da46be935233c32884257d2c22d78.jpg", + "text": "$$\n\\frac { 1 } { T } \\sum _ { t = 1 } ^ { T } \\Big | \\Big | \\nabla h ( w _ { t } ) \\Big | \\Big | _ { 2 } ^ { 2 } \\le \\frac { 2 } { T } \\sum _ { t = 1 } ^ { T } \\Big | \\Big | \\nabla f _ { p } ( w _ { t } ) \\Big | \\Big | _ { 2 } ^ { 2 } + \\frac { 2 } { T } \\sum _ { t = 1 } ^ { T } \\Big | \\Big | \\nabla f ( w _ { t } ) \\Big | \\Big | _ { 2 } ^ { 2 }\n$$", + "text_format": "latex", + "bbox": [ + 282, + 623, + 715, + 666 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "also converges at rate $O ( \\log T / \\sqrt { T } )$ because each item in the RHS converges at rate $O ( \\log T \\sqrt { T } )$ ", + "bbox": [ + 176, + 669, + 821, + 690 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "A.5 PROOF OF COROLLARY. 5.2.1 ", + "text_level": 1, + "bbox": [ + 176, + 714, + 423, + 729 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Using the results from Thm. 5.2, with probability at least $1 - a$ , we have ", + "bbox": [ + 173, + 739, + 650, + 756 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/d6d6735f956239c6a406a6da86ec2a4df677c0ee86a8a43e7d5255ade5c7a3d0.jpg", + "text": "$$\n\\mathbb { E } _ { w \\sim \\mathcal { Q } } \\mathbb { E } _ { { x } } f ( w , { x } ) \\le \\mathbb { E } _ { w \\sim \\mathcal { Q } } \\widehat { f } ( w ) + 4 \\sqrt { \\frac { K L ( \\mathcal { Q } | | \\mathcal { P } ) + \\log \\frac { 2 m } { a } } { m } }\n$$", + "text_format": "latex", + "bbox": [ + 297, + 760, + 700, + 803 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Assume $\\delta \\sim \\mathcal { N } ( 0 , b ^ { 2 } I _ { k } )$ where $k$ is the dimension of model parameters, hence $\\delta ^ { 2 }$ (element-wise square) follows a a Chi-square distribution. By Lemma.1 in Laurent $\\&$ Massart (2000), we have ", + "bbox": [ + 173, + 806, + 820, + 835 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/9a80a6dc4791072d1c9581bc88f146f071e5194bae61c7457c7879f3b159b60f.jpg", + "text": "$$\n\\mathbb { P } \\big ( | | \\delta | | _ { 2 } ^ { 2 } - k b ^ { 2 } \\geq 2 b ^ { 2 } \\sqrt { k t } + 2 t b ^ { 2 } \\big ) \\leq e x p ( - t )\n$$", + "text_format": "latex", + "bbox": [ + 346, + 839, + 651, + 861 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "hence with probability at least $1 - 1 / \\sqrt { n }$ , we have ", + "bbox": [ + 173, + 864, + 506, + 881 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/c8756a1c2e1f10ff50b62494132fb4b8385d315d97819b7b44cba0caa6909518.jpg", + "text": "$$\n| | \\delta | | _ { 2 } ^ { 2 } \\leq b ^ { 2 } \\Bigg ( 2 \\log \\sqrt { n } + k + 2 \\sqrt { k \\log \\sqrt { n } } \\Bigg ) \\leq 2 b ^ { 2 } k \\Bigg ( 1 + \\sqrt { \\frac { \\log \\sqrt { n } } { k } } \\Bigg ) ^ { 2 } \\leq \\rho ^ { 2 }\n$$", + "text_format": "latex", + "bbox": [ + 243, + 883, + 754, + 929 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Therefore, with probability at least $\\begin{array} { r } { 1 - 1 / \\sqrt { n } = 1 - e x p \\bigg ( - \\big ( \\frac { \\rho } { \\sqrt { 2 } b } - \\sqrt { k } \\big ) ^ { 2 } \\bigg ) } \\end{array}$ ", + "bbox": [ + 171, + 99, + 681, + 126 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/9873e3fcbf6cb330d2f6e915e7f9e0f3f4fd8105ce05f24031966ce7a85c13c4.jpg", + "text": "$$\n\\begin{array} { r } { \\mathbb { E } _ { \\delta } \\widehat { f } ( w + \\delta ) \\leq \\operatorname* { m a x } _ { | | \\delta | | _ { 2 } \\leq \\rho } \\widehat { f } ( w + \\delta ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 374, + 132, + 622, + 154 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Combine Eq. 65 and Eq. 67, subtract the same constant $C$ on both sides, and under the same assumption as in (Foret et al., 2020) that $\\begin{array} { r } { \\mathbb { E } _ { w \\sim \\mathcal { Q } } \\mathbb { E } _ { x } f ( w , x ) \\le \\mathbb { E } _ { \\delta \\sim \\mathcal { N } ( 0 , b ^ { 2 } I ^ { k } ) } \\mathbb { E } _ { w \\sim \\mathcal { Q } } \\mathbb { E } _ { x } f ( w + \\delta , x ) \\mathrm { w e } } \\end{array}$ finish the proof. \u0003 ", + "bbox": [ + 173, + 160, + 820, + 203 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "A.6 PROOF OF THM. 5.3 ", + "text_level": 1, + "bbox": [ + 176, + 219, + 357, + 234 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "STEP 1: A SUFFICIENT CONDITION THAT THE LOSS GAP IS EXPECTED TO DECREASE FOR EACH STEP ", + "bbox": [ + 173, + 246, + 785, + 273 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Take Taylor expansion, then the expected change of loss gap caused by descent step is ", + "bbox": [ + 174, + 284, + 736, + 299 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/b4b058537c3b909a6bfeec92f872fe96de9cb6edebcae36ae3fb1198ad6f5cdc.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathbb { E } \\langle \\nabla f _ { p } ( w _ { t } ) - \\nabla f ( w _ { t } ) , - \\eta _ { t } \\nabla f _ { p } ( w _ { t } ) \\rangle } \\\\ & { \\Big ( w h e r e \\mathbb { E } g _ { \\perp } = \\nabla f _ { \\perp } ( w _ { t } ) \\Big ) } \\\\ & { = \\eta _ { t } \\Bigg [ - | | \\nabla f _ { p } ( w _ { t } ) | | _ { 2 } ^ { 2 } + | | \\nabla f _ { p } ( w _ { t } ) | | _ { 2 } \\big | | \\nabla f ( w _ { t } ) | | _ { 2 } \\cos \\theta _ { t } \\Bigg ] } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 299, + 325, + 697, + 417 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "where $\\theta _ { t }$ is the angle between vector $\\nabla f _ { p } ( w _ { t } )$ and $\\nabla f ( w _ { t } )$ . ", + "bbox": [ + 173, + 422, + 566, + 438 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "The expected change of loss gap caused by ascent step is ", + "bbox": [ + 169, + 438, + 562, + 452 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/66412a0182b9ce0f9b598bc60dfeec285823de2b3ebd5c0497afcee9ca5d6d06.jpg", + "text": "$$\n\\begin{array} { r } { \\mathbb { E } \\langle \\nabla f _ { p } ( w _ { t } ) - \\nabla f ( w _ { t } ) , \\alpha \\eta _ { t } \\nabla f _ { \\perp } ( w _ { t } ) \\rangle = - \\alpha \\eta _ { t } \\big \\lvert \\big \\rvert \\nabla f _ { \\perp } ( w _ { t } ) \\big \\rvert \\big \\rvert _ { 2 } ^ { 2 } < 0 } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 281, + 459, + 717, + 482 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Above results demonstrate that ascent step decreases the loss gap, while descent step might increase the loss gap. A sufficient (but not necessary) condition for $\\mathbb { E } \\langle \\nabla h ( w _ { t } ) , d t \\rangle \\leq 0$ requires $\\alpha$ to be large or $| \\big | \\nabla f ( w _ { t } ) \\big | \\big | _ { 2 } \\cos \\theta _ { t } \\leq \\big | \\big | \\nabla f _ { p } ( w _ { t } ) \\big | \\big |$ . In practice, the perturbation amplitude $\\rho$ is small and we can assume $\\theta _ { t }$ is close to 0 and $\\big | \\big | \\nabla f _ { p } ( w _ { t } ) \\big | \\big |$ is close to $\\left| \\left| \\nabla f ( w _ { t } ) \\right| \\right|$ , we can also set the parameter $\\alpha$ to be large in order to decrease the loss gap. ", + "bbox": [ + 173, + 487, + 825, + 563 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "STEP 2: UPPER AND LOWER BOUND OF DECREASE IN LOSS GAP (BY THE ASCENT STEP IN ORTHOGONAL GRADIENT DIRECTION) COMPARED TO SAM. ", + "bbox": [ + 171, + 577, + 794, + 606 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Next we give an estimate of the decrease in $\\widehat { h }$ caused by our ascent step. We refer to Eq. 69 and Eq. 70 to analyze the change in loss gap caused by the descent and ascent (orthogonally) respectively. It can be seen that gradient descent step might not decrease loss gap, in fact they often increase loss gap in practice; while the ascent step is guaranteed to decrease the loss gap. ", + "bbox": [ + 174, + 616, + 825, + 674 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "The decrease in loss gap is: ", + "bbox": [ + 174, + 680, + 356, + 695 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/fa6c0955fb0387b100d8943ef1b910167707cba694d39fbfabc0fac42d1e8085.jpg", + "text": "$$\n\\begin{array} { r l r } & { } & { \\Delta \\widehat { h } _ { t } = - \\langle \\nabla \\widehat { f } _ { p } ( w _ { t } ) - \\nabla \\widehat { f } ( w _ { t } ) , \\alpha \\eta _ { t } \\nabla \\widehat { f } _ { \\bot } ( w _ { t } ) \\rangle = \\alpha \\eta _ { t } \\big \\lvert \\big \\lvert \\nabla \\widehat { f } _ { \\bot } ( w _ { t } ) \\big \\rvert \\big \\rvert _ { 2 } ^ { 2 } } \\\\ & { } & { = \\alpha \\eta _ { t } \\big \\lvert \\big \\lvert \\nabla \\widehat { f } ( w _ { t } ) \\big \\rvert \\big \\rvert _ { 2 } ^ { 2 } \\big \\lvert \\tan \\theta _ { t } \\big \\rvert ^ { 2 } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 251, + 702, + 745, + 751 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/4a0ef4c803ae793da7989d8e8d891269cdb1cd8b9e0c4c9dad4d9240d0158448.jpg", + "text": "$$\n\\begin{array} { r l r } { { \\sum _ { t = 1 } ^ { T } \\Delta \\widehat { h } _ { t } \\le \\sum _ { t = 1 } ^ { T } \\alpha L ^ { 2 } \\eta _ { t } \\rho _ { t } ^ { 2 } } } \\\\ & { } & { \\Big ( \\mathrm { B y ~ E q . ~ 4 9 } \\Big ) } \\\\ & { } & { \\le \\displaystyle \\sum _ { t = 1 } ^ { T } \\alpha L ^ { 2 } \\eta _ { 0 } \\rho _ { 0 } ^ { 2 } \\frac { 1 } { t ^ { 3 / 2 } } } \\\\ & { } & { \\le 2 . 7 \\alpha L ^ { 2 } \\eta _ { 0 } \\rho _ { 0 } ^ { 2 } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 398, + 763, + 598, + 900 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Hence we derive an upper bound for $\\Sigma _ { t = 1 } ^ { T } \\Delta \\widehat { h } _ { t }$ ", + "bbox": [ + 173, + 906, + 493, + 926 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Next we derive a lower bound for $\\Sigma _ { t = 1 } ^ { T } \\Delta \\widehat { h } _ { t }$ Note that when $\\rho _ { t }$ is small, by Taylor expansion ", + "bbox": [ + 173, + 101, + 787, + 119 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/59ade6d0d63ed89543502716ce38d5928cc2d54dc37fcf85b46d5afc568464d1.jpg", + "text": "$$\n\\nabla \\widehat { f } _ { p } ( w _ { t } ) = \\nabla \\widehat { f } ( w _ { t } + \\delta _ { t } ) = \\nabla \\widehat { f } ( w _ { t } ) + \\frac { \\rho _ { t } } { | | \\nabla \\widehat { f } ( w _ { t } ) | | } \\widehat { H } ( w _ { t } ) \\nabla \\widehat { f } ( w _ { t } ) + O ( \\rho _ { t } ^ { 2 } )\n$$", + "text_format": "latex", + "bbox": [ + 246, + 122, + 750, + 155 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "where $\\widehat { H } ( w _ { t } )$ is the Hessian evaluated on training samples. Also when $\\rho _ { t }$ is small, the angle $\\theta _ { t }$ between $\\nabla \\widehat { f } _ { p } ( w _ { t } )$ and $\\nabla \\widehat { f } ( \\boldsymbol { w } _ { t } )$ is small, by the limit that ", + "bbox": [ + 174, + 161, + 823, + 194 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/0b552bcdaad8255fabc0717f3bb85c2ee31af43aefa366826e25da796e1352a9.jpg", + "text": "$$\n\\begin{array} { l } { \\tan x = x + O ( x ^ { 2 } ) , x \\to 0 } \\\\ { \\sin x = x + O ( x ^ { 2 } ) , x \\to 0 } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 406, + 196, + 593, + 239 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "We have ", + "bbox": [ + 173, + 239, + 233, + 253 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/63c13d186321c4ea50bcb20e5bd2c7fff2fe82d158749d4941ca4247bc1a28ff.jpg", + "text": "$$\n\\left| \\tan \\theta _ { t } \\right| = \\left| \\sin \\theta _ { t } \\right| + O ( \\theta _ { t } ^ { 2 } ) = | \\theta _ { t } | + O ( \\theta _ { t } ^ { 2 } )\n$$", + "text_format": "latex", + "bbox": [ + 352, + 251, + 645, + 270 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Omitting high order term, we have ", + "bbox": [ + 174, + 270, + 403, + 285 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/042d990fe9c2542b4f7cac44cb240740149d52dcd6da21645ab887890c29e706.jpg", + "text": "$$\n| \\tan \\theta _ { t } | \\approx | \\theta _ { t } | = \\frac { | | \\nabla \\widehat { f } _ { p } ( w _ { t } ) - \\nabla \\widehat { f } ( w _ { t } ) | | } { | | \\widehat { f } ( w _ { t } ) | | } = \\frac { | | \\rho _ { t } \\widehat { H } ( w _ { t } ) + O ( \\rho _ { t } ^ { 2 } ) | | } { | | \\nabla \\widehat { f } ( w _ { t } ) | | } \\geq \\frac { \\rho _ { t } | \\sigma | _ { m i n } } { G }\n$$", + "text_format": "latex", + "bbox": [ + 236, + 287, + 759, + 328 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "where $G$ is the upper-bound on norm of gradient, $| \\sigma | _ { m i n }$ is the minimum absolute eigenvalue of the Hessian. The intuition is that as perturbation amplitude decreases, the angle $\\theta _ { t }$ decreases at a similar rate, though the scale constant might be different. Hence we have ", + "bbox": [ + 176, + 332, + 821, + 373 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/696775696d71ffbfb2af20956be032a8254a55afd9eb75d5eb8751a605040322.jpg", + "text": "$$\n\\begin{array} { r l } { { \\sum _ { t = 1 } ^ { T } \\Delta \\hat { h } _ { t } = \\sum _ { t = 1 } ^ { T } \\alpha \\eta _ { t } \\vert \\big \\vert \\nabla \\widehat { f } ( w _ { t } ) \\big \\vert \\big \\vert _ { 2 } ^ { 2 } \\vert \\tan \\theta _ { t } \\vert ^ { 2 } + O ( \\theta _ { t } ^ { 4 } ) } \\ ~ } \\\\ & { \\geq \\sum _ { t = 1 } ^ { T } \\alpha \\eta _ { t } c ^ { 2 } \\Big ( \\frac { \\rho _ { t } \\vert \\sigma \\vert _ { m i n } } { G } \\Big ) ^ { 2 } } \\\\ & { = \\frac { \\alpha c ^ { 2 } \\rho _ { 0 } ^ { 2 } \\eta _ { 0 } \\vert \\sigma \\vert _ { m i n } ^ { 2 } } { G ^ { 2 } } \\sum _ { t = 1 } ^ { T } \\frac { 1 } { t ^ { 3 / 2 } } } \\\\ & { \\geq \\frac { \\alpha c ^ { 2 } \\rho _ { 0 } ^ { 2 } \\eta _ { 0 } \\vert \\sigma \\vert _ { m i n } ^ { 2 } } { G ^ { 2 } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 331, + 377, + 665, + 541 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "where $c ^ { 2 }$ is the lower bound of $| | \\nabla \\widehat { f } | | ^ { 2 }$ (e.g. due to noise in data and gradient observation). Results above indicate that the decrease in loss gap caused by the ascent step is non-trivial, hence our proposed method efficiently improves generalization compared with SAM. \u0003 ", + "bbox": [ + 173, + 546, + 825, + 590 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "A.7 DISCUSSION ON COROLLARY 5.2.1 ", + "text_level": 1, + "bbox": [ + 176, + 604, + 460, + 619 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "The comment “‘The corollary gives a bound on the risk in terms of the perturbed training loss if one removes $C$ from both sides”’ is correct. But there is a misunderstanding in the statement “‘the perturbed training loss is small then the model has a small risk”’: it’s only true when $\\rho _ { t r a i n }$ for training equals its real value $\\rho _ { t r u e }$ determined by the data distribution; in practice, we never know $\\rho _ { t r u e }$ . In the following we show that the minimization of both $h$ and $f _ { p }$ is better than simply minimizing $f _ { p }$ when $\\rho _ { t r u e } \\neq \\rho _ { t r a i n }$ . ", + "bbox": [ + 173, + 631, + 826, + 717 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "1. First, we re-write the conclusion of Corollary 5.2.1 as ", + "bbox": [ + 176, + 722, + 544, + 736 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/cdaf2ab0abf25e025e63bc79a18bd9a65ace8a902a290b154e1abfb27914e5e9.jpg", + "text": "$$\n\\begin{array} { r } { \\mathbb { E } _ { w } \\mathbb { E } _ { x } f ( w , x ) \\le f _ { p } + R = C + \\widehat { h } + R = C + \\rho ^ { 2 } \\sigma / 2 + R + O ( \\rho ^ { 3 } ) } \\\\ { w i t h p r o b a b i l i t y \\left( 1 - a \\right) [ 1 - e ^ { - ( \\frac { \\rho } { \\sqrt { 2 } b } - \\sqrt { k } ) ^ { 2 } } ] } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 272, + 739, + 725, + 785 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "where $R$ is the regularization term, $C$ is the training loss, $\\sigma$ is the dominant eigenvalue of Hessian. As in lemma 3.3, we perform Taylor-expansion and can ignore the high-order term $O ( \\rho ^ { 3 } )$ . We focus on ", + "bbox": [ + 174, + 787, + 821, + 829 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/a4c4871c27e9cd5aff90d0bbfaebf1c51736abe97c45a1184e05d9ae4fff9a26.jpg", + "text": "$$\nf _ { p } = C + \\widehat { h } = C + \\rho ^ { 2 } \\sigma / 2\n$$", + "text_format": "latex", + "bbox": [ + 408, + 825, + 589, + 845 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "2. When $\\rho _ { t r u e } \\neq \\rho _ { t r a i n }$ , minimizing $h$ achieves a lower risk than only minimizing $f _ { p }$ . (1) Note that after training, $C$ (training loss) is fixed, but $h$ could vary with $\\rho$ (e.g. when training on dataset A and testing on an unrelated dataset B, the training loss remains unchanged, but the risk would be huge and a large $\\rho$ is required for a valid bound). (2) With an example, we show a low $f _ { p }$ is insufficient for generalization, and a low $\\sigma$ is necessary: ", + "bbox": [ + 173, + 853, + 825, + 924 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "A Suppose we use $\\rho _ { t r a i n }$ for training, and consider two solutions with $C _ { 1 } , \\sigma _ { 1 }$ (SAM) and $C _ { 2 } , \\sigma _ { 2 }$ (GSAM). Suppose they have the same $f _ { p }$ during training for some $\\rho _ { t r a i n }$ , so $f _ { p 1 } = C _ { 1 } + \\sigma _ { 1 } / 2 \\times \\rho _ { t r a i n } ^ { 2 } = C _ { 2 } + \\sigma _ { 2 } / 2 \\times \\rho _ { t r a i n } ^ { 2 } = f _ { p 2 }$ Suppose $C _ { 1 } < C _ { 2 }$ so $\\sigma _ { 1 } > \\sigma _ { 2 }$ . \nB When $\\rho _ { t r u e } > \\rho _ { t r a i n }$ , we have risk bound $\\begin{array} { r } { - 1 = C _ { 1 } + \\sigma _ { 1 } / 2 \\times \\rho _ { t r u e } ^ { 2 } + R > \\mathrm { r i s k . b o u n d . } 2 = C _ { 2 } + \\sigma _ { 2 } / 2 \\times \\rho _ { t r u e } ^ { 2 } + R } \\end{array}$ This implies that a small $\\sigma$ helps generalization, but only a low $f _ { p 1 }$ (caused by a low $C _ { 1 }$ and high $\\sigma _ { 1 }$ ) is insufficient for a good generalization. \nC Note that $\\rho _ { t r a i n }$ is fixed during training, so minimizing $h _ { t r a i n }$ during training is equivalently minimizing $\\sigma$ by Lemma 3.3 ", + "bbox": [ + 209, + 102, + 826, + 290 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "3. Why we are often unlucky to have $\\rho _ { t r u e } ~ > ~ \\rho _ { t r a i n }$ (1) First, the test sets are almost surely outside the convex hull of the training set because “‘interpolation almost surely never occurs in high-dimensional $( > ~ 1 0 0 )$ cases”’ Balestriero et al. (2021). As a result, the variability of (train $^ +$ test) sets is almost surely larger than the variability of (train) set. Since $\\rho$ increases with data variability (see point 4 below), we have $\\rho _ { t r u e } > \\rho _ { t r a i n - s e t }$ almost surely. (2) Second, we don’t know the value of $\\rho _ { t r u e }$ and can only guess it. In practice, we often guess a small value because training often diverges with large $\\rho$ (as observed in Foret et al. (2020); Chen et al. (2021)). ", + "bbox": [ + 173, + 303, + 825, + 401 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "4. Why $\\rho$ increases with data variability. In Corollary 5.2.1, we assume weight perturbation $\\delta \\sim \\mathcal { N } ( 0 , b ^ { 2 } I ^ { k } )$ . The meaning of $b$ is the following. If we can randomly sample a fixed number of samples from the underlying distribution, then training the model from scratch (with a fixed seed for random initialization) gives rise to a set of weights. Repeating this process, we get many sets of weights, and their standard deviation is $b$ . Since the number of training samples is limited and fixed, the more variability in data, the more variability in weights, and the larger $b$ . Note that Corollary stated that the bound holds with probability proportional to $[ 1 - e ^ { - ( \\frac { \\rho } { \\sqrt { 2 } b } - \\sqrt { k } ) ^ { 2 } } ]$ . In order for the result to hold with a fixed probability, $\\rho$ must stay proportional to $b$ , hence $\\rho$ also increases with the variability of data. ", + "bbox": [ + 173, + 406, + 825, + 537 + ], + "page_idx": 20 + }, + { + "type": "table", + "img_path": "images/39cf1fa71daf880c7592c481853b96eb2a6664e17bfa442a306dff59d064b0b1.jpg", + "table_caption": [ + "Table 4: Hyper-parameters to reproduce experimental results " + ], + "table_footnote": [], + "table_body": "
ModelPmaxPminαlrmaxlrminWeight DecayBase OptimizerEpochsWarmup StepsLR schedule
ResNet500.040.020.011.61.6e-20.3SGD905kLinear
ResNet1010.040.020.011.61.6e-20.3SGD905kLinear
ResNet5120.040.020.0051.61.6e-20.3SGD905kLinear
ViT-S/320.60.00.43e-33e-50.3AdamW30010kLinear
ViT-S/160.60.01.03e-33e-50.3AdamW30010kLinear
ViT-B/320.60.10.63e-33e-50.3AdamW30010kLinear
ViT-B/160.60.20.43e-33e-50.3AdamW30010kLinear
Mixer-S/320.50.00.23e-33e-50.3AdamW30010kLinear
Mixer-S/160.50.00.63e-33e-50.3AdamW30010kLinear
Mixer-S/80.50.10.13e-33e-50.3AdamW30010kLinear
Mixer-B/320.70.20.053e-33e-50.3AdamW30010kLinear
Mixer-B/160.50.20.013e-33e-50.3AdamW30010kLinear
", + "bbox": [ + 176, + 126, + 897, + 275 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "B EXPERIMENTAL DETAILS ", + "text_level": 1, + "bbox": [ + 176, + 300, + 419, + 316 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "B.1 TRAINING DETAILS ", + "text_level": 1, + "bbox": [ + 176, + 332, + 352, + 347 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "For ViT and Mixer, we search the learning rate in $\\{ 1 \\mathrm { e } { - } 3 , 3 \\mathrm { e } { - } 3 , 1 \\mathrm { e } { - } 2 , 3 \\mathrm { e } { - } 3 \\}$ , and search weight decay in $\\{ 0 . 0 0 3 , 0 . 0 3 , 0 . 3 \\}$ . For ResNet, we search the learning rate in $\\left. 1 . 6 , 0 . 1 6 , 0 . 0 1 6 \\right.$ , and search the weight decay in $\\{ 0 . 0 0 1 , 0 . 0 1 , 0 . 1 \\}$ . For ViT and Mixer, we use the AdamW optimizer with $\\beta _ { 1 } = 0 . 9 , \\beta _ { 2 } = 0 . 9 9 $ ; for ResNet we use SGD with momentum $\\ l = 0 . 9$ . We train ResNets for 90 epochs, and train ViTs and Mixers for 300 epochs following the settings in (Chen et al., 2021) and (Dosovitskiy et al., 2020). Considering that SAM and GSAM uses twice the computation of vanilla training for each step, for vanilla training we try $2 \\times$ longer training, and does not find significant improvement as in Table. 5. ", + "bbox": [ + 173, + 358, + 825, + 469 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "We first search the optimal learning rate and weight decay for vanilla training, and keep these two hyper-parameters fixed for SAM and GSAM. For ViT and Mixer, we search $\\rho$ in $\\{ 0 . 1 , \\bar { 0 . 2 } , 0 . 3 , 0 . 4 ,$ $0 . 5 , 0 . 6 \\}$ for SAM and GSAM; for ResNet, we search $\\rho$ from 0.01 to 0.05 with a stepsize 0.01. For ASAM, we amplify $\\rho$ by $1 0 \\times$ compared to SAM, as recommended by Kwon et al. (2021). For GSAM, we search $\\alpha$ in $\\{ 0 . 1 , 0 . 2 , 0 . 3 \\}$ throughout the paper. We report the best configuration of each individual model in Table. 4. ", + "bbox": [ + 174, + 477, + 825, + 560 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "B.2 TRANSFER LEARNING EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 178, + 578, + 472, + 592 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Using weights trained on ImageNet-1k, we finetune models with SGD on downstream tasks including the CIFAR10/CIFAR100 (Krizhevsky et al., 2009), Oxford-flowers (Nilsback & Zisserman, 2008) and Oxford-IITPets (Parkhi et al., 2012). For all experiments, we use the SGD optimizer with no weight decay under a linear learning rate schedule and gradient clipping with global norm 1. We search the maximum learning rate in $\\left. 0 . 0 0 1 , 0 . 0 0 3 , 0 . 0 1 , 0 . 0 3 \\right.$ . On Cifar datasets, we train models for 10k steps with a warmup step of 500; on Oxford datasets, we train models for 500 steps with a wamup step of 100. ", + "bbox": [ + 174, + 604, + 825, + 702 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "B.3 EXPERIMENTAL SETUP WITH ABLATION STUDIES ON DATA AUGMENTATION ", + "text_level": 1, + "bbox": [ + 178, + 720, + 735, + 734 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "We follow the settings in (Tolstikhin et al., 2021) to perform ablation studies on data augmentation. In the left subfigure of Fig. 6, “Light” refers to Inception-style data augmentation with random flip and crop of images, “Medium” refers to the mixup augmentation with probability 0.2 and RandAug magnitude 10; “Strong” refers to the mixup augmentation with probability 0.2 and RandAug magnitude 15. ", + "bbox": [ + 174, + 746, + 825, + 815 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "C ABLATION STUDIES AND DISCUSSIONS ", + "text_level": 1, + "bbox": [ + 174, + 838, + 532, + 853 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "C.1 INFLUENCE OF $\\rho$ AND $\\alpha$ ", + "text_level": 1, + "bbox": [ + 176, + 868, + 382, + 883 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "We plot the performance of a ViT-B/32 model varying with $\\rho$ (Fig. 7a) and $\\alpha$ (Fig. 7b). We empirically validate that fine-tuning $\\rho$ in SAM can not achieve comparable performance with GSAM, as ", + "bbox": [ + 174, + 895, + 823, + 924 + ], + "page_idx": 21 + }, + { + "type": "image", + "img_path": "images/24eba187fef4c92575a570adcf3b277bd800c56050a22796582502f9883ba719.jpg", + "image_caption": [ + "Figure 7: Performance of GSAM varying with $\\rho$ and $\\alpha$ " + ], + "image_footnote": [], + "bbox": [ + 176, + 102, + 823, + 296 + ], + "page_idx": 22 + }, + { + "type": "table", + "img_path": "images/94b5e0557e32f4de068eaf1676e623436219e5c38ae6893c5f93c736e1d05370.jpg", + "table_caption": [ + "Table 5: Top-1 accuracy of ViT-B/32 on ImageNet with Inception-style data augmentation. For vanilla training we report results for training 300 epochs and 600 epochs, for GSAM we report the results for 300 epochs. " + ], + "table_footnote": [], + "table_body": "
MethodEpochsImageNetImageNet-RealImageNet-v2ImageNet-R
Vanilla30071.477.557.523.4
60072.078.257.923.6
GSAM30076.882.763.025.1
", + "bbox": [ + 222, + 397, + 774, + 458 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "shown in Fig. 7a. Considering that GSAM has one more parameter $\\alpha$ , we plot the accuracy varying with $\\alpha$ in Fig. 7b, and show that GSAM consistently outperforms SAM and vanilla training. ", + "bbox": [ + 174, + 489, + 825, + 518 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "C.2 CONSTANT $\\rho$ V.S. DECAYED $\\rho _ { t }$ SCHEDULE ", + "text_level": 1, + "bbox": [ + 173, + 544, + 506, + 558 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Note that Thm. 5.1 assumes $\\rho _ { t }$ to decay with $t$ in order to prove the convergence, while SAM uses a constant $\\rho$ during training. To eliminate the influence of $\\rho _ { t }$ schedule, we conduct ablation study as in Table. 6. The ascent step in GSAM can be applied to both constant $\\rho$ or a decayed $\\rho _ { t }$ schedule, and improves accuracy for both cases. Without ascent step, constant $\\rho$ and decayed $\\rho _ { t }$ achieve similar performance. Results in Table. 6 implies that the ascent step in GSAM is the main reason for improvement of generalization performance. ", + "bbox": [ + 173, + 571, + 826, + 656 + ], + "page_idx": 22 + }, + { + "type": "image", + "img_path": "images/9b390274fa28b5ba47c294dfef588e29c76201cc8670a2da5a44ef3d22931fcd.jpg", + "image_caption": [ + "Figure 8: The value of $\\cos \\theta _ { t }$ varying with training steps, where $\\theta _ { t }$ is the angle between $\\nabla f ( w _ { t } )$ and $\\nabla f _ { p } ( w _ { t } )$ as in Fig. 2. " + ], + "image_footnote": [], + "bbox": [ + 178, + 689, + 490, + 863 + ], + "page_idx": 22 + }, + { + "type": "image", + "img_path": "images/c33dd62d4567c39ae55b11964e44b74a04f56f0f5e2ffdc6a47b45abc7f9be5f.jpg", + "image_caption": [ + "Figure 9: Surrogate gap curve under different $\\alpha$ values. " + ], + "image_footnote": [], + "bbox": [ + 509, + 695, + 818, + 869 + ], + "page_idx": 22 + }, + { + "type": "table", + "img_path": "images/163292e6a8281310681be67d4f4d1e3727cf3cd40f7fba58ee488688c9368395.jpg", + "table_caption": [ + "Table 6: Top-1 Accuracy on ViT-B/32 on ImageNet. Ablation studies on constant $\\rho$ or a decayed $\\rho _ { t }$ " + ], + "table_footnote": [], + "table_body": "
VanillaConstant p (SAM)Constant ρ+ascentDecayed ptIDecayed pt+ascent
72.075.876.275.876.8
", + "bbox": [ + 171, + 126, + 779, + 160 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "C.3 VISUALIZE THE TRAINING PROCESS ", + "text_level": 1, + "bbox": [ + 178, + 184, + 464, + 199 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "In the proof of Thm. 5.3, our analysis relies on assumption that $\\theta _ { t }$ is small. We empirically validated this assumption by plotting $\\cos \\theta _ { t }$ in Fig. 8, where $\\theta _ { t }$ is the angle between $\\nabla f ( \\bar { w } _ { t } )$ and $\\nabla f _ { p } ( w _ { t } )$ . Note that the cosine value is calculated in the parameter space of dimension $8 . 8 \\times 1 0 ^ { 7 }$ , and in high-dimensional space two random vectors are highly likely to be perpendicular. In Fig. 8 the cosine value is always above 0.9, indicating that $\\nabla f ( w _ { t } )$ and $\\nabla f _ { p } ( w _ { t } )$ point to very close directions considering the high dimension of parameters. This empirically validates our assumption that $\\theta _ { t }$ is small during training. ", + "bbox": [ + 174, + 210, + 825, + 309 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "We also plot the surrogate gap during training in Fig. 9. As $\\alpha$ increases, the surrogate gap decreases, validating that the ascent step in GSAM efficiently minimizes the surrogate gap. Furthermore, the surrogate gap increases with training steps for any fixed $\\alpha$ , indicating that the training process gradually falls into local minimum in order to minimize the training loss. ", + "bbox": [ + 174, + 315, + 825, + 372 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "D RELATED WORKS ", + "text_level": 1, + "bbox": [ + 174, + 392, + 354, + 407 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Besides SAM and ASAM, other methods were proposed in the literature to improve generalization: Lin et al. (2020) proposed extrapolation of gradient, Xie et al. (2021) proposed to manipulate the noise in gradient, and Damian et al. (2021) proved label noise improves generalization, Yue et al. (2020) proposed to adjust learning rate according to sharpness, and Zheng et al. (2021) proposed model perturbation with similar idea to SAM. Izmailov et al. (2018) proposed averaging weights to improve generalization, and Heo et al. (2020) restricted the norm of updated weights to improve generalization. Many of aforementioned methods can be combined with GSAM to further improve generalization. ", + "bbox": [ + 174, + 424, + 825, + 535 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Besides modified training schemes, there are other two types of techniques to improve generalization: data augmentation and model regularization. Data augmentation typically generates new data from training samples; besides standard data augmentation such as flipping or rotation of images, recent data augmentations include label smoothing (Muller et al., 2019) and mixup (M ¨ uller et al., ¨ 2019) which trains on convex combinations of both inputs and labels, automatically learned augmentation (Cubuk et al., 2018), and cutout (DeVries & Taylor, 2017) which randomly masks out parts of an image. Model regularization typically applies auxiliary losses besides the training loss such as weight decay (Loshchilov & Hutter, 2017), other methods randomly modify the model architecture during training, such as dropout (Srivastava et al., 2014) and shake-shake regularization (Gastaldi, 2017). Note that the data augmentation and model regularization literature mentioned here typically train with the standard back-propagation (Rumelhart et al., 1985) and first-order gradient optimizers, and both techniques can be combined with GSAM. ", + "bbox": [ + 174, + 542, + 825, + 708 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Besides SGD, Adam and AdaBelief, GSAM can be combined with other first-order gradient optimizers, such as AdaBound (Luo et al., 2019), RAdam (Liu et al., 2019), Yogi (Zaheer et al., 2018), AdaGrad (Duchi et al., 2011), AMSGrad (Reddi et al., 2019) and AdaDelta (Zeiler, 2012). 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However, we show that both sharp and flat", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 323, + 470, + 336 + ], + "spans": [ + { + "bbox": [ + 141, + 323, + 470, + 336 + ], + "score": 1.0, + "content": "minima can have a low perturbed loss, implying that SAM does not always prefer", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 140, + 333, + 470, + 348 + ], + "spans": [ + { + "bbox": [ + 140, + 333, + 470, + 348 + ], + "score": 1.0, + "content": "flat minima. Instead, we define a surrogate gap, a measure equivalent to the dom-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 345, + 470, + 357 + ], + "spans": [ + { + "bbox": [ + 141, + 345, + 470, + 357 + ], + "score": 1.0, + "content": "inant eigenvalue of Hessian at a local minimum when the radius of neighborhood", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 356, + 470, + 369 + ], + "spans": [ + { + "bbox": [ + 141, + 356, + 470, + 369 + ], + "score": 1.0, + "content": "(to derive the perturbed loss) is small. The surrogate gap is easy to compute and", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 366, + 470, + 379 + ], + "spans": [ + { + "bbox": [ + 141, + 366, + 470, + 379 + ], + "score": 1.0, + "content": "feasible for direct minimization during training. Based on the above observations,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 378, + 470, + 391 + ], + "spans": [ + { + "bbox": [ + 141, + 378, + 470, + 391 + ], + "score": 1.0, + "content": "we propose Surrogate Gap Guided Sharpness-Aware Minimization (GSAM), a", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 389, + 469, + 401 + ], + "spans": [ + { + "bbox": [ + 141, + 389, + 469, + 401 + ], + "score": 1.0, + "content": "novel improvement over SAM with negligible computation overhead. Conceptu-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 141, + 399, + 470, + 412 + ], + "spans": [ + { + "bbox": [ + 141, + 399, + 470, + 412 + ], + "score": 1.0, + "content": "ally, GSAM consists of two steps: 1) a gradient descent like SAM to minimize", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 141, + 410, + 470, + 423 + ], + "spans": [ + { + "bbox": [ + 141, + 410, + 470, + 423 + ], + "score": 1.0, + "content": "the perturbed loss, and 2) an ascent step in the orthogonal direction (after gradi-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 141, + 421, + 470, + 434 + ], + "spans": [ + { + "bbox": [ + 141, + 421, + 470, + 434 + ], + "score": 1.0, + "content": "ent decomposition) to minimize the surrogate gap and yet not affect the perturbed", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 141, + 432, + 469, + 446 + ], + "spans": [ + { + "bbox": [ + 141, + 432, + 469, + 446 + ], + "score": 1.0, + "content": "loss. GSAM seeks a region with both small loss (by step 1) and low sharpness (by", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 142, + 444, + 469, + 456 + ], + "spans": [ + { + "bbox": [ + 142, + 444, + 469, + 456 + ], + "score": 1.0, + "content": "step 2), giving rise to a model with high generalization capabilities. Theoretically,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 141, + 454, + 470, + 467 + ], + "spans": [ + { + "bbox": [ + 141, + 454, + 470, + 467 + ], + "score": 1.0, + "content": "we show the convergence of GSAM and provably better generalization than SAM.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 141, + 465, + 469, + 478 + ], + "spans": [ + { + "bbox": [ + 141, + 465, + 396, + 478 + ], + "score": 1.0, + "content": "Empirically, GSAM consistently improves generalization (e.g.,", + "type": "text" + }, + { + "bbox": [ + 396, + 465, + 424, + 476 + ], + "score": 0.88, + "content": "+ 3 . 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 465, + 469, + 478 + ], + "score": 1.0, + "content": "over SAM", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 141, + 476, + 469, + 488 + ], + "spans": [ + { + "bbox": [ + 141, + 476, + 160, + 488 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 160, + 477, + 188, + 487 + ], + "score": 0.9, + "content": "+ 5 . 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 476, + 469, + 488 + ], + "score": 1.0, + "content": "over AdamW on ImageNet top-1 accuracy for ViT-B/32). Code is", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 141, + 487, + 468, + 500 + ], + "spans": [ + { + "bbox": [ + 141, + 487, + 468, + 500 + ], + "score": 1.0, + "content": "released at https://sites.google.com/view/gsam-iclr22/home.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 21 + }, + { + "type": "title", + "bbox": [ + 108, + 519, + 206, + 531 + ], + "lines": [ + { + "bbox": [ + 105, + 518, + 208, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 208, + 534 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 543, + 505, + 653 + ], + "lines": [ + { + "bbox": [ + 105, + 542, + 506, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 506, + 557 + ], + "score": 1.0, + "content": "Modern neural networks are typically highly over-parameterized and easy to overfit to training data,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 554, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 505, + 568 + ], + "score": 1.0, + "content": "yet the generalization performances on unseen data (test set) often suffer a gap from the training", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 565, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 505, + 579 + ], + "score": 1.0, + "content": "performance (Zhang et al., 2017a). Many studies try to understand the generalization of machine", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 576, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 505, + 590 + ], + "score": 1.0, + "content": "learning models, including the Bayesian perspective (McAllester, 1999; Neyshabur et al., 2017),", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 587, + 506, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 506, + 600 + ], + "score": 1.0, + "content": "the information perspective (Liang et al., 2019), the loss surface geometry perspective (Hochreiter", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 597, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 505, + 612 + ], + "score": 1.0, + "content": "& Schmidhuber, 1995; Jiang et al., 2019) and the kernel perspective (Jacot et al., 2018; Wei et al.,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 609, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 505, + 622 + ], + "score": 1.0, + "content": "2019). Besides analyzing the properties of a model after training, some works study the influence", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 621, + 505, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 505, + 633 + ], + "score": 1.0, + "content": "of training and the optimization process, such as the implicit regularization of stochastic gradient", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 631, + 506, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 506, + 644 + ], + "score": 1.0, + "content": "descent (SGD) (Bottou, 2010; Zhou et al., 2020), the learning rate’s regularization effect (Li et al.,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 642, + 358, + 654 + ], + "spans": [ + { + "bbox": [ + 106, + 642, + 358, + 654 + ], + "score": 1.0, + "content": "2019), and the influence of the batch size (Keskar et al., 2016).", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 36.5 + }, + { + "type": "text", + "bbox": [ + 108, + 659, + 504, + 714 + ], + "lines": [ + { + "bbox": [ + 105, + 658, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 658, + 505, + 672 + ], + "score": 1.0, + "content": "These studies have led to various modifications to the training process to improve generalization.", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 670, + 506, + 684 + ], + "spans": [ + { + "bbox": [ + 106, + 670, + 506, + 684 + ], + "score": 1.0, + "content": "Keskar & Socher (2017) proposed to use Adam in early training phases for fast convergence and", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 681, + 506, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 506, + 694 + ], + "score": 1.0, + "content": "then switch to SGD in late phases for better generalization. Izmailov et al. (2018) proposed to", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 104, + 690, + 506, + 707 + ], + "spans": [ + { + "bbox": [ + 104, + 690, + 506, + 707 + ], + "score": 1.0, + "content": "average weights to achieve a wider local minimum, which is expected to generalize better than sharp", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 703, + 505, + 716 + ], + "spans": [ + { + "bbox": [ + 105, + 703, + 505, + 716 + ], + "score": 1.0, + "content": "minima. A similar idea was later used in Lookahead (Zhang et al., 2019). 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However, we show that both sharp and flat", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 323, + 470, + 336 + ], + "spans": [ + { + "bbox": [ + 141, + 323, + 470, + 336 + ], + "score": 1.0, + "content": "minima can have a low perturbed loss, implying that SAM does not always prefer", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 140, + 333, + 470, + 348 + ], + "spans": [ + { + "bbox": [ + 140, + 333, + 470, + 348 + ], + "score": 1.0, + "content": "flat minima. Instead, we define a surrogate gap, a measure equivalent to the dom-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 345, + 470, + 357 + ], + "spans": [ + { + "bbox": [ + 141, + 345, + 470, + 357 + ], + "score": 1.0, + "content": "inant eigenvalue of Hessian at a local minimum when the radius of neighborhood", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 356, + 470, + 369 + ], + "spans": [ + { + "bbox": [ + 141, + 356, + 470, + 369 + ], + "score": 1.0, + "content": "(to derive the perturbed loss) is small. The surrogate gap is easy to compute and", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 366, + 470, + 379 + ], + "spans": [ + { + "bbox": [ + 141, + 366, + 470, + 379 + ], + "score": 1.0, + "content": "feasible for direct minimization during training. Based on the above observations,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 378, + 470, + 391 + ], + "spans": [ + { + "bbox": [ + 141, + 378, + 470, + 391 + ], + "score": 1.0, + "content": "we propose Surrogate Gap Guided Sharpness-Aware Minimization (GSAM), a", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 389, + 469, + 401 + ], + "spans": [ + { + "bbox": [ + 141, + 389, + 469, + 401 + ], + "score": 1.0, + "content": "novel improvement over SAM with negligible computation overhead. Conceptu-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 141, + 399, + 470, + 412 + ], + "spans": [ + { + "bbox": [ + 141, + 399, + 470, + 412 + ], + "score": 1.0, + "content": "ally, GSAM consists of two steps: 1) a gradient descent like SAM to minimize", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 141, + 410, + 470, + 423 + ], + "spans": [ + { + "bbox": [ + 141, + 410, + 470, + 423 + ], + "score": 1.0, + "content": "the perturbed loss, and 2) an ascent step in the orthogonal direction (after gradi-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 141, + 421, + 470, + 434 + ], + "spans": [ + { + "bbox": [ + 141, + 421, + 470, + 434 + ], + "score": 1.0, + "content": "ent decomposition) to minimize the surrogate gap and yet not affect the perturbed", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 141, + 432, + 469, + 446 + ], + "spans": [ + { + "bbox": [ + 141, + 432, + 469, + 446 + ], + "score": 1.0, + "content": "loss. GSAM seeks a region with both small loss (by step 1) and low sharpness (by", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 142, + 444, + 469, + 456 + ], + "spans": [ + { + "bbox": [ + 142, + 444, + 469, + 456 + ], + "score": 1.0, + "content": "step 2), giving rise to a model with high generalization capabilities. Theoretically,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 141, + 454, + 470, + 467 + ], + "spans": [ + { + "bbox": [ + 141, + 454, + 470, + 467 + ], + "score": 1.0, + "content": "we show the convergence of GSAM and provably better generalization than SAM.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 141, + 465, + 469, + 478 + ], + "spans": [ + { + "bbox": [ + 141, + 465, + 396, + 478 + ], + "score": 1.0, + "content": "Empirically, GSAM consistently improves generalization (e.g.,", + "type": "text" + }, + { + "bbox": [ + 396, + 465, + 424, + 476 + ], + "score": 0.88, + "content": "+ 3 . 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 465, + 469, + 478 + ], + "score": 1.0, + "content": "over SAM", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 141, + 476, + 469, + 488 + ], + "spans": [ + { + "bbox": [ + 141, + 476, + 160, + 488 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 160, + 477, + 188, + 487 + ], + "score": 0.9, + "content": "+ 5 . 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 476, + 469, + 488 + ], + "score": 1.0, + "content": "over AdamW on ImageNet top-1 accuracy for ViT-B/32). Code is", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 141, + 487, + 468, + 500 + ], + "spans": [ + { + "bbox": [ + 141, + 487, + 468, + 500 + ], + "score": 1.0, + "content": "released at https://sites.google.com/view/gsam-iclr22/home.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 21, + "bbox_fs": [ + 140, + 289, + 470, + 500 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 519, + 206, + 531 + ], + "lines": [ + { + "bbox": [ + 105, + 518, + 208, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 208, + 534 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 543, + 505, + 653 + ], + "lines": [ + { + "bbox": [ + 105, + 542, + 506, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 506, + 557 + ], + "score": 1.0, + "content": "Modern neural networks are typically highly over-parameterized and easy to overfit to training data,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 554, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 505, + 568 + ], + "score": 1.0, + "content": "yet the generalization performances on unseen data (test set) often suffer a gap from the training", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 565, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 505, + 579 + ], + "score": 1.0, + "content": "performance (Zhang et al., 2017a). Many studies try to understand the generalization of machine", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 576, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 505, + 590 + ], + "score": 1.0, + "content": "learning models, including the Bayesian perspective (McAllester, 1999; Neyshabur et al., 2017),", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 587, + 506, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 506, + 600 + ], + "score": 1.0, + "content": "the information perspective (Liang et al., 2019), the loss surface geometry perspective (Hochreiter", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 597, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 505, + 612 + ], + "score": 1.0, + "content": "& Schmidhuber, 1995; Jiang et al., 2019) and the kernel perspective (Jacot et al., 2018; Wei et al.,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 609, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 505, + 622 + ], + "score": 1.0, + "content": "2019). Besides analyzing the properties of a model after training, some works study the influence", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 621, + 505, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 505, + 633 + ], + "score": 1.0, + "content": "of training and the optimization process, such as the implicit regularization of stochastic gradient", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 631, + 506, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 506, + 644 + ], + "score": 1.0, + "content": "descent (SGD) (Bottou, 2010; Zhou et al., 2020), the learning rate’s regularization effect (Li et al.,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 642, + 358, + 654 + ], + "spans": [ + { + "bbox": [ + 106, + 642, + 358, + 654 + ], + "score": 1.0, + "content": "2019), and the influence of the batch size (Keskar et al., 2016).", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 36.5, + "bbox_fs": [ + 105, + 542, + 506, + 654 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 659, + 504, + 714 + ], + "lines": [ + { + "bbox": [ + 105, + 658, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 658, + 505, + 672 + ], + "score": 1.0, + "content": "These studies have led to various modifications to the training process to improve generalization.", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 670, + 506, + 684 + ], + "spans": [ + { + "bbox": [ + 106, + 670, + 506, + 684 + ], + "score": 1.0, + "content": "Keskar & Socher (2017) proposed to use Adam in early training phases for fast convergence and", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 681, + 506, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 506, + 694 + ], + "score": 1.0, + "content": "then switch to SGD in late phases for better generalization. Izmailov et al. (2018) proposed to", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 104, + 690, + 506, + 707 + ], + "spans": [ + { + "bbox": [ + 104, + 690, + 506, + 707 + ], + "score": 1.0, + "content": "average weights to achieve a wider local minimum, which is expected to generalize better than sharp", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 703, + 505, + 716 + ], + "spans": [ + { + "bbox": [ + 105, + 703, + 505, + 716 + ], + "score": 1.0, + "content": "minima. A similar idea was later used in Lookahead (Zhang et al., 2019). Entropy-SGD (Chaudhari", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 82, + 504, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 504, + 95 + ], + "score": 1.0, + "content": "et al., 2019) derived the gradient of local entropy to avoid solutions in sharp valleys. Entropy-SGD", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 484, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 484, + 107 + ], + "score": 1.0, + "content": "has a nested Langevin iteration, inducing much higher computation costs than vanilla training.", + "type": "text", + "cross_page": true + } + ], + "index": 1 + } + ], + "index": 44, + "bbox_fs": [ + 104, + 658, + 506, + 716 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 82, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 504, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 504, + 95 + ], + "score": 1.0, + "content": "et al., 2019) derived the gradient of local entropy to avoid solutions in sharp valleys. Entropy-SGD", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 484, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 484, + 107 + ], + "score": 1.0, + "content": "has a nested Langevin iteration, inducing much higher computation costs than vanilla training.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 106, + 110, + 505, + 212 + ], + "lines": [ + { + "bbox": [ + 105, + 109, + 505, + 124 + ], + "spans": [ + { + "bbox": [ + 105, + 109, + 505, + 124 + ], + "score": 1.0, + "content": "The recently proposed Sharpness-Aware Minimization (SAM) (Foret et al., 2020) is a generic train-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 122, + 506, + 133 + ], + "spans": [ + { + "bbox": [ + 106, + 122, + 506, + 133 + ], + "score": 1.0, + "content": "ing scheme that improves generalization and has been shown especially effective for Vision Trans-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 132, + 505, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 505, + 145 + ], + "score": 1.0, + "content": "formers (Dosovitskiy et al., 2020) when large-scale pre-training is unavailable (Chen et al., 2021).", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 143, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 268, + 156 + ], + "score": 1.0, + "content": "Suppose vanilla training minimizes loss", + "type": "text" + }, + { + "bbox": [ + 268, + 144, + 290, + 155 + ], + "score": 0.92, + "content": "f ( w )", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 143, + 505, + 156 + ], + "score": 1.0, + "content": "(e.g., the cross-entropy loss for classification), where", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 154, + 507, + 172 + ], + "spans": [ + { + "bbox": [ + 106, + 158, + 115, + 166 + ], + "score": 0.72, + "content": "w", + "type": "inline_equation" + }, + { + "bbox": [ + 116, + 154, + 374, + 172 + ], + "score": 1.0, + "content": "is the parameter. SAM minimizes a perturbed loss defined as", + "type": "text" + }, + { + "bbox": [ + 375, + 155, + 501, + 169 + ], + "score": 0.91, + "content": "f _ { p } ( w ) \\triangleq \\operatorname* { m a x } _ { | | \\delta | | \\leq \\rho } f ( w + \\delta )", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 154, + 507, + 172 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 167, + 506, + 179 + ], + "spans": [ + { + "bbox": [ + 106, + 167, + 274, + 179 + ], + "score": 1.0, + "content": "which is the maximum loss within radius", + "type": "text" + }, + { + "bbox": [ + 274, + 169, + 281, + 178 + ], + "score": 0.8, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 167, + 415, + 179 + ], + "score": 1.0, + "content": "centered at the model parameter", + "type": "text" + }, + { + "bbox": [ + 415, + 169, + 424, + 177 + ], + "score": 0.71, + "content": "w", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 167, + 506, + 179 + ], + "score": 1.0, + "content": ". Intuitively, vanilla", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 178, + 505, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 178, + 505, + 190 + ], + "score": 1.0, + "content": "training seeks a single point with a low loss, while SAM searches for a neighborhood within which", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 189, + 506, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 189, + 392, + 201 + ], + "score": 1.0, + "content": "the maximum loss is low. However, we show that a low perturbed loss", + "type": "text" + }, + { + "bbox": [ + 393, + 189, + 403, + 201 + ], + "score": 0.89, + "content": "f _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 189, + 506, + 201 + ], + "score": 1.0, + "content": "could appear in both flat", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 200, + 445, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 200, + 307, + 213 + ], + "score": 1.0, + "content": "and sharp minima, implying that only minimizing", + "type": "text" + }, + { + "bbox": [ + 308, + 200, + 318, + 212 + ], + "score": 0.89, + "content": "f _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 200, + 445, + 213 + ], + "score": 1.0, + "content": "is not always sharpness-aware.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 106, + 216, + 505, + 307 + ], + "lines": [ + { + "bbox": [ + 105, + 216, + 506, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 221, + 231 + ], + "score": 1.0, + "content": "Although the perturbed loss", + "type": "text" + }, + { + "bbox": [ + 221, + 217, + 246, + 229 + ], + "score": 0.93, + "content": "f _ { p } ( w )", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 216, + 506, + 231 + ], + "score": 1.0, + "content": "might disagree with sharpness, we find a surrogate gap defined", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 228, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 229, + 118, + 244 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 118, + 228, + 214, + 242 + ], + "score": 0.92, + "content": "h ( w ) \\triangleq f _ { p } ( w ) - f ( w )", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 229, + 487, + 244 + ], + "score": 1.0, + "content": "agrees with sharpness — Lemma 3.3 shows that the surrogate gap", + "type": "text" + }, + { + "bbox": [ + 487, + 230, + 494, + 240 + ], + "score": 0.76, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 229, + 506, + 244 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 241, + 505, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 505, + 253 + ], + "score": 1.0, + "content": "an equivalent measure of the dominant eigenvalue of Hessian at a local minimum. Inspired by this", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 251, + 505, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 505, + 264 + ], + "score": 1.0, + "content": "observation, we propose the Surrogate Gap Guided Sharpness Aware Minimization (GSAM) which", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 263, + 506, + 276 + ], + "spans": [ + { + "bbox": [ + 104, + 263, + 254, + 276 + ], + "score": 1.0, + "content": "jointly minimizes the perturbed loss", + "type": "text" + }, + { + "bbox": [ + 255, + 263, + 266, + 275 + ], + "score": 0.89, + "content": "f _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 263, + 357, + 276 + ], + "score": 1.0, + "content": "and the surrogate gap", + "type": "text" + }, + { + "bbox": [ + 357, + 263, + 364, + 272 + ], + "score": 0.71, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 263, + 454, + 276 + ], + "score": 1.0, + "content": ": a low perturbed 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test performances (Thm. 5.3).", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 295, + 460, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 460, + 308 + ], + "score": 1.0, + "content": "When both criteria are satisfied, we find a generalizable model with good performances.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 107, + 312, + 505, + 369 + ], + "lines": [ + { + "bbox": [ + 105, + 310, + 506, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 310, + 366, + 326 + ], + "score": 1.0, + "content": "GSAM consists of two steps for each update: 1) descend gradient", + "type": "text" + }, + { + "bbox": [ + 366, + 312, + 399, + 325 + ], + "score": 0.93, + "content": "\\nabla f _ { p } ( w )", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 310, + 506, + 326 + ], + "score": 1.0, + "content": "to minimize the perturbed", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 322, 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"\\nabla _ { \\perp } f ( w )", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 345, + 436, + 360 + ], + "score": 1.0, + "content": "to minimize the surrogate gap", + "type": "text" + }, + { + "bbox": [ + 437, + 347, + 458, + 358 + ], + "score": 0.9, + "content": "h ( w )", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 345, + 506, + 360 + ], + "score": 1.0, + "content": ". Note that", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 356, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 308, + 371 + ], + "score": 1.0, + "content": "this ascent step does not change the perturbed loss", + "type": "text" + }, + { + "bbox": [ + 309, + 358, + 320, + 370 + ], + "score": 0.89, + "content": "f _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 356, + 354, + 371 + ], + "score": 1.0, + "content": "because", + "type": "text" + }, + { + "bbox": [ + 355, + 357, + 437, + 370 + ], + "score": 0.92, + "content": "\\nabla f _ { \\perp } ( w ) \\perp \\mathbf { \\bar { \\nabla } } \\nabla f _ { p } ( \\bar { w } )", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 356, + 505, + 371 + ], + "score": 1.0, + "content": "by construction.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 109, + 374, + 279, + 385 + ], + "lines": [ + { + "bbox": [ + 107, + 373, + 281, + 387 + ], + "spans": [ + { + "bbox": [ + 107, + 373, + 281, + 387 + ], + "score": 1.0, + "content": "We summarize our contribution as follows:", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 117, + 394, + 506, + 485 + ], + "lines": [ + { + "bbox": [ + 115, + 394, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 115, + 394, + 505, + 408 + ], + "score": 1.0, + "content": "• We define surrogate gap, which measures the sharpness at local minima and is easy to compute.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 115, + 408, + 506, + 423 + ], + "spans": [ + { + "bbox": [ + 115, + 408, + 506, + 423 + ], + "score": 1.0, + "content": "• We propose the GSAM method to improve the generalization of neural networks. GSAM is", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 126, + 421, + 455, + 434 + ], + "spans": [ + { + "bbox": [ + 126, + 421, + 455, + 434 + ], + "score": 1.0, + "content": "widely applicable and incurs negligible computation overhead compared to SAM.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 116, + 435, + 497, + 447 + ], + "spans": [ + { + "bbox": [ + 116, + 435, + 497, + 447 + ], + "score": 1.0, + "content": "• We demonstrate the convergence of GSAM and its provably better generalization than SAM.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 116, + 450, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 116, + 450, + 505, + 463 + ], + "score": 1.0, + "content": "• We empirically validate GSAM over image classification tasks with various neural architec-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 126, + 461, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 126, + 461, + 506, + 474 + ], + "score": 1.0, + "content": "tures, including ResNets (He et al., 2016), Vision Transformers (Dosovitskiy et al., 2020), and", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 126, + 473, + 278, + 484 + ], + "spans": [ + { + "bbox": [ + 126, + 473, + 278, + 484 + ], + "score": 1.0, + "content": "MLP-Mixers (Tolstikhin et al., 2021).", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28 + }, + { + "type": "title", + "bbox": [ + 107, + 500, + 208, + 513 + ], + "lines": [ + { + "bbox": [ + 104, + 498, + 209, + 515 + ], + "spans": [ + { + "bbox": [ + 104, + 498, + 209, + 515 + ], + "score": 1.0, + "content": "2 PRELIMINARIES", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "title", + "bbox": [ + 107, + 524, + 182, + 536 + ], + "lines": [ + { + "bbox": [ + 105, + 523, + 183, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 183, + 538 + ], + "score": 1.0, + "content": "2.1 NOTATIONS", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 115, + 542, + 470, + 554 + ], + "lines": [ + { + "bbox": [ + 115, + 540, + 472, + 555 + ], + "spans": [ + { + "bbox": [ + 115, + 540, + 126, + 555 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 127, + 542, + 149, + 554 + ], + "score": 0.9, + "content": "f ( w )", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 540, + 217, + 555 + ], + "score": 1.0, + "content": ": A loss function", + "type": "text" + }, + { + "bbox": [ + 217, + 542, + 224, + 554 + ], + "score": 0.84, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 540, + 288, + 555 + ], + "score": 1.0, + "content": "with parameter", + "type": "text" + }, + { + "bbox": [ + 288, + 541, + 320, + 552 + ], + "score": 0.92, + "content": "w \\in \\mathbb { R } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 540, + 351, + 555 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 351, + 542, + 358, + 552 + ], + "score": 0.83, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 540, + 472, + 555 + ], + "score": 1.0, + "content": "is the parameter dimension.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 117, + 557, + 416, + 569 + ], + "lines": [ + { + "bbox": [ + 115, + 556, + 417, + 571 + ], + "spans": [ + { + "bbox": [ + 115, + 556, + 126, + 571 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 126, + 557, + 155, + 569 + ], + "score": 0.89, + "content": "\\rho _ { t } \\in \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 556, + 413, + 571 + ], + "score": 1.0, + "content": ": A scalar value controlling the amplitude of perturbation at step", + "type": "text" + }, + { + "bbox": [ + 413, + 558, + 417, + 567 + ], + "score": 0.65, + "content": "t", + "type": "inline_equation" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 117, + 573, + 446, + 585 + ], + "lines": [ + { + "bbox": [ + 115, + 571, + 446, + 587 + ], + "spans": [ + { + "bbox": [ + 115, + 571, + 126, + 587 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 126, + 573, + 151, + 583 + ], + "score": 0.88, + "content": "\\epsilon \\in \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 571, + 346, + 587 + ], + "score": 1.0, + "content": ": A small positive constant (to avoid division by", + "type": "text" + }, + { + "bbox": [ + 346, + 572, + 396, + 584 + ], + "score": 0.89, + "content": ") , \\epsilon = 1 0 ^ { - 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Since", + "type": "text" + }, + { + "bbox": [ + 357, + 223, + 476, + 235 + ], + "score": 0.91, + "content": "f _ { p } ( w _ { 1 } ) = f _ { p } ( w _ { 3 } ) < f _ { p } ( w _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 223, + 505, + 236 + ], + "score": 1.0, + "content": ", SAM", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 233, + 505, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 233, + 136, + 246 + ], + "score": 1.0, + "content": "prefers", + "type": "text" + }, + { + "bbox": [ + 137, + 235, + 149, + 245 + ], + "score": 0.87, + "content": "w _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 233, + 167, + 246 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 168, + 236, + 180, + 245 + ], + "score": 0.85, + "content": "w _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 233, + 192, + 246 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 192, + 236, + 204, + 245 + ], + "score": 0.84, + "content": "w _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 233, + 273, + 246 + ], + "score": 1.0, + "content": ". 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However, this vanilla training", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 368, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 368, + 505, + 381 + ], + "score": 1.0, + "content": "often falls into a sharp valley of the loss surface, resulting in inferior generalization performance", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 379, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 505, + 392 + ], + "score": 1.0, + "content": "(Chaudhari et al., 2019). 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This perturbed loss", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 449, + 506, + 463 + ], + "spans": [ + { + "bbox": [ + 106, + 450, + 117, + 462 + ], + "score": 0.88, + "content": "f _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 118, + 449, + 168, + 463 + ], + "score": 1.0, + "content": "induced by", + "type": "text" + }, + { + "bbox": [ + 168, + 450, + 191, + 461 + ], + "score": 0.92, + "content": "f ( w )", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 449, + 506, + 463 + ], + "score": 1.0, + "content": "is the maximum loss within the neighborhood. When the perturbed loss is", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 459, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 495, + 474 + ], + "score": 1.0, + "content": "minimized, the neighborhood corresponds to low losses (below the perturbed loss). 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For", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "score": 1.0, + "content": "simplicity, we do not consider the influence of reparameterization on the geometry of loss surfaces,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 720, + 427, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 427, + 732 + ], + "score": 1.0, + "content": "which is thoroughly discussed in (Laurent & Massart, 2000; Kwon et al., 2021).", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 39 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 105, + 25, + 294, + 39 + ], + "spans": [ + { + "bbox": [ + 105, + 25, + 294, + 39 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "score": 1.0, + "content": "3", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 124, + 76, + 485, + 179 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 124, + 76, + 485, + 179 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 125, + 76, + 483, + 179 + ], + "spans": [ + { + "bbox": [ + 125, + 76, + 483, + 179 + ], + "score": 0.887, + "type": "image", + "image_path": "3396788a9126ef1da5043f16c4c9713656da2cf3b5a9e09a6b9971d5556bacff.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 124, + 76, + 485, + 110.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 124, + 110.33333333333334, + 485, + 144.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 124, + 144.66666666666669, + 485, + 179.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 106, + 186, + 506, + 268 + ], + "lines": [ + { + "bbox": [ + 105, + 186, + 506, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 233, + 201 + ], + "score": 1.0, + "content": "Figure 1: Consider original loss", + "type": "text" + }, + { + "bbox": [ + 234, + 188, + 240, + 199 + ], + "score": 0.86, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 186, + 346, + 201 + ], + "score": 1.0, + "content": "(solid line), perturbed loss", + "type": "text" + }, + { + "bbox": [ + 347, + 186, + 448, + 200 + ], + "score": 0.91, + "content": "\\begin{array} { r } { f _ { p } \\triangleq \\operatorname* { m a x } _ { | | \\delta | | \\leq \\rho } f ( w + \\delta ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 186, + 506, + 201 + ], + "score": 1.0, + "content": "(dashed line),", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 200, + 506, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 201, + 180, + 213 + ], + "score": 1.0, + "content": "and surrogate gap", + "type": "text" + }, + { + "bbox": [ + 180, + 200, + 274, + 213 + ], + "score": 0.92, + "content": "h ( w ) \\triangleq f _ { p } ( w ) - f ( w )", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 201, + 324, + 213 + ], + "score": 1.0, + "content": ". 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However, this vanilla training", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 368, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 368, + 505, + 381 + ], + "score": 1.0, + "content": "often falls into a sharp valley of the loss surface, resulting in inferior generalization performance", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 379, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 505, + 392 + ], + "score": 1.0, + "content": "(Chaudhari et al., 2019). 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For some fixed", + "type": "text" + }, + { + "bbox": [ + 227, + 290, + 234, + 299 + ], + "score": 0.67, + "content": "\\rho _ { ; }", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 286, + 350, + 301 + ], + "score": 1.0, + "content": ", consider two local minima", + "type": "text" + }, + { + "bbox": [ + 351, + 289, + 363, + 299 + ], + "score": 0.84, + "content": "w _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 286, + 384, + 301 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 384, + 289, + 396, + 299 + ], + "score": 0.77, + "content": "w _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 286, + 402, + 301 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 402, + 287, + 502, + 300 + ], + "score": 0.88, + "content": "f _ { p } ( w _ { 1 } ) \\leq f _ { p } ( w _ { 2 } ) \\neq", + "type": "inline_equation" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 298, + 430, + 312 + ], + "spans": [ + { + "bbox": [ + 106, + 298, + 204, + 311 + ], + "score": 0.92, + "content": "\\sigma _ { m a x } ( w _ { 1 } ) \\leq \\sigma _ { m a x } ( w _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 298, + 234, + 312 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 234, + 300, + 257, + 310 + ], + "score": 0.87, + "content": "\\sigma _ { m a x }", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 298, + 430, + 312 + ], + "score": 1.0, + "content": "is the dominant eigenvalue of the Hessian.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19.5 + }, + { + "type": "text", + "bbox": [ + 106, + 318, + 505, + 398 + ], + "lines": [ + { + "bbox": [ + 106, + 318, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 505, + 331 + ], + "score": 1.0, + "content": "We leave the proof to Appendix. Fig. 1 illustrates Lemma 3.1 with an example. Consider three", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 329, + 504, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 207, + 343 + ], + "score": 1.0, + "content": "local minima denoted as", + "type": "text" + }, + { + "bbox": [ + 207, + 332, + 219, + 341 + ], + "score": 0.87, + "content": "w _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 329, + 231, + 343 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 232, + 332, + 244, + 341 + ], + "score": 0.86, + "content": "w _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 329, + 491, + 343 + ], + "score": 1.0, + "content": ", and suppose the corresponding loss surfaces are flatter from", + "type": "text" + }, + { + "bbox": [ + 491, + 331, + 504, + 341 + ], + "score": 0.83, + "content": "w _ { 1 }", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 341, + 505, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 117, + 356 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 117, + 343, + 129, + 353 + ], + "score": 0.84, + "content": "w _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 130, + 342, + 198, + 356 + ], + "score": 1.0, + "content": ". For some fixed", + "type": "text" + }, + { + "bbox": [ + 198, + 344, + 205, + 354 + ], + "score": 0.8, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 342, + 316, + 356 + ], + "score": 1.0, + "content": ", we plot the perturbed loss", + "type": "text" + }, + { + "bbox": [ + 316, + 342, + 327, + 354 + ], + "score": 0.89, + "content": "f _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 342, + 402, + 356 + ], + "score": 1.0, + "content": "and surrogate gap", + "type": "text" + }, + { + "bbox": [ + 403, + 341, + 452, + 354 + ], + "score": 0.89, + "content": "h \\triangleq f _ { p } - f", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 342, + 505, + 356 + ], + "score": 1.0, + "content": "around each", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 353, + 505, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 195, + 367 + ], + "score": 1.0, + "content": "solution. Comparing", + "type": "text" + }, + { + "bbox": [ + 195, + 355, + 208, + 364 + ], + "score": 0.86, + "content": "w _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 353, + 231, + 367 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 232, + 354, + 244, + 364 + ], + "score": 0.86, + "content": "w _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 353, + 411, + 367 + ], + "score": 1.0, + "content": ": Suppose their vanilla losses are equal,", + "type": "text" + }, + { + "bbox": [ + 412, + 353, + 480, + 365 + ], + "score": 0.92, + "content": "f ( \\tilde { w _ { 2 } } ) = f ( w _ { 3 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 353, + 505, + 367 + ], + "score": 1.0, + "content": ", then", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 107, + 363, + 504, + 378 + ], + "spans": [ + { + "bbox": [ + 107, + 364, + 180, + 376 + ], + "score": 0.92, + "content": "f _ { p } ( w _ { 2 } ) > f _ { p } ( \\bar { w } _ { 3 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 363, + 347, + 378 + ], + "score": 1.0, + "content": "because the loss surface is flatter around", + "type": "text" + }, + { + "bbox": [ + 348, + 365, + 360, + 375 + ], + "score": 0.85, + "content": "w _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 363, + 491, + 378 + ], + "score": 1.0, + "content": ", implying that SAM will prefer", + "type": "text" + }, + { + "bbox": [ + 491, + 366, + 504, + 375 + ], + "score": 0.83, + "content": "w _ { 3 }", + "type": "inline_equation" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 375, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 117, + 389 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 117, + 378, + 129, + 386 + ], + "score": 0.86, + "content": "w _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 130, + 375, + 182, + 389 + ], + "score": 1.0, + "content": ". 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The surrogate gap has the following properties.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5 + }, + { + "type": "text", + "bbox": [ + 106, + 490, + 505, + 513 + ], + "lines": [ + { + "bbox": [ + 105, + 488, + 506, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 308, + 504 + ], + "score": 1.0, + "content": "Lemma 3.2. Suppose the perturbation amplitude", + "type": "text" + }, + { + "bbox": [ + 309, + 492, + 316, + 501 + ], + "score": 0.75, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 488, + 506, + 504 + ], + "score": 1.0, + "content": "is sufficiently small, then the approximation to", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 500, + 452, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 309, + 515 + ], + "score": 1.0, + "content": "the surrogate gap in Eq. 4 is always non-negative,", + "type": "text" + }, + { + "bbox": [ + 309, + 501, + 447, + 514 + ], + "score": 0.93, + "content": "h ( w ) \\approx f ( w ^ { a d v } ) - f ( w ) \\geq 0 , \\forall w", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 500, + 452, + 515 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.5 + }, + { + "type": "text", + "bbox": [ + 107, + 515, + 504, + 549 + ], + "lines": [ + { + "bbox": [ + 105, + 515, + 506, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 251, + 528 + ], + "score": 1.0, + "content": "Lemma 3.3. For a local minimum", + "type": "text" + }, + { + "bbox": [ + 252, + 516, + 264, + 525 + ], + "score": 0.85, + "content": "w ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 515, + 408, + 528 + ], + "score": 1.0, + "content": ", consider the dominate eigenvalue", + "type": "text" + }, + { + "bbox": [ + 408, + 517, + 431, + 527 + ], + "score": 0.87, + "content": "\\sigma _ { m a x }", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 515, + 506, + 528 + ], + "score": 1.0, + "content": "of the Hessian of", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 526, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 106, + 526, + 124, + 538 + ], + "score": 1.0, + "content": "loss", + "type": "text" + }, + { + "bbox": [ + 125, + 527, + 131, + 538 + ], + "score": 0.82, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 526, + 412, + 538 + ], + "score": 1.0, + "content": "as a measure of sharpness. 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Lemma 3.2 tells that the surrogate gap is non-negative, and Lemma 3.3", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 568, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 106, + 568, + 261, + 581 + ], + "score": 1.0, + "content": "shows that the loss surface is flatter as", + "type": "text" + }, + { + "bbox": [ + 261, + 569, + 268, + 578 + ], + "score": 0.81, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 568, + 505, + 581 + ], + "score": 1.0, + "content": "gets closer to 0. The two lemmas together indicate that we", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 579, + 424, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 399, + 592 + ], + "score": 1.0, + "content": "can find a region with a flat loss surface by minimizing the surrogate gap", + "type": "text" + }, + { + "bbox": [ + 399, + 579, + 421, + 591 + ], + "score": 0.92, + "content": "h ( w )", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 579, + 424, + 592 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39 + }, + { + "type": "title", + "bbox": [ + 107, + 606, + 446, + 620 + ], + "lines": [ + { + "bbox": [ + 104, + 605, + 447, + 622 + ], + "spans": [ + { + "bbox": [ + 104, + 605, + 447, + 622 + ], + "score": 1.0, + "content": "4 SURROGATE GAP GUIDED SHARPNESS-AWARE MINIMIZATION", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 107, + 631, + 497, + 653 + ], + "lines": [ + { + "bbox": [ + 105, + 630, + 499, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 499, + 644 + ], + "score": 1.0, + "content": "4.1 GENERAL IDEA: SIMULTANEOUSLY MINIMIZE THE PERTURBED LOSS AND SURROGATE", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 129, + 642, + 151, + 654 + ], + "spans": [ + { + "bbox": [ + 129, + 642, + 151, + 654 + ], + "score": 1.0, + "content": "GAP", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 42.5 + }, + { + "type": "text", + "bbox": [ + 106, + 662, + 504, + 695 + ], + "lines": [ + { + "bbox": [ + 105, + 662, + 506, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 662, + 506, + 675 + ], + "score": 1.0, + "content": "Inspired by the analysis in Section 3, we propose Surrogate Gap Guided Sharpness-Aware", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 672, + 506, + 687 + ], + "spans": [ + { + "bbox": [ + 105, + 672, + 459, + 687 + ], + "score": 1.0, + "content": "Minimzation (GSAM) to simultaneously minimize two objectives, the perturbed loss", + "type": "text" + }, + { + "bbox": [ + 459, + 674, + 470, + 686 + ], + "score": 0.89, + "content": "f _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 672, + 506, + 687 + ], + "score": 1.0, + "content": "and the", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 685, + 174, + 698 + ], + "spans": [ + { + "bbox": [ + 106, + 685, + 163, + 698 + ], + "score": 1.0, + "content": "surrogate gap", + "type": "text" + }, + { + "bbox": [ + 163, + 685, + 170, + 695 + ], + "score": 0.76, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 170, + 685, + 174, + 698 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 45 + }, + { + "type": "interline_equation", + "bbox": [ + 263, + 694, + 349, + 709 + ], + "lines": [ + { + "bbox": [ + 263, + 694, + 349, + 709 + ], + "spans": [ + { + "bbox": [ + 263, + 694, + 349, + 709 + ], + "score": 0.91, + "content": "\\operatorname* { m i n } _ { w } \\left( f _ { p } ( w ) , h ( w ) \\right)", + "type": "interline_equation", + "image_path": "9792aae0717fc879fbc94f0fb316680ebdfdaa6f830266912be569c38ffd7482.jpg" + } + ] + } + ], + "index": 47, + "virtual_lines": [ + { + "bbox": [ + 263, + 694, + 349, + 709 + ], + "spans": [], + "index": 47 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 709, + 507, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 211, + 722 + ], + "score": 1.0, + "content": "Intuitively, by minimizng", + "type": "text" + }, + { + "bbox": [ + 211, + 710, + 222, + 722 + ], + "score": 0.89, + "content": "f _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "we search for a region with a low perturbed loss similar to SAM, and", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 719, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 168, + 733 + ], + "score": 1.0, + "content": "by minimizing", + "type": "text" + }, + { + "bbox": [ + 168, + 721, + 176, + 730 + ], + "score": 0.79, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 719, + 505, + 733 + ], + "score": 1.0, + "content": "we search for a local minimum with a flat surface. 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For some fixed", + "type": "text" + }, + { + "bbox": [ + 227, + 290, + 234, + 299 + ], + "score": 0.67, + "content": "\\rho _ { ; }", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 286, + 350, + 301 + ], + "score": 1.0, + "content": ", consider two local minima", + "type": "text" + }, + { + "bbox": [ + 351, + 289, + 363, + 299 + ], + "score": 0.84, + "content": "w _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 286, + 384, + 301 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 384, + 289, + 396, + 299 + ], + "score": 0.77, + "content": "w _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 286, + 402, + 301 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 402, + 287, + 502, + 300 + ], + "score": 0.88, + "content": "f _ { p } ( w _ { 1 } ) \\leq f _ { p } ( w _ { 2 } ) \\neq", + "type": "inline_equation" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 298, + 430, + 312 + ], + "spans": [ + { + "bbox": [ + 106, + 298, + 204, + 311 + ], + "score": 0.92, + "content": "\\sigma _ { m a x } ( w _ { 1 } ) \\leq \\sigma _ { m a x } ( w _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 298, + 234, + 312 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 234, + 300, + 257, + 310 + ], + "score": 0.87, + "content": "\\sigma _ { m a x }", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 298, + 430, + 312 + ], + "score": 1.0, + "content": "is the dominant eigenvalue of the Hessian.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19.5, + "bbox_fs": [ + 105, + 286, + 502, + 312 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 318, + 505, + 398 + ], + "lines": [ + { + "bbox": [ + 106, + 318, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 505, + 331 + ], + "score": 1.0, + "content": "We leave the proof to Appendix. Fig. 1 illustrates Lemma 3.1 with an example. Consider three", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 329, + 504, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 207, + 343 + ], + "score": 1.0, + "content": "local minima denoted as", + "type": "text" + }, + { + "bbox": [ + 207, + 332, + 219, + 341 + ], + "score": 0.87, + "content": "w _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 329, + 231, + 343 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 232, + 332, + 244, + 341 + ], + "score": 0.86, + "content": "w _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 329, + 491, + 343 + ], + "score": 1.0, + "content": ", and suppose the corresponding loss surfaces are flatter from", + "type": "text" + }, + { + "bbox": [ + 491, + 331, + 504, + 341 + ], + "score": 0.83, + "content": "w _ { 1 }", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 341, + 505, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 117, + 356 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 117, + 343, + 129, + 353 + ], + "score": 0.84, + "content": "w _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 130, + 342, + 198, + 356 + ], + "score": 1.0, + "content": ". For some fixed", + "type": "text" + }, + { + "bbox": [ + 198, + 344, + 205, + 354 + ], + "score": 0.8, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 342, + 316, + 356 + ], + "score": 1.0, + "content": ", we plot the perturbed loss", + "type": "text" + }, + { + "bbox": [ + 316, + 342, + 327, + 354 + ], + "score": 0.89, + "content": "f _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 342, + 402, + 356 + ], + "score": 1.0, + "content": "and surrogate gap", + "type": "text" + }, + { + "bbox": [ + 403, + 341, + 452, + 354 + ], + "score": 0.89, + "content": "h \\triangleq f _ { p } - f", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 342, + 505, + 356 + ], + "score": 1.0, + "content": "around each", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 353, + 505, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 195, + 367 + ], + "score": 1.0, + "content": "solution. Comparing", + "type": "text" + }, + { + "bbox": [ + 195, + 355, + 208, + 364 + ], + "score": 0.86, + "content": "w _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 353, + 231, + 367 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 232, + 354, + 244, + 364 + ], + "score": 0.86, + "content": "w _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 353, + 411, + 367 + ], + "score": 1.0, + "content": ": Suppose their vanilla losses are equal,", + "type": "text" + }, + { + "bbox": [ + 412, + 353, + 480, + 365 + ], + "score": 0.92, + "content": "f ( \\tilde { w _ { 2 } } ) = f ( w _ { 3 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 353, + 505, + 367 + ], + "score": 1.0, + "content": ", then", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 107, + 363, + 504, + 378 + ], + "spans": [ + { + "bbox": [ + 107, + 364, + 180, + 376 + ], + "score": 0.92, + "content": "f _ { p } ( w _ { 2 } ) > f _ { p } ( \\bar { w } _ { 3 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 363, + 347, + 378 + ], + "score": 1.0, + "content": "because the loss surface is flatter around", + "type": "text" + }, + { + "bbox": [ + 348, + 365, + 360, + 375 + ], + "score": 0.85, + "content": "w _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 363, + 491, + 378 + ], + "score": 1.0, + "content": ", implying that SAM will prefer", + "type": "text" + }, + { + "bbox": [ + 491, + 366, + 504, + 375 + ], + "score": 0.83, + "content": "w _ { 3 }", + "type": "inline_equation" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 375, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 117, + 389 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 117, + 378, + 129, + 386 + ], + "score": 0.86, + "content": "w _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 130, + 375, + 182, + 389 + ], + "score": 1.0, + "content": ". Comparing", + "type": "text" + }, + { + "bbox": [ + 183, + 377, + 196, + 386 + ], + "score": 0.86, + "content": "w _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 375, + 214, + 389 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 214, + 376, + 227, + 386 + ], + "score": 0.84, + "content": "w _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 375, + 232, + 389 + ], + "score": 1.0, + "content": ":", + "type": "text" + }, + { + "bbox": [ + 233, + 375, + 306, + 388 + ], + "score": 0.92, + "content": "f _ { p } ( w _ { 1 } ) < f _ { p } ( w _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 375, + 393, + 389 + ], + "score": 1.0, + "content": ", and SAM will favor", + "type": "text" + }, + { + "bbox": [ + 393, + 376, + 406, + 386 + ], + "score": 0.86, + "content": "w _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 375, + 427, + 389 + ], + "score": 1.0, + "content": "over", + "type": "text" + }, + { + "bbox": [ + 428, + 376, + 441, + 386 + ], + "score": 0.86, + "content": "w _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 375, + 505, + 389 + ], + "score": 1.0, + "content": "because it only", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 385, + 478, + 400 + ], + "spans": [ + { + "bbox": [ + 104, + 385, + 227, + 400 + ], + "score": 1.0, + "content": "cares about the perturbed loss", + "type": "text" + }, + { + "bbox": [ + 227, + 387, + 238, + 398 + ], + "score": 0.89, + "content": "f _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 385, + 428, + 400 + ], + "score": 1.0, + "content": ", even though the loss surface is sharper around", + "type": "text" + }, + { + "bbox": [ + 428, + 388, + 441, + 397 + ], + "score": 0.87, + "content": "w _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 385, + 461, + 400 + ], + "score": 1.0, + "content": "than", + "type": "text" + }, + { + "bbox": [ + 462, + 388, + 474, + 397 + ], + "score": 0.85, + "content": "w _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 385, + 478, + 400 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24, + "bbox_fs": [ + 104, + 318, + 505, + 400 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 410, + 338, + 422 + ], + "lines": [ + { + "bbox": [ + 105, + 410, + 338, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 338, + 423 + ], + "score": 1.0, + "content": "3.2 THE SURROGATE GAP AGREES WITH SHARPNESS", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 105, + 431, + 387, + 443 + ], + "lines": [ + { + "bbox": [ + 106, + 430, + 388, + 444 + ], + "spans": [ + { + "bbox": [ + 106, + 430, + 388, + 444 + ], + "score": 1.0, + "content": "We introduce the surrogate gap that agrees with sharpness, defined as:", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29, + "bbox_fs": [ + 106, + 430, + 388, + 444 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 189, + 446, + 423, + 462 + ], + "lines": [ + { + "bbox": [ + 189, + 446, + 423, + 462 + ], + "spans": [ + { + "bbox": [ + 189, + 446, + 423, + 462 + ], + "score": 0.9, + "content": "\\begin{array} { r } { h ( w ) \\triangleq \\operatorname* { m a x } _ { | | \\delta | | \\leq \\rho } f ( w + \\delta ) - f ( w ) \\approx f ( w ^ { a d v } ) - f ( w ) } \\end{array}", + "type": "interline_equation", + "image_path": "d0dbbd89e8de3b43b04b536ddabb4b177de49bb9d5a399b105fb3fea337bdabb.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 189, + 446, + 423, + 462 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 465, + 503, + 488 + ], + "lines": [ + { + "bbox": [ + 106, + 464, + 504, + 477 + ], + "spans": [ + { + "bbox": [ + 106, + 464, + 504, + 477 + ], + "score": 1.0, + "content": "Intuitively, the surrogate gap represents the difference between the maximum loss within the neigh-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 475, + 460, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 460, + 489 + ], + "score": 1.0, + "content": "borhood and the loss at the center point. The surrogate gap has the following properties.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 464, + 504, + 489 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 490, + 505, + 513 + ], + "lines": [ + { + "bbox": [ + 105, + 488, + 506, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 308, + 504 + ], + "score": 1.0, + "content": "Lemma 3.2. Suppose the perturbation amplitude", + "type": "text" + }, + { + "bbox": [ + 309, + 492, + 316, + 501 + ], + "score": 0.75, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 488, + 506, + 504 + ], + "score": 1.0, + "content": "is sufficiently small, then the approximation to", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 500, + 452, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 309, + 515 + ], + "score": 1.0, + "content": "the surrogate gap in Eq. 4 is always non-negative,", + "type": "text" + }, + { + "bbox": [ + 309, + 501, + 447, + 514 + ], + "score": 0.93, + "content": "h ( w ) \\approx f ( w ^ { a d v } ) - f ( w ) \\geq 0 , \\forall w", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 500, + 452, + 515 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 488, + 506, + 515 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 515, + 504, + 549 + ], + "lines": [ + { + "bbox": [ + 105, + 515, + 506, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 251, + 528 + ], + "score": 1.0, + "content": "Lemma 3.3. For a local minimum", + "type": "text" + }, + { + "bbox": [ + 252, + 516, + 264, + 525 + ], + "score": 0.85, + "content": "w ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 515, + 408, + 528 + ], + "score": 1.0, + "content": ", consider the dominate eigenvalue", + "type": "text" + }, + { + "bbox": [ + 408, + 517, + 431, + 527 + ], + "score": 0.87, + "content": "\\sigma _ { m a x }", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 515, + 506, + 528 + ], + "score": 1.0, + "content": "of the Hessian of", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 526, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 106, + 526, + 124, + 538 + ], + "score": 1.0, + "content": "loss", + "type": "text" + }, + { + "bbox": [ + 125, + 527, + 131, + 538 + ], + "score": 0.82, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 526, + 412, + 538 + ], + "score": 1.0, + "content": "as a measure of sharpness. Considering the neighborhood centered at", + "type": "text" + }, + { + "bbox": [ + 413, + 527, + 425, + 536 + ], + "score": 0.86, + "content": "w ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 526, + 505, + 538 + ], + "score": 1.0, + "content": "with a small radius", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 107, + 537, + 475, + 550 + ], + "spans": [ + { + "bbox": [ + 107, + 540, + 113, + 549 + ], + "score": 0.47, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 113, + 537, + 189, + 550 + ], + "score": 1.0, + "content": ", the surrogate gap", + "type": "text" + }, + { + "bbox": [ + 190, + 537, + 216, + 549 + ], + "score": 0.92, + "content": "h ( w ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 537, + 390, + 550 + ], + "score": 1.0, + "content": "is an equivalent measure of the sharpness:", + "type": "text" + }, + { + "bbox": [ + 390, + 537, + 471, + 549 + ], + "score": 0.92, + "content": "\\sigma _ { m a x } \\approx 2 h ( w ^ { * } ) / \\rho ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 537, + 475, + 550 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36, + "bbox_fs": [ + 105, + 515, + 506, + 550 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 557, + 505, + 591 + ], + "lines": [ + { + "bbox": [ + 106, + 556, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 106, + 556, + 505, + 570 + ], + "score": 1.0, + "content": "The proof is in Appendix. Lemma 3.2 tells that the surrogate gap is non-negative, and Lemma 3.3", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 568, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 106, + 568, + 261, + 581 + ], + "score": 1.0, + "content": "shows that the loss surface is flatter as", + "type": "text" + }, + { + "bbox": [ + 261, + 569, + 268, + 578 + ], + "score": 0.81, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 568, + 505, + 581 + ], + "score": 1.0, + "content": "gets closer to 0. The two lemmas together indicate that we", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 579, + 424, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 399, + 592 + ], + "score": 1.0, + "content": "can find a region with a flat loss surface by minimizing the surrogate gap", + "type": "text" + }, + { + "bbox": [ + 399, + 579, + 421, + 591 + ], + "score": 0.92, + "content": "h ( w )", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 579, + 424, + 592 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39, + "bbox_fs": [ + 105, + 556, + 505, + 592 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 606, + 446, + 620 + ], + "lines": [ + { + "bbox": [ + 104, + 605, + 447, + 622 + ], + "spans": [ + { + "bbox": [ + 104, + 605, + 447, + 622 + ], + "score": 1.0, + "content": "4 SURROGATE GAP GUIDED SHARPNESS-AWARE MINIMIZATION", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 107, + 631, + 497, + 653 + ], + "lines": [ + { + "bbox": [ + 105, + 630, + 499, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 499, + 644 + ], + "score": 1.0, + "content": "4.1 GENERAL IDEA: SIMULTANEOUSLY MINIMIZE THE PERTURBED LOSS AND SURROGATE", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 129, + 642, + 151, + 654 + ], + "spans": [ + { + "bbox": [ + 129, + 642, + 151, + 654 + ], + "score": 1.0, + "content": "GAP", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 42.5, + "bbox_fs": [ + 105, + 630, + 499, + 654 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 662, + 504, + 695 + ], + "lines": [ + { + "bbox": [ + 105, + 662, + 506, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 662, + 506, + 675 + ], + "score": 1.0, + "content": "Inspired by the analysis in Section 3, we propose Surrogate Gap Guided Sharpness-Aware", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 672, + 506, + 687 + ], + "spans": [ + { + "bbox": [ + 105, + 672, + 459, + 687 + ], + "score": 1.0, + "content": "Minimzation (GSAM) to simultaneously minimize two objectives, the perturbed loss", + "type": "text" + }, + { + "bbox": [ + 459, + 674, + 470, + 686 + ], + "score": 0.89, + "content": "f _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 672, + 506, + 687 + ], + "score": 1.0, + "content": "and the", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 685, + 174, + 698 + ], + "spans": [ + { + "bbox": [ + 106, + 685, + 163, + 698 + ], + "score": 1.0, + "content": "surrogate gap", + "type": "text" + }, + { + "bbox": [ + 163, + 685, + 170, + 695 + ], + "score": 0.76, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 170, + 685, + 174, + 698 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 45, + "bbox_fs": [ + 105, + 662, + 506, + 698 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 263, + 694, + 349, + 709 + ], + "lines": [ + { + "bbox": [ + 263, + 694, + 349, + 709 + ], + "spans": [ + { + "bbox": [ + 263, + 694, + 349, + 709 + ], + "score": 0.91, + "content": "\\operatorname* { m i n } _ { w } \\left( f _ { p } ( w ) , h ( w ) \\right)", + "type": "interline_equation", + "image_path": "9792aae0717fc879fbc94f0fb316680ebdfdaa6f830266912be569c38ffd7482.jpg" + } + ] + } + ], + "index": 47, + "virtual_lines": [ + { + "bbox": [ + 263, + 694, + 349, + 709 + ], + "spans": [], + "index": 47 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 709, + 507, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 211, + 722 + ], + "score": 1.0, + "content": "Intuitively, by minimizng", + "type": "text" + }, + { + "bbox": [ + 211, + 710, + 222, + 722 + ], + "score": 0.89, + "content": "f _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "we search for a region with a low perturbed loss similar to SAM, and", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 719, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 168, + 733 + ], + "score": 1.0, + "content": "by minimizing", + "type": "text" + }, + { + "bbox": [ + 168, + 721, + 176, + 730 + ], + "score": 0.79, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 719, + 505, + 733 + ], + "score": 1.0, + "content": "we search for a local minimum with a flat surface. A low perturbed loss implies", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 80, + 506, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 506, + 97 + ], + "score": 1.0, + "content": "low training losses within the neighborhood, and a flat loss surface reduces the generalization gap", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "between training and test performances (Chaudhari et al., 2019). When both are minimized, the", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 350, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 350, + 117 + ], + "score": 1.0, + "content": "solution gives rise to high accuracy and good generalization.", + "type": "text", + "cross_page": true + } + ], + "index": 2 + } + ], + "index": 48.5, + "bbox_fs": [ + 105, + 709, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 116 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 506, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 506, + 97 + ], + "score": 1.0, + "content": "low training losses within the neighborhood, and a flat loss surface reduces the generalization gap", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "between training and test performances (Chaudhari et al., 2019). When both are minimized, the", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 350, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 350, + 117 + ], + "score": 1.0, + "content": "solution gives rise to high accuracy and good generalization.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 106, + 121, + 505, + 199 + ], + "lines": [ + { + "bbox": [ + 105, + 121, + 505, + 135 + ], + "spans": [ + { + "bbox": [ + 105, + 121, + 505, + 135 + ], + "score": 1.0, + "content": "Potential caveat in optimization It is tempting and yet sub-optimal to combine the objectives in", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 132, + 506, + 146 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 172, + 146 + ], + "score": 1.0, + "content": "Eq. 5 to arrive at", + "type": "text" + }, + { + "bbox": [ + 173, + 133, + 196, + 144 + ], + "score": 0.64, + "content": "\\mathrm { m i n } _ { w }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 132, + 258, + 145 + ], + "score": 0.84, + "content": "f _ { p } ( w ) + \\lambda h ( w )", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 132, + 287, + 146 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 288, + 133, + 294, + 142 + ], + "score": 0.78, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 132, + 506, + 146 + ], + "score": 1.0, + "content": "is some positive scalar. One caveat when solving this", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 143, + 504, + 157 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 469, + 157 + ], + "score": 1.0, + "content": "weighted combination is the potential conflict between the gradients of the two terms, i.e.,", + "type": "text" + }, + { + "bbox": [ + 470, + 143, + 504, + 156 + ], + "score": 0.92, + "content": "\\nabla f _ { p } ( w )", + "type": "inline_equation" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 152, + 506, + 169 + ], + "spans": [ + { + "bbox": [ + 104, + 152, + 124, + 169 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 154, + 154, + 166 + ], + "score": 0.92, + "content": "\\overline { { \\nabla } h } ( \\boldsymbol { w } )", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 152, + 340, + 169 + ], + "score": 1.0, + "content": ". We illustrate this conflict by Fig. 2, where", + "type": "text" + }, + { + "bbox": [ + 340, + 154, + 464, + 166 + ], + "score": 0.91, + "content": "\\bar { \\nabla } h ( w ) = \\nabla f _ { p } ( w ) - \\nabla f ( w )", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 152, + 506, + 169 + ], + "score": 1.0, + "content": "(the grey", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 506, + 179 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 300, + 179 + ], + "score": 1.0, + "content": "dashed arrow) has a negative inner product with", + "type": "text" + }, + { + "bbox": [ + 301, + 165, + 335, + 177 + ], + "score": 0.93, + "content": "\\nabla f _ { p } ( w )", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 165, + 353, + 179 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 353, + 166, + 384, + 177 + ], + "score": 0.92, + "content": "\\nabla f ( w )", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 165, + 506, + 179 + ], + "score": 1.0, + "content": ". Hence, the gradient descent", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 175, + 506, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 175, + 332, + 190 + ], + "score": 1.0, + "content": "for the surrogate gap could potentially increase the loss", + "type": "text" + }, + { + "bbox": [ + 332, + 177, + 343, + 189 + ], + "score": 0.88, + "content": "f _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 175, + 506, + 190 + ], + "score": 1.0, + "content": ", harming the model’s performance. We", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 187, + 293, + 200 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 293, + 200 + ], + "score": 1.0, + "content": "empirically validate this argument in Sec. 6.4.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6 + }, + { + "type": "title", + "bbox": [ + 107, + 222, + 493, + 234 + ], + "lines": [ + { + "bbox": [ + 106, + 222, + 494, + 234 + ], + "spans": [ + { + "bbox": [ + 106, + 222, + 494, + 234 + ], + "score": 1.0, + "content": "4.2 GRADIENT DECOMPOSITION AND ASCENT FOR THE MULTI-OBJECTIVE OPTIMIZATION", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 246, + 505, + 281 + ], + "lines": [ + { + "bbox": [ + 105, + 246, + 505, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 241, + 260 + ], + "score": 1.0, + "content": "Our primary goal is to minimize", + "type": "text" + }, + { + "bbox": [ + 242, + 248, + 253, + 259 + ], + "score": 0.89, + "content": "f _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 246, + 505, + 260 + ], + "score": 1.0, + "content": "because otherwise a flat solution of high loss is meaningless,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 258, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 205, + 270 + ], + "score": 1.0, + "content": "and the minimization of", + "type": "text" + }, + { + "bbox": [ + 205, + 258, + 213, + 268 + ], + "score": 0.83, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 258, + 294, + 270 + ], + "score": 1.0, + "content": "should not increase", + "type": "text" + }, + { + "bbox": [ + 294, + 258, + 304, + 270 + ], + "score": 0.86, + "content": "f _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 258, + 417, + 270 + ], + "score": 1.0, + "content": ". We propose to decompose", + "type": "text" + }, + { + "bbox": [ + 418, + 258, + 451, + 270 + ], + "score": 0.93, + "content": "\\nabla f ( w _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 258, + 470, + 270 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 470, + 258, + 486, + 268 + ], + "score": 0.86, + "content": "\\nabla h", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 258, + 505, + 270 + ], + "score": 1.0, + "content": "into", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 268, + 437, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 294, + 282 + ], + "score": 1.0, + "content": "components that are parallel and orthogonal to", + "type": "text" + }, + { + "bbox": [ + 295, + 269, + 331, + 281 + ], + "score": 0.92, + "content": "\\dot { \\nabla } f _ { p } ( \\boldsymbol { w } _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 268, + 437, + 282 + ], + "score": 1.0, + "content": ", respectively (see Fig. 2):", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12 + }, + { + "type": "interline_equation", + "bbox": [ + 234, + 293, + 377, + 341 + ], + "lines": [ + { + "bbox": [ + 234, + 293, + 377, + 341 + ], + "spans": [ + { + "bbox": [ + 234, + 293, + 377, + 341 + ], + "score": 0.93, + "content": "\\begin{array} { r l } & { \\nabla f ( w _ { t } ) = \\nabla f _ { \\parallel } ( w _ { t } ) + \\nabla f _ { \\perp } ( w _ { t } ) } \\\\ & { \\nabla h ( w _ { t } ) = \\nabla h _ { \\parallel } ( w _ { t } ) + \\nabla h _ { \\perp } ( w _ { t } ) } \\\\ & { \\nabla h _ { \\perp } ( w _ { t } ) = - \\nabla f _ { \\perp } ( w _ { t } ) } \\end{array}", + "type": "interline_equation", + "image_path": "bfe642ad9dc93e7e579b1ee21ac9a7dda338ecd01d5984fbaa9ec0612a2953b5.jpg" + } + ] + } + ], + "index": 14.5, + "virtual_lines": [ + { + "bbox": [ + 234, + 293, + 377, + 317.0 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 234, + 317.0, + 377, + 341.0 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 352, + 505, + 420 + ], + "lines": [ + { + "bbox": [ + 106, + 352, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 276, + 365 + ], + "score": 1.0, + "content": "The key is that updating in the direction of", + "type": "text" + }, + { + "bbox": [ + 277, + 353, + 316, + 365 + ], + "score": 0.93, + "content": "\\nabla h _ { \\perp } ( w _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 352, + 505, + 365 + ], + "score": 1.0, + "content": "does not change the value of the perturbed loss", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 107, + 362, + 506, + 378 + ], + "spans": [ + { + "bbox": [ + 107, + 364, + 135, + 376 + ], + "score": 0.91, + "content": "f _ { p } ( w _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 362, + 171, + 378 + ], + "score": 1.0, + "content": "because", + "type": "text" + }, + { + "bbox": [ + 172, + 364, + 225, + 376 + ], + "score": 0.91, + "content": "\\nabla h _ { \\perp } \\perp \\nabla f _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 362, + 506, + 378 + ], + "score": 1.0, + "content": "by construction. 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f ( w _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 395, + 506, + 411 + ], + "score": 1.0, + "content": "(by increasing", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 107, + 407, + 227, + 421 + ], + "spans": [ + { + "bbox": [ + 107, + 408, + 132, + 420 + ], + "score": 0.92, + "content": "f ( w _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 407, + 190, + 421 + ], + "score": 1.0, + "content": "and not affect", + "type": "text" + }, + { + "bbox": [ + 191, + 408, + 221, + 420 + ], + "score": 0.91, + "content": "f _ { p } ( w _ { t } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 407, + 227, + 421 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 106, + 425, + 506, + 559 + ], + "lines": [ + { + "bbox": [ + 104, + 425, + 507, + 442 + ], + "spans": [ + { + "bbox": [ + 104, + 425, + 368, + 442 + ], + "score": 1.0, + "content": "The full GSAM Algorithm is shown in Algo. 1 and Fig. 2, where", + "type": "text" + }, + { + "bbox": [ + 368, + 425, + 403, + 440 + ], + "score": 0.84, + "content": "g ^ { ( t ) } , g _ { p } ^ { ( t ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 425, + 507, + 442 + ], + "score": 1.0, + "content": "are noisy observations of", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 439, + 506, + 458 + ], + "spans": [ + { + "bbox": [ + 106, + 442, + 140, + 454 + ], + "score": 0.92, + "content": "\\nabla f ( w _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 439, + 158, + 458 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 158, + 442, + 195, + 455 + ], + "score": 0.93, + "content": "\\nabla f _ { p } ( w _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 439, + 268, + 458 + ], + "score": 1.0, + "content": ", respectively, and", + "type": "text" + }, + { + "bbox": [ + 268, + 439, + 303, + 456 + ], + "score": 0.95, + "content": "g _ { \\parallel } ^ { ( t ) } , g _ { \\perp } ^ { ( t ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 439, + 406, + 458 + ], + "score": 1.0, + "content": "are noisy observations of", + "type": "text" + }, + { + "bbox": [ + 407, + 442, + 443, + 455 + ], + "score": 0.94, + "content": "\\nabla f _ { \\parallel } ( w _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 439, + 462, + 458 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 462, + 442, + 501, + 454 + ], + "score": 0.92, + "content": "\\nabla f _ { \\perp } ( w _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 439, + 506, + 458 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 102, + 453, + 508, + 473 + ], + "spans": [ + { + "bbox": [ + 102, + 453, + 217, + 473 + ], + "score": 1.0, + "content": "respectively, by projecting", + "type": "text" + }, + { + "bbox": [ + 217, + 457, + 233, + 470 + ], + "score": 0.91, + "content": "g ^ { ( t ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 453, + 257, + 473 + ], + "score": 1.0, + "content": "onto", + "type": "text" + }, + { + "bbox": [ + 257, + 456, + 273, + 471 + ], + "score": 0.91, + "content": "g _ { p } ^ { ( t ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 453, + 383, + 473 + ], + "score": 1.0, + "content": ". 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Steps 1) to 2) are the same as SAM: At current point", + "type": "text" + }, + { + "bbox": [ + 368, + 471, + 379, + 480 + ], + "score": 0.84, + "content": "w _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 469, + 506, + 483 + ], + "score": 1.0, + "content": ", step 1) takes a gradient ascent", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 479, + 507, + 499 + ], + "spans": [ + { + "bbox": [ + 104, + 479, + 117, + 499 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 118, + 482, + 139, + 495 + ], + "score": 0.91, + "content": "w _ { t } ^ { a d v }", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 479, + 320, + 499 + ], + "score": 1.0, + "content": "followed by step 2) evaluating the gradient", + "type": "text" + }, + { + "bbox": [ + 321, + 480, + 336, + 495 + ], + "score": 0.92, + "content": "g _ { p } ^ { ( t ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 479, + 349, + 499 + ], + "score": 1.0, + "content": "at", + "type": "text" + }, + { + "bbox": [ + 349, + 482, + 371, + 495 + ], + "score": 0.91, + "content": "w _ { t } ^ { a d v }", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 479, + 446, + 499 + ], + "score": 1.0, + "content": ". Step 3) projects", + "type": "text" + }, + { + "bbox": [ + 446, + 482, + 462, + 495 + ], + "score": 0.9, + "content": "g ^ { ( t ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 479, + 485, + 499 + ], + "score": 1.0, + "content": "onto", + "type": "text" + }, + { + "bbox": [ + 485, + 480, + 501, + 495 + ], + "score": 0.9, + "content": "g _ { p } ^ { ( t ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 479, + 507, + 499 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 493, + 506, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 506, + 507 + ], + "score": 1.0, + "content": "which requires negligible computation compared to the forward and backward passes. In step 4),", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 505, + 506, + 521 + ], + "spans": [ + { + "bbox": [ + 106, + 505, + 138, + 519 + ], + "score": 0.92, + "content": "- \\eta _ { t } g _ { p } ^ { ( t ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 505, + 367, + 521 + ], + "score": 1.0, + "content": "is the same as in SAM and minimizes the perturbed loss", + "type": "text" + }, + { + "bbox": [ + 367, + 507, + 396, + 520 + ], + "score": 0.92, + "content": "f _ { p } ( w _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 505, + 506, + 521 + ], + "score": 1.0, + "content": "with gradient descent, and", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 100, + 517, + 508, + 539 + ], + "spans": [ + { + "bbox": [ + 100, + 517, + 144, + 539 + ], + "score": 1.0, + "content": "αηtg(t)⊥", + "type": "text" + }, + { + "bbox": [ + 138, + 517, + 362, + 537 + ], + "score": 1.0, + "content": "performs an ascent step in the orthogonal direction of", + "type": "text" + }, + { + "bbox": [ + 363, + 520, + 378, + 534 + ], + "score": 0.94, + "content": "g _ { p } ^ { ( t ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 517, + 508, + 537 + ], + "score": 1.0, + "content": "to minimize the surrogate gap", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 533, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 106, + 533, + 132, + 545 + ], + "score": 0.9, + "content": "h ( w _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 533, + 224, + 546 + ], + "score": 1.0, + "content": "( equivalently increase", + "type": "text" + }, + { + "bbox": [ + 225, + 533, + 250, + 545 + ], + "score": 0.92, + "content": "f ( w _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 533, + 289, + 546 + ], + "score": 1.0, + "content": "and keep", + "type": "text" + }, + { + "bbox": [ + 289, + 533, + 317, + 545 + ], + "score": 0.93, + "content": "f _ { p } ( w _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 533, + 505, + 546 + ], + "score": 1.0, + "content": "intact). In coding, GSAM feeds the “surrogate", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 101, + 542, + 483, + 564 + ], + "spans": [ + { + "bbox": [ + 101, + 542, + 204, + 564 + ], + "score": 1.0, + "content": "gradient” ∇fGSAMt , g", + "type": "text" + }, + { + "bbox": [ + 146, + 544, + 249, + 560 + ], + "score": 0.93, + "content": "\\nabla f _ { t } ^ { G S A M } \\triangleq g _ { p } ^ { ( t ) } - \\alpha g _ { \\perp } ^ { ( t ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 544, + 483, + 561 + ], + "score": 1.0, + "content": "to first-order gradient optimizers such as SGD and Adam.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 26.5 + }, + { + "type": "text", + "bbox": [ + 107, + 565, + 505, + 626 + ], + "lines": [ + { + "bbox": [ + 106, + 564, + 504, + 583 + ], + "spans": [ + { + "bbox": [ + 106, + 564, + 204, + 583 + ], + "score": 1.0, + "content": "The ascent step along", + "type": "text" + }, + { + "bbox": [ + 204, + 564, + 220, + 580 + ], + "score": 0.91, + "content": "g _ { \\perp } ^ { ( t ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 564, + 493, + 583 + ], + "score": 1.0, + "content": "does not harm convergence SAM demonstrates that minimizing", + "type": "text" + }, + { + "bbox": [ + 493, + 568, + 504, + 579 + ], + "score": 0.86, + "content": "f _ { p }", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 578, + 504, + 597 + ], + "spans": [ + { + "bbox": [ + 104, + 578, + 327, + 597 + ], + "score": 1.0, + "content": "makes the network generalize better than minimizing", + "type": "text" + }, + { + "bbox": [ + 328, + 582, + 335, + 594 + ], + "score": 0.83, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 578, + 487, + 597 + ], + "score": 1.0, + "content": ". Even though our ascent step along", + "type": "text" + }, + { + "bbox": [ + 488, + 580, + 504, + 594 + ], + "score": 0.9, + "content": "g _ { \\perp } ^ { ( t ) }", + "type": "inline_equation" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 591, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 146, + 606 + ], + "score": 1.0, + "content": "increases", + "type": "text" + }, + { + "bbox": [ + 146, + 592, + 168, + 604 + ], + "score": 0.91, + "content": "f ( w )", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 591, + 245, + 606 + ], + "score": 1.0, + "content": ", it does not affect", + "type": "text" + }, + { + "bbox": [ + 245, + 593, + 271, + 605 + ], + "score": 0.93, + "content": "f _ { p } ( w )", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 591, + 454, + 606 + ], + "score": 1.0, + "content": ", so GSAM still decreases the perturbed loss", + "type": "text" + }, + { + "bbox": [ + 455, + 593, + 465, + 605 + ], + "score": 0.89, + "content": "f _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 591, + 505, + 606 + ], + "score": 1.0, + "content": "in a way", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 603, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 505, + 617 + ], + "score": 1.0, + "content": "similar to SAM. In Thm. 5.1, we formally prove the convergence of GSAM. In Sec. 6 and Appendix", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 613, + 455, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 455, + 628 + ], + "score": 1.0, + "content": "C, we empirically validate that the loss decreases and accuracy increases with training.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 106, + 631, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 632, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 106, + 632, + 505, + 644 + ], + "score": 1.0, + "content": "Illustration with a toy example We demonstrate different algorithms by a numerical toy example", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 642, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 505, + 655 + ], + "score": 1.0, + "content": "shown in Fig. 3. The trajectory of GSAM is closer to the ridge and tends to find a flat minimum.", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 654, + 505, + 666 + ], + "spans": [ + { + "bbox": [ + 106, + 654, + 505, + 666 + ], + "score": 1.0, + "content": "Intuitively, since the loss surface is smoother along the ridge than in sharp local minima, the sur-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 664, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 104, + 664, + 151, + 678 + ], + "score": 1.0, + "content": "rogate gap", + "type": "text" + }, + { + "bbox": [ + 152, + 664, + 174, + 677 + ], + "score": 0.92, + "content": "h ( w )", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 664, + 440, + 678 + ], + "score": 1.0, + "content": "is small near the ridge, and the ascent step in GSAM minimizes", + "type": "text" + }, + { + "bbox": [ + 440, + 665, + 447, + 675 + ], + "score": 0.75, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 664, + 506, + 678 + ], + "score": 1.0, + "content": "to pushes the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 675, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 675, + 294, + 689 + ], + "score": 1.0, + "content": "trajectory closer to the ridge. More concretely,", + "type": "text" + }, + { + "bbox": [ + 294, + 675, + 327, + 687 + ], + "score": 0.92, + "content": "\\nabla f ( w _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 675, + 506, + 689 + ], + "score": 1.0, + "content": "points to a sharp local solution and deviates", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 104, + 684, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 104, + 684, + 217, + 700 + ], + "score": 1.0, + "content": "from the ridge; in contrast,", + "type": "text" + }, + { + "bbox": [ + 217, + 686, + 239, + 698 + ], + "score": 0.92, + "content": "w _ { t } ^ { a d v }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 684, + 343, + 700 + ], + "score": 1.0, + "content": "is closer to the ridge and", + "type": "text" + }, + { + "bbox": [ + 343, + 687, + 386, + 699 + ], + "score": 0.93, + "content": "\\nabla f ( w _ { t } ^ { a d v } )", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 684, + 506, + 700 + ], + "score": 1.0, + "content": "is closer to the ridge descent", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 104, + 693, + 504, + 712 + ], + "spans": [ + { + "bbox": [ + 104, + 693, + 165, + 712 + ], + "score": 1.0, + "content": "direction than", + "type": "text" + }, + { + "bbox": [ + 165, + 697, + 199, + 709 + ], + "score": 0.93, + "content": "\\nabla f ( w _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 693, + 245, + 712 + ], + "score": 1.0, + "content": ". Note that", + "type": "text" + }, + { + "bbox": [ + 245, + 697, + 286, + 709 + ], + "score": 0.9, + "content": "\\nabla f _ { t } ^ { G S A M }", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 693, + 306, + 712 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 306, + 698, + 339, + 709 + ], + "score": 0.93, + "content": "\\nabla f ( w _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 693, + 466, + 712 + ], + "score": 1.0, + "content": "always lie at different sides of", + "type": "text" + }, + { + "bbox": [ + 466, + 697, + 504, + 709 + ], + "score": 0.92, + "content": "\\nabla f _ { p } ( w _ { t } )", + "type": "inline_equation" + } + ], + "index": 43 + }, + { + "bbox": [ + 103, + 707, + 504, + 725 + ], + "spans": [ + { + "bbox": [ + 103, + 707, + 246, + 725 + ], + "score": 1.0, + "content": "by construction (see Fig. 2), hence", + "type": "text" + }, + { + "bbox": [ + 246, + 709, + 287, + 721 + ], + "score": 0.9, + "content": "\\nabla f _ { t } ^ { G S A M }", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 707, + 466, + 725 + ], + "score": 1.0, + "content": "pushes the trajectory closer to the ridge than", + "type": "text" + }, + { + "bbox": [ + 467, + 710, + 504, + 722 + ], + "score": 0.91, + "content": "\\nabla f _ { p } ( w _ { t } )", + "type": "inline_equation" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 721, + 472, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 472, + 734 + ], + "score": 1.0, + "content": "does. 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One caveat when solving this", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 143, + 504, + 157 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 469, + 157 + ], + "score": 1.0, + "content": "weighted combination is the potential conflict between the gradients of the two terms, i.e.,", + "type": "text" + }, + { + "bbox": [ + 470, + 143, + 504, + 156 + ], + "score": 0.92, + "content": "\\nabla f _ { p } ( w )", + "type": "inline_equation" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 152, + 506, + 169 + ], + "spans": [ + { + "bbox": [ + 104, + 152, + 124, + 169 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 154, + 154, + 166 + ], + "score": 0.92, + "content": "\\overline { { \\nabla } h } ( \\boldsymbol { w } )", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 152, + 340, + 169 + ], + "score": 1.0, + "content": ". We illustrate this conflict by Fig. 2, where", + "type": "text" + }, + { + "bbox": [ + 340, + 154, + 464, + 166 + ], + "score": 0.91, + "content": "\\bar { \\nabla } h ( w ) = \\nabla f _ { p } ( w ) - \\nabla f ( w )", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 152, + 506, + 169 + ], + "score": 1.0, + "content": "(the grey", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 506, + 179 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 300, + 179 + ], + "score": 1.0, + "content": "dashed arrow) has a negative inner product with", + "type": "text" + }, + { + "bbox": [ + 301, + 165, + 335, + 177 + ], + "score": 0.93, + "content": "\\nabla f _ { p } ( w )", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 165, + 353, + 179 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 353, + 166, + 384, + 177 + ], + "score": 0.92, + "content": "\\nabla f ( w )", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 165, + 506, + 179 + ], + "score": 1.0, + "content": ". Hence, the gradient descent", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 175, + 506, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 175, + 332, + 190 + ], + "score": 1.0, + "content": "for the surrogate gap could potentially increase the loss", + "type": "text" + }, + { + "bbox": [ + 332, + 177, + 343, + 189 + ], + "score": 0.88, + "content": "f _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 175, + 506, + 190 + ], + "score": 1.0, + "content": ", harming the model’s performance. We", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 187, + 293, + 200 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 293, + 200 + ], + "score": 1.0, + "content": "empirically validate this argument in Sec. 6.4.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6, + "bbox_fs": [ + 104, + 121, + 506, + 200 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 222, + 493, + 234 + ], + "lines": [ + { + "bbox": [ + 106, + 222, + 494, + 234 + ], + "spans": [ + { + "bbox": [ + 106, + 222, + 494, + 234 + ], + "score": 1.0, + "content": "4.2 GRADIENT DECOMPOSITION AND ASCENT FOR THE MULTI-OBJECTIVE OPTIMIZATION", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 246, + 505, + 281 + ], + "lines": [ + { + "bbox": [ + 105, + 246, + 505, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 241, + 260 + ], + "score": 1.0, + "content": "Our primary goal is to minimize", + "type": "text" + }, + { + "bbox": [ + 242, + 248, + 253, + 259 + ], + "score": 0.89, + "content": "f _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 246, + 505, + 260 + ], + "score": 1.0, + "content": "because otherwise a flat solution of high loss is meaningless,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 258, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 205, + 270 + ], + "score": 1.0, + "content": "and the minimization of", + "type": "text" + }, + { + "bbox": [ + 205, + 258, + 213, + 268 + ], + "score": 0.83, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 258, + 294, + 270 + ], + "score": 1.0, + "content": "should not increase", + "type": "text" + }, + { + "bbox": [ + 294, + 258, + 304, + 270 + ], + "score": 0.86, + "content": "f _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 258, + 417, + 270 + ], + "score": 1.0, + "content": ". 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f ( w _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 395, + 506, + 411 + ], + "score": 1.0, + "content": "(by increasing", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 107, + 407, + 227, + 421 + ], + "spans": [ + { + "bbox": [ + 107, + 408, + 132, + 420 + ], + "score": 0.92, + "content": "f ( w _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 407, + 190, + 421 + ], + "score": 1.0, + "content": "and not affect", + "type": "text" + }, + { + "bbox": [ + 191, + 408, + 221, + 420 + ], + "score": 0.91, + "content": "f _ { p } ( w _ { t } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 407, + 227, + 421 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18.5, + "bbox_fs": [ + 104, + 352, + 506, + 421 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 425, + 506, + 559 + ], + "lines": [ + { + "bbox": [ + 104, + 425, + 507, + 442 + ], + "spans": [ + { + "bbox": [ + 104, + 425, + 368, + 442 + ], + "score": 1.0, + "content": "The full GSAM Algorithm is shown in Algo. 1 and Fig. 2, where", + "type": "text" + }, + { + "bbox": [ + 368, + 425, + 403, + 440 + ], + "score": 0.84, + "content": "g ^ { ( t ) } , g _ { p } ^ { ( t ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 425, + 507, + 442 + ], + "score": 1.0, + "content": "are noisy observations of", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 439, + 506, + 458 + ], + "spans": [ + { + "bbox": [ + 106, + 442, + 140, + 454 + ], + "score": 0.92, + "content": "\\nabla f ( w _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 439, + 158, + 458 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 158, + 442, + 195, + 455 + ], + "score": 0.93, + "content": "\\nabla f _ { p } ( w _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 439, + 268, + 458 + ], + "score": 1.0, + "content": ", respectively, and", + "type": "text" + }, + { + "bbox": [ + 268, + 439, + 303, + 456 + ], + "score": 0.95, + "content": "g _ { \\parallel } ^ { ( t ) } , g _ { \\perp } ^ { ( t ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 439, + 406, + 458 + ], + "score": 1.0, + "content": "are noisy observations of", + "type": "text" + }, + { + "bbox": [ + 407, + 442, + 443, + 455 + ], + "score": 0.94, + "content": "\\nabla f _ { \\parallel } ( w _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 439, + 462, + 458 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 462, + 442, + 501, + 454 + ], + "score": 0.92, + "content": "\\nabla f _ { \\perp } ( w _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 439, + 506, + 458 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 102, + 453, + 508, + 473 + ], + "spans": [ + { + "bbox": [ + 102, + 453, + 217, + 473 + ], + "score": 1.0, + "content": "respectively, by projecting", + "type": "text" + }, + { + "bbox": [ + 217, + 457, + 233, + 470 + ], + "score": 0.91, + "content": "g ^ { ( t ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 453, + 257, + 473 + ], + "score": 1.0, + "content": "onto", + "type": "text" + }, + { + "bbox": [ + 257, + 456, + 273, + 471 + ], + "score": 0.91, + "content": "g _ { p } ^ { ( t ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 453, + 383, + 473 + ], + "score": 1.0, + "content": ". 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Step 3) projects", + "type": "text" + }, + { + "bbox": [ + 446, + 482, + 462, + 495 + ], + "score": 0.9, + "content": "g ^ { ( t ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 479, + 485, + 499 + ], + "score": 1.0, + "content": "onto", + "type": "text" + }, + { + "bbox": [ + 485, + 480, + 501, + 495 + ], + "score": 0.9, + "content": "g _ { p } ^ { ( t ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 479, + 507, + 499 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 493, + 506, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 506, + 507 + ], + "score": 1.0, + "content": "which requires negligible computation compared to the forward and backward passes. 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In coding, GSAM feeds the “surrogate", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 101, + 542, + 483, + 564 + ], + "spans": [ + { + "bbox": [ + 101, + 542, + 204, + 564 + ], + "score": 1.0, + "content": "gradient” ∇fGSAMt , g", + "type": "text" + }, + { + "bbox": [ + 146, + 544, + 249, + 560 + ], + "score": 0.93, + "content": "\\nabla f _ { t } ^ { G S A M } \\triangleq g _ { p } ^ { ( t ) } - \\alpha g _ { \\perp } ^ { ( t ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 544, + 483, + 561 + ], + "score": 1.0, + "content": "to first-order gradient optimizers such as SGD and Adam.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 26.5, + "bbox_fs": [ + 100, + 425, + 508, + 564 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 565, + 505, + 626 + ], + "lines": [ + { + "bbox": [ + 106, + 564, + 504, + 583 + ], + "spans": [ + { + "bbox": [ + 106, + 564, + 204, + 583 + ], + "score": 1.0, + "content": "The ascent step along", + "type": "text" + }, + { + "bbox": [ + 204, + 564, + 220, + 580 + ], + "score": 0.91, + "content": "g _ { \\perp } ^ { ( t ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 564, + 493, + 583 + ], + "score": 1.0, + "content": "does not harm convergence SAM demonstrates that minimizing", + "type": "text" + }, + { + "bbox": [ + 493, + 568, + 504, + 579 + ], + "score": 0.86, + "content": "f _ { p }", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 578, + 504, + 597 + ], + "spans": [ + { + "bbox": [ + 104, + 578, + 327, + 597 + ], + "score": 1.0, + "content": "makes the network generalize better than minimizing", + "type": "text" + }, + { + "bbox": [ + 328, + 582, + 335, + 594 + ], + "score": 0.83, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 578, + 487, + 597 + ], + "score": 1.0, + "content": ". Even though our ascent step along", + "type": "text" + }, + { + "bbox": [ + 488, + 580, + 504, + 594 + ], + "score": 0.9, + "content": "g _ { \\perp } ^ { ( t ) }", + "type": "inline_equation" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 591, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 146, + 606 + ], + "score": 1.0, + "content": "increases", + "type": "text" + }, + { + "bbox": [ + 146, + 592, + 168, + 604 + ], + "score": 0.91, + "content": "f ( w )", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 591, + 245, + 606 + ], + "score": 1.0, + "content": ", it does not affect", + "type": "text" + }, + { + "bbox": [ + 245, + 593, + 271, + 605 + ], + "score": 0.93, + "content": "f _ { p } ( w )", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 591, + 454, + 606 + ], + "score": 1.0, + "content": ", so GSAM still decreases the perturbed loss", + "type": "text" + }, + { + "bbox": [ + 455, + 593, + 465, + 605 + ], + "score": 0.89, + "content": "f _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 591, + 505, + 606 + ], + "score": 1.0, + "content": "in a way", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 603, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 505, + 617 + ], + "score": 1.0, + "content": "similar to SAM. In Thm. 5.1, we formally prove the convergence of GSAM. In Sec. 6 and Appendix", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 613, + 455, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 455, + 628 + ], + "score": 1.0, + "content": "C, we empirically validate that the loss decreases and accuracy increases with training.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34, + "bbox_fs": [ + 104, + 564, + 505, + 628 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 631, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 632, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 106, + 632, + 505, + 644 + ], + "score": 1.0, + "content": "Illustration with a toy example We demonstrate different algorithms by a numerical toy example", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 642, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 505, + 655 + ], + "score": 1.0, + "content": "shown in Fig. 3. The trajectory of GSAM is closer to the ridge and tends to find a flat minimum.", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 654, + 505, + 666 + ], + "spans": [ + { + "bbox": [ + 106, + 654, + 505, + 666 + ], + "score": 1.0, + "content": "Intuitively, since the loss surface is smoother along the ridge than in sharp local minima, the sur-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 664, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 104, + 664, + 151, + 678 + ], + "score": 1.0, + "content": "rogate gap", + "type": "text" + }, + { + "bbox": [ + 152, + 664, + 174, + 677 + ], + "score": 0.92, + "content": "h ( w )", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 664, + 440, + 678 + ], + "score": 1.0, + "content": "is small near the ridge, and the ascent step in GSAM minimizes", + "type": "text" + }, + { + "bbox": [ + 440, + 665, + 447, + 675 + ], + "score": 0.75, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 664, + 506, + 678 + ], + "score": 1.0, + "content": "to pushes the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 675, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 675, + 294, + 689 + ], + "score": 1.0, + "content": "trajectory closer to the ridge. More concretely,", + "type": "text" + }, + { + "bbox": [ + 294, + 675, + 327, + 687 + ], + "score": 0.92, + "content": "\\nabla f ( w _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 675, + 506, + 689 + ], + "score": 1.0, + "content": "points to a sharp local solution and deviates", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 104, + 684, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 104, + 684, + 217, + 700 + ], + "score": 1.0, + "content": "from the ridge; in contrast,", + "type": "text" + }, + { + "bbox": [ + 217, + 686, + 239, + 698 + ], + "score": 0.92, + "content": "w _ { t } ^ { a d v }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 684, + 343, + 700 + ], + "score": 1.0, + "content": "is closer to the ridge and", + "type": "text" + }, + { + "bbox": [ + 343, + 687, + 386, + 699 + ], + "score": 0.93, + "content": "\\nabla f ( w _ { t } ^ { a d v } )", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 684, + 506, + 700 + ], + "score": 1.0, + "content": "is closer to the ridge descent", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 104, + 693, + 504, + 712 + ], + "spans": [ + { + "bbox": [ + 104, + 693, + 165, + 712 + ], + "score": 1.0, + "content": "direction than", + "type": "text" + }, + { + "bbox": [ + 165, + 697, + 199, + 709 + ], + "score": 0.93, + "content": "\\nabla f ( w _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 693, + 245, + 712 + ], + "score": 1.0, + "content": ". Note that", + "type": "text" + }, + { + "bbox": [ + 245, + 697, + 286, + 709 + ], + "score": 0.9, + "content": "\\nabla f _ { t } ^ { G S A M }", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 693, + 306, + 712 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 306, + 698, + 339, + 709 + ], + "score": 0.93, + "content": "\\nabla f ( w _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 693, + 466, + 712 + ], + "score": 1.0, + "content": "always lie at different sides of", + "type": "text" + }, + { + "bbox": [ + 466, + 697, + 504, + 709 + ], + "score": 0.92, + "content": "\\nabla f _ { p } ( w _ { t } )", + "type": "inline_equation" + } + ], + "index": 43 + }, + { + "bbox": [ + 103, + 707, + 504, + 725 + ], + "spans": [ + { + "bbox": [ + 103, + 707, + 246, + 725 + ], + "score": 1.0, + "content": "by construction (see Fig. 2), hence", + "type": "text" + }, + { + "bbox": [ + 246, + 709, + 287, + 721 + ], + "score": 0.9, + "content": "\\nabla f _ { t } ^ { G S A M }", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 707, + 466, + 725 + ], + "score": 1.0, + "content": "pushes the trajectory closer to the ridge than", + "type": "text" + }, + { + "bbox": [ + 467, + 710, + 504, + 722 + ], + "score": 0.91, + "content": "\\nabla f _ { p } ( w _ { t } )", + "type": "inline_equation" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 721, + 472, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 472, + 734 + ], + "score": 1.0, + "content": "does. 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Left: Overview of the procedures", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 210, + 506, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 210, + 329, + 225 + ], + "score": 1.0, + "content": "of SGD, SAM and GSAM. SGD takes a descent step at", + "type": "text" + }, + { + "bbox": [ + 330, + 213, + 342, + 222 + ], + "score": 0.86, + "content": "w _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 210, + 367, + 225 + ], + "score": 1.0, + "content": "using", + "type": "text" + }, + { + "bbox": [ + 367, + 211, + 401, + 223 + ], + "score": 0.94, + "content": "\\nabla f ( w _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 210, + 506, + 225 + ], + "score": 1.0, + "content": "(orange), which points to", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 103, + 220, + 504, + 236 + ], + "spans": [ + { + "bbox": [ + 103, + 220, + 412, + 236 + ], + "score": 1.0, + "content": "a sharp local minima. SAM first performs gradient ascent in the direction of", + "type": "text" + }, + { + "bbox": [ + 413, + 222, + 446, + 234 + ], + "score": 0.93, + "content": "\\nabla f ( w _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 220, + 482, + 236 + ], + "score": 1.0, + "content": "to reach", + "type": "text" + }, + { + "bbox": [ + 482, + 222, + 504, + 234 + ], + "score": 0.91, + "content": "w _ { t } ^ { a d v }", + "type": "inline_equation" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 231, + 506, + 247 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 327, + 247 + ], + "score": 1.0, + "content": "with a higher loss, followed by descent with gradient", + "type": "text" + }, + { + "bbox": [ + 327, + 233, + 371, + 245 + ], + "score": 0.93, + "content": "\\nabla f ( w _ { t } ^ { a d v } )", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 231, + 506, + 247 + ], + "score": 1.0, + "content": "(green) at the perturbed weight.", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 242, + 506, + 257 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 147, + 257 + ], + "score": 1.0, + "content": "Based on", + "type": "text" + }, + { + "bbox": [ + 147, + 244, + 181, + 256 + ], + "score": 0.93, + "content": "\\nabla f ( w _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 242, + 201, + 257 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 201, + 244, + 244, + 256 + ], + "score": 0.93, + "content": "\\nabla f ( w _ { t } ^ { a d v } )", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 242, + 506, + 257 + ], + "score": 1.0, + "content": ", GSAM updates in a new direction (red) that points to a flatter", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 255, + 505, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 255, + 505, + 267 + ], + "score": 1.0, + "content": "region. Right: Trajectories by different methods. SGD and SAM fall into different sharp local", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 266, + 496, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 496, + 279 + ], + "score": 1.0, + "content": "minima, while GSAM reaches a flat region. A video is in the supplement for better visualization.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6 + } + ], + "index": 3.5 + }, + { + "type": "title", + "bbox": [ + 107, + 297, + 321, + 310 + ], + "lines": [ + { + "bbox": [ + 105, + 297, + 322, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 322, + 311 + ], + "score": 1.0, + "content": "5 THEORETICAL PROPERTIES OF GSAM", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "title", + "bbox": [ + 108, + 321, + 279, + 333 + ], + "lines": [ + { + "bbox": [ + 105, + 321, + 280, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 280, + 334 + ], + "score": 1.0, + "content": "5.1 CONVERGENCE DURING TRAINING", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 106, + 341, + 506, + 392 + ], + "lines": [ + { + "bbox": [ + 105, + 341, + 506, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 300, + 356 + ], + "score": 1.0, + "content": "Theorem 5.1. Consider a non-convex function", + "type": "text" + }, + { + "bbox": [ + 300, + 342, + 322, + 354 + ], + "score": 0.92, + "content": "f ( w )", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 341, + 452, + 356 + ], + "score": 1.0, + "content": "with Lipschitz-smooth constant", + "type": "text" + }, + { + "bbox": [ + 452, + 343, + 460, + 352 + ], + "score": 0.8, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 341, + 506, + 356 + ], + "score": 1.0, + "content": "and lower", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 353, + 506, + 368 + ], + "spans": [ + { + "bbox": [ + 104, + 353, + 135, + 368 + ], + "score": 1.0, + "content": "bound", + "type": "text" + }, + { + "bbox": [ + 135, + 355, + 155, + 367 + ], + "score": 0.88, + "content": "f _ { m i n }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 353, + 383, + 368 + ], + "score": 1.0, + "content": ". 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For some constant", + "type": "text" + }, + { + "bbox": [ + 343, + 370, + 351, + 377 + ], + "score": 0.68, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 366, + 432, + 380 + ], + "score": 1.0, + "content": ", with learning rate", + "type": "text" + }, + { + "bbox": [ + 432, + 367, + 483, + 379 + ], + "score": 0.92, + "content": "\\eta _ { t } = \\eta _ { 0 } / \\sqrt { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 366, + 506, + 380 + ], + "score": 1.0, + "content": ", and", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 379, + 456, + 392 + ], + "spans": [ + { + "bbox": [ + 104, + 379, + 201, + 392 + ], + "score": 1.0, + "content": "perturbation amplitude", + "type": "text" + }, + { + "bbox": [ + 202, + 381, + 211, + 391 + ], + "score": 0.83, + "content": "\\rho _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 379, + 367, + 392 + ], + "score": 1.0, + "content": "proportional to the learning rate, e.g.,", + "type": "text" + }, + { + "bbox": [ + 367, + 379, + 416, + 392 + ], + "score": 0.93, + "content": "\\rho _ { t } = \\rho _ { 0 } / \\sqrt { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 379, + 456, + 392 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13.5 + }, + { + "type": "interline_equation", + "bbox": [ + 136, + 396, + 476, + 430 + ], + "lines": [ + { + "bbox": [ + 136, + 396, + 476, + 430 + ], + "spans": [ + { + "bbox": [ + 136, + 396, + 476, + 430 + ], + "score": 0.94, + "content": "\\frac { 1 } { T } \\sum _ { t = 1 } ^ { T } \\mathbb { E } \\Big | \\Big | \\nabla f _ { p } ( w _ { t } ) \\Big | \\Big | _ { 2 } ^ { 2 } \\leq \\frac { C _ { 1 } + C _ { 2 } \\log T } { \\sqrt { T } } , \\quad \\frac { 1 } { T } \\sum _ { t = 1 } ^ { T } \\mathbb { E } \\Big | \\Big | \\nabla f ( w _ { t } ) \\Big | \\Big | _ { 2 } ^ { 2 } \\leq \\frac { C _ { 3 } + C _ { 4 } \\log T } { \\sqrt { T } }", + 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constants.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 106, + 454, + 505, + 478 + ], + "lines": [ + { + "bbox": [ + 106, + 454, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 106, + 454, + 197, + 469 + ], + "score": 1.0, + "content": "Thm. 5.1 implies both", + "type": "text" + }, + { + "bbox": [ + 197, + 456, + 208, + 468 + ], + "score": 0.88, + "content": "f _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 454, + 225, + 469 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 226, + 456, + 233, + 467 + ], + "score": 0.86, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 454, + 340, + 469 + ], + "score": 1.0, + "content": "converge in GSAM at rate", + "type": "text" + }, + { + "bbox": [ + 341, + 454, + 399, + 468 + ], + "score": 0.93, + "content": "O ( \\log T / \\sqrt { T } )", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 454, + 505, + 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provably lower than SAM’s, so GSAM is", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 523, + 367, + 535 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 367, + 535 + ], + "score": 1.0, + "content": "expected to find a smoother minimum with better generalization.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 106, + 536, + 505, + 595 + ], + "lines": [ + { + "bbox": [ + 106, + 537, + 504, + 549 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 504, + 549 + ], + "score": 1.0, + "content": "Theorem 5.2 (PAC-Bayesian Theorem (McAllester, 2003)). Suppose the training set has m ele-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 548, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 469, + 563 + ], + "score": 1.0, + "content": "ments drawn i.i.d. from the true distribution, and denote the loss on the training set as", + "type": "text" + }, + { + "bbox": [ + 469, + 548, + 505, + 562 + ], + "score": 0.91, + "content": "{ \\widehat { f } } ( w ) =", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 115, + 559, + 507, + 576 + ], + "spans": [ + { + "bbox": [ + 115, + 559, + 507, + 576 + ], + "score": 1.0, + "content": "Pmi=1 f (w, xi), where we use xi to denote the (input, target) pair of the i-th element. Let w be", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 573, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 573, + 264, + 585 + ], + "score": 1.0, + "content": "learned from the training set. Suppose", + "type": "text" + }, + { + "bbox": [ + 264, + 574, + 272, + 582 + ], + "score": 0.55, + "content": "w", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 573, + 421, + 585 + ], + "score": 1.0, + "content": "is drawn from posterior distribution", + "type": "text" + }, + { + "bbox": [ + 422, + 573, + 431, + 583 + ], + "score": 0.59, + "content": "\\mathcal { Q } .", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 573, + 505, + 585 + ], + "score": 1.0, + "content": "Denote the prior", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 584, + 302, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 584, + 269, + 595 + ], + "score": 1.0, + "content": "distribution (independent of training) as", + "type": "text" + }, + { + "bbox": [ + 269, + 584, + 278, + 594 + ], + "score": 0.76, + "content": "\\mathcal { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 584, + 302, + 595 + ], + "score": 1.0, + "content": ", then", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27 + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 599, + 512, + 626 + ], + "lines": [ + { + "bbox": [ + 111, + 599, + 512, + 626 + ], + "spans": [ + { + "bbox": [ + 111, + 599, + 512, + 626 + ], + "score": 0.59, + "content": "\\mathbb { E } _ { w \\sim Q } \\mathbb { E } _ { x } f ( w , x ) \\le \\mathbb { E } _ { w \\sim Q } \\widehat { f } ( w ) + 4 \\sqrt { \\Big ( K L ( Q | | \\mathcal { P } ) + \\log \\frac { 2 m } { a } \\Big ) / m } w i t h \\ p r o b a b i l i t y \\ a t \\ l e a s t 1 - a", + "type": "interline_equation", + "image_path": "40e4313f846d41c34d8f7d3b9fbf975b19216b7aae4dd612155f8d60d593487b.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 111, + 599, + 512, + 626 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 631, + 505, + 666 + ], + "lines": [ + { + "bbox": [ + 106, + 629, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 629, + 267, + 646 + ], + "score": 1.0, + "content": "Corollary 5.2.1. 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Right: Trajectories by different methods. SGD and SAM fall into different sharp local", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 266, + 496, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 496, + 279 + ], + "score": 1.0, + "content": "minima, while GSAM reaches a flat region. A video is in the supplement for better visualization.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6 + } + ], + "index": 3.5 + }, + { + "type": "title", + "bbox": [ + 107, + 297, + 321, + 310 + ], + "lines": [ + { + "bbox": [ + 105, + 297, + 322, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 322, + 311 + ], + "score": 1.0, + "content": "5 THEORETICAL PROPERTIES OF GSAM", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "title", + "bbox": [ + 108, + 321, + 279, + 333 + ], + "lines": [ + { + "bbox": [ + 105, + 321, + 280, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 280, + 334 + ], + "score": 1.0, + "content": "5.1 CONVERGENCE DURING TRAINING", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 106, + 341, + 506, + 392 + ], + "lines": [ + { + "bbox": [ + 105, + 341, + 506, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 300, + 356 + ], + "score": 1.0, + "content": "Theorem 5.1. 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Suppose the training set has m ele-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 548, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 469, + 563 + ], + "score": 1.0, + "content": "ments drawn i.i.d. from the true distribution, and denote the loss on the training set as", + "type": "text" + }, + { + "bbox": [ + 469, + 548, + 505, + 562 + ], + "score": 0.91, + "content": "{ \\widehat { f } } ( w ) =", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 115, + 559, + 507, + 576 + ], + "spans": [ + { + "bbox": [ + 115, + 559, + 507, + 576 + ], + "score": 1.0, + "content": "Pmi=1 f (w, xi), where we use xi to denote the (input, target) pair of the i-th element. Let w be", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 573, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 573, + 264, + 585 + ], + "score": 1.0, + "content": "learned from the training set. 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Compared to SAM, the proposed method generates a", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 103, + 270, + 367, + 293 + ], + "spans": [ + { + "bbox": [ + 103, + 270, + 233, + 293 + ], + "score": 1.0, + "content": "total decrease in surrogate gap", + "type": "text" + }, + { + "bbox": [ + 233, + 273, + 278, + 288 + ], + "score": 0.93, + "content": "\\Sigma _ { t = 1 } ^ { T } \\Delta \\widehat { h } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 270, + 367, + 293 + ], + "score": 1.0, + "content": ", which is bounded by", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 12 + }, + { + "type": "interline_equation", + "bbox": [ + 206, + 291, + 405, + 326 + ], + "lines": [ + { + "bbox": [ + 206, + 291, + 405, + 326 + ], + "spans": [ + { + "bbox": [ + 206, + 291, + 405, + 326 + ], + "score": 0.93, + "content": "\\frac { \\alpha c ^ { 2 } \\rho _ { 0 } ^ { 2 } \\eta _ { 0 } | \\sigma | _ { m i n } ^ { 2 } } { G ^ { 2 } } \\le \\operatorname* { l i m } _ { T \\to \\infty } \\sum _ { t = 1 } ^ { T } \\Delta \\widehat { h } _ { t } \\le 2 . 7 \\alpha L ^ { 2 } \\eta _ { 0 } \\rho _ { 0 } ^ { 2 }", + "type": "interline_equation", + "image_path": "3f670afed15392088428627fda19523afa663ca466b07b6112b86954f1322bf2.jpg" + } + ] + } + ], + "index": 16.5, + "virtual_lines": [ + { + "bbox": [ + 206, + 291, + 405, + 308.5 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 206, + 308.5, + 405, + 326.0 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 336, + 505, + 372 + ], + "lines": [ + { + "bbox": [ + 101, + 331, + 510, + 363 + ], + "spans": [ + { + "bbox": [ + 101, + 331, + 329, + 363 + ], + "score": 1.0, + "content": "We provide proof in the appendix. The lower-bound of provably non-trivial decrease in the surrogate gap. Com", + "type": "text" + }, + { + "bbox": [ + 329, + 336, + 373, + 352 + ], + "score": 0.93, + "content": "\\Sigma _ { t = 1 } ^ { T } \\Delta \\widehat { h } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 331, + 510, + 363 + ], + "score": 1.0, + "content": "indicates that GSAM achieves aCorollary 5.2.1, GSAM provably", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 361, + 318, + 373 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 318, + 373 + ], + "score": 1.0, + "content": "improves the generalization performance over SAM.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5 + }, + { + "type": "title", + "bbox": [ + 108, + 388, + 200, + 401 + ], + "lines": [ + { + "bbox": [ + 105, + 387, + 201, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 201, + 402 + ], + "score": 1.0, + "content": "6 EXPERIMENTS", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "title", + "bbox": [ + 106, + 408, + 461, + 419 + ], + "lines": [ + { + "bbox": [ + 106, + 408, + 462, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 408, + 462, + 419 + ], + "score": 1.0, + "content": "6.1 GSAM IMPROVES TEST PERFORMANCE ON VARIOUS MODEL ARCHITECTURES", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 106, + 427, + 505, + 560 + ], + "lines": [ + { + "bbox": [ + 105, + 426, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 505, + 441 + ], + "score": 1.0, + "content": "We conduct experiments with ResNets (He et al., 2016), Vision Transformers (ViTs) (Dosovitskiy", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 438, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 505, + 451 + ], + "score": 1.0, + "content": "et al., 2020) and MLP-Mixers (Tolstikhin et al., 2021). Following the settings by Chen et al. (2021),", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 449, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 104, + 449, + 505, + 463 + ], + "score": 1.0, + "content": "we train on the ImageNet-1k (Deng et al., 2009) training set using the Inception-style (Szegedy", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 461, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 506, + 474 + ], + "score": 1.0, + "content": "et al., 2015) pre-processing without extra training data or strong augmentation. For all models, we", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 472, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 505, + 484 + ], + "score": 1.0, + "content": "search for the best learning rate and weight decay for vanilla training, and then use the same values", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 483, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 401, + 495 + ], + "score": 1.0, + "content": "for the experiments with SAM and GSAM. For ResNets, we search for", + "type": "text" + }, + { + "bbox": [ + 401, + 485, + 408, + 495 + ], + "score": 0.8, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 483, + 505, + 495 + ], + "score": 1.0, + "content": "from 0.01 to 0.05 with", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 494, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 325, + 506 + ], + "score": 1.0, + "content": "a stepsize 0.01. For ViTs and Mixers, we search for", + "type": "text" + }, + { + "bbox": [ + 325, + 496, + 333, + 505 + ], + "score": 0.8, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 494, + 505, + 506 + ], + "score": 1.0, + "content": "from 0.05 to 0.6 with a stepsize 0.05. In", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 505, + 506, + 517 + ], + "spans": [ + { + "bbox": [ + 106, + 505, + 200, + 517 + ], + "score": 1.0, + "content": "GSAM, we search for", + "type": "text" + }, + { + "bbox": [ + 201, + 506, + 209, + 515 + ], + "score": 0.76, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 505, + 222, + 517 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 223, + 505, + 295, + 517 + ], + "score": 0.89, + "content": "\\{ 0 . 0 1 , 0 . 0 2 , 0 . 0 3 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 505, + 367, + 517 + ], + "score": 1.0, + "content": "for ResNets and", + "type": "text" + }, + { + "bbox": [ + 368, + 507, + 376, + 515 + ], + "score": 0.73, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 505, + 389, + 517 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 390, + 505, + 448, + 517 + ], + "score": 0.89, + "content": "\\{ 0 . 1 , 0 . 2 , 0 . 3 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 505, + 506, + 517 + ], + "score": 1.0, + "content": "for ViTs and", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 515, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 506, + 529 + ], + "score": 1.0, + "content": "Mixers. Considering that each step in SAM and GSAM requires twice the computation of vanilla", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 527, + 506, + 540 + ], + "spans": [ + { + "bbox": [ + 104, + 527, + 506, + 540 + ], + "score": 1.0, + "content": "training, we experiment with the vanilla training for twice the epochs of SAM and GSAM, but we", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 538, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 505, + 550 + ], + "score": 1.0, + "content": "observe no significant improvements from the longer training (Table 5 in appendix). We summarize", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 549, + 338, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 549, + 338, + 561 + ], + "score": 1.0, + "content": "the best hyper-parameters for each model in Appendix B.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 27.5 + }, + { + "type": "text", + "bbox": [ + 107, + 565, + 505, + 654 + ], + "lines": [ + { + "bbox": [ + 105, + 564, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 506, + 578 + ], + "score": 1.0, + "content": "We report the performances on ImageNet (Deng et al., 2009), ImageNet-v2 (Recht et al., 2019) and", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 577, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 505, + 589 + ], + "score": 1.0, + "content": "ImageNet-Real (Beyer et al., 2020) in Table 1. GSAM consistently improves over SAM and vanilla", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 587, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 363, + 600 + ], + "score": 1.0, + "content": "training (with SGD or AdamW): on ViT-B/32, GSAM achieves", + "type": "text" + }, + { + "bbox": [ + 363, + 588, + 392, + 598 + ], + "score": 0.9, + "content": "+ 5 . 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 587, + 505, + 600 + ], + "score": 1.0, + "content": "improvement over AdamW", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 598, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 124, + 610 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 599, + 152, + 609 + ], + "score": 0.89, + "content": "+ 3 . 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 598, + 412, + 610 + ], + "score": 1.0, + "content": "over SAM in top-1 accuracy; on Mixer-B/32, GSAM achieves", + "type": "text" + }, + { + "bbox": [ + 413, + 599, + 446, + 609 + ], + "score": 0.9, + "content": "+ 1 1 . 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 598, + 505, + 610 + ], + "score": 1.0, + "content": "over AdamW", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 609, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 123, + 622 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 610, + 152, + 620 + ], + "score": 0.87, + "content": "+ 1 . 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 610, + 464, + 622 + ], + "score": 1.0, + "content": "over SAM. We ignore the standard deviation since it is typically negligible", + "type": "text" + }, + { + "bbox": [ + 464, + 609, + 502, + 621 + ], + "score": 0.86, + "content": "( < 0 . 1 \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 610, + 505, + 622 + ], + "score": 0.0, + "content": "", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 620, + 505, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 505, + 633 + ], + "score": 1.0, + "content": "compared to the improvements. We also test the generalization performance on out-of-distribution", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 630, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 506, + 645 + ], + "score": 1.0, + "content": "data (ImageNet-R and ImageNet-C), and the observation is consistent with that on ImageNet, e.g.,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 642, + 375, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 642, + 134, + 653 + ], + "score": 0.89, + "content": "+ 5 . 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 642, + 216, + 655 + ], + "score": 1.0, + "content": "on ImageNet-R and", + "type": "text" + }, + { + "bbox": [ + 216, + 642, + 244, + 653 + ], + "score": 0.89, + "content": "+ 5 . 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 642, + 375, + 655 + ], + "score": 1.0, + "content": "on ImageNet-C for Mixer-B/32.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 37.5 + }, + { + "type": "title", + "bbox": [ + 107, + 667, + 482, + 678 + ], + "lines": [ + { + "bbox": [ + 106, + 667, + 482, + 679 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 482, + 679 + ], + "score": 1.0, + "content": "6.2 GSAM FINDS A MINIMUM WHOSE HESSIAN HAS SMALL DOMINANT EIGENVALUES", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 283, + 700 + ], + "score": 1.0, + "content": "Lemma 3.3 indicates that the surrogate gap", + "type": "text" + }, + { + "bbox": [ + 283, + 688, + 290, + 698 + ], + "score": 0.8, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "is an equivalent measure of the dominant eigenvalue", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 232, + 711 + ], + "score": 1.0, + "content": "of the Hessian, and minimizing", + "type": "text" + }, + { + "bbox": [ + 233, + 699, + 240, + 709 + ], + "score": 0.79, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "equivalently searches for a flat minimum. We empirically validate", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 347, + 722 + ], + "score": 1.0, + "content": "this in Fig. 4. As shown in the left subfigure, for some fixed", + "type": "text" + }, + { + "bbox": [ + 347, + 712, + 353, + 721 + ], + "score": 0.76, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 709, + 400, + 722 + ], + "score": 1.0, + "content": ", increasing", + "type": "text" + }, + { + "bbox": [ + 401, + 712, + 408, + 720 + ], + "score": 0.72, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "decreases the dominant", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 721, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 732 + ], + "score": 1.0, + "content": "value and improves generalization (test accuracy). In the middle subfigure, we plot the dominant", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 44.5 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 104, + 79, + 506, + 97 + ], + "spans": [ + { + "bbox": [ + 104, + 79, + 133, + 97 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 80, + 176, + 95 + ], + "score": 0.93, + "content": "C = { \\widehat { f } } ( w )", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 79, + 313, + 97 + ], + "score": 1.0, + "content": "is the empirical training loss, and", + "type": "text" + }, + { + "bbox": [ + 313, + 80, + 321, + 93 + ], + "score": 0.78, + "content": "\\widehat { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 79, + 506, + 97 + ], + "score": 1.0, + "content": "is the surrogate gap evaluated on the training", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 123, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 123, + 106 + ], + "score": 1.0, + "content": "set.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 104, + 79, + 506, + 106 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 112, + 505, + 196 + ], + "lines": [ + { + "bbox": [ + 105, + 112, + 506, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 113, + 267, + 127 + ], + "score": 1.0, + "content": "Corollary 5.2.1 implies that minimizing", + "type": "text" + }, + { + "bbox": [ + 267, + 112, + 275, + 124 + ], + "score": 0.82, + "content": "\\widehat { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 113, + 506, + 127 + ], + "score": 1.0, + "content": "(right hand side of Eq. 7) is expected to achieve a tighter", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 124, + 506, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 124, + 506, + 138 + ], + "score": 1.0, + "content": "upper bound of the generalization performance (left hand side of Eq. 7). The third term on the right", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 136, + 505, + 149 + ], + "spans": [ + { + "bbox": [ + 106, + 136, + 350, + 149 + ], + "score": 1.0, + "content": "of Eq. 7 is typically hard to analyze and often simplified to", + "type": "text" + }, + { + "bbox": [ + 351, + 136, + 364, + 146 + ], + "score": 0.67, + "content": "L 2", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 136, + 505, + 149 + ], + "score": 1.0, + "content": "regularization (Foret et al., 2020).", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 147, + 506, + 163 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 146, + 163 + ], + "score": 1.0, + "content": "Note that", + "type": "text" + }, + { + "bbox": [ + 146, + 147, + 195, + 161 + ], + "score": 0.93, + "content": "f _ { p } = C + \\widehat { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 148, + 264, + 163 + ], + "score": 1.0, + "content": "only holds when", + "type": "text" + }, + { + "bbox": [ + 265, + 150, + 290, + 161 + ], + "score": 0.87, + "content": "\\rho _ { t r a i n }", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 148, + 506, + 163 + ], + "score": 1.0, + "content": "(the perturbation amplitude specified by users during", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 160, + 505, + 173 + ], + "spans": [ + { + "bbox": [ + 106, + 160, + 173, + 173 + ], + "score": 1.0, + "content": "training) equals", + "type": "text" + }, + { + "bbox": [ + 173, + 162, + 195, + 172 + ], + "score": 0.77, + "content": "\\rho _ { t r u e }", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 160, + 505, + 173 + ], + "score": 1.0, + "content": "(the ground truth value determined by underlying data distribution); when", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 171, + 506, + 186 + ], + "spans": [ + { + "bbox": [ + 106, + 173, + 168, + 185 + ], + "score": 0.75, + "content": "\\rho _ { t r a i n } \\neq \\rho _ { t r u e }", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 172, + 173, + 186 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 173, + 171, + 219, + 185 + ], + "score": 0.83, + "content": "m i n ( f _ { p } , \\widehat { h } )", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 172, + 311, + 186 + ], + "score": 1.0, + "content": "is more effective than", + "type": "text" + }, + { + "bbox": [ + 312, + 172, + 348, + 185 + ], + "score": 0.89, + "content": "m i n ( f _ { p } )", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 172, + 506, + 186 + ], + "score": 1.0, + "content": "in terms of minimizing generalization", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 183, + 297, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 183, + 297, + 196 + ], + "score": 1.0, + "content": "loss. A detailed discussion is in Appendix A.7.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5, + "bbox_fs": [ + 105, + 112, + 506, + 196 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 198, + 505, + 288 + ], + "lines": [ + { + "bbox": [ + 106, + 197, + 505, + 211 + ], + "spans": [ + { + "bbox": [ + 106, + 197, + 505, + 211 + ], + "score": 1.0, + "content": "Theorem 5.3 (Unlike SAM, GSAM decreases the surrogate gap). 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The lower-bound of provably non-trivial decrease in the surrogate gap. Com", + "type": "text" + }, + { + "bbox": [ + 329, + 336, + 373, + 352 + ], + "score": 0.93, + "content": "\\Sigma _ { t = 1 } ^ { T } \\Delta \\widehat { h } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 331, + 510, + 363 + ], + "score": 1.0, + "content": "indicates that GSAM achieves aCorollary 5.2.1, GSAM provably", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 361, + 318, + 373 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 318, + 373 + ], + "score": 1.0, + "content": "improves the generalization performance over SAM.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5, + "bbox_fs": [ + 101, + 331, + 510, + 373 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 388, + 200, + 401 + ], + "lines": [ + { + "bbox": [ + 105, + 387, + 201, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 201, + 402 + ], + "score": 1.0, + "content": "6 EXPERIMENTS", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "title", + "bbox": [ + 106, + 408, + 461, + 419 + ], + "lines": [ + { + "bbox": [ + 106, + 408, + 462, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 408, + 462, + 419 + ], + "score": 1.0, + "content": "6.1 GSAM IMPROVES TEST PERFORMANCE ON VARIOUS MODEL ARCHITECTURES", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 106, + 427, + 505, + 560 + ], + "lines": [ + { + "bbox": [ + 105, + 426, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 505, + 441 + ], + "score": 1.0, + "content": "We conduct experiments with ResNets (He et al., 2016), Vision Transformers (ViTs) (Dosovitskiy", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 438, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 505, + 451 + ], + "score": 1.0, + "content": "et al., 2020) and MLP-Mixers (Tolstikhin et al., 2021). Following the settings by Chen et al. (2021),", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 449, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 104, + 449, + 505, + 463 + ], + "score": 1.0, + "content": "we train on the ImageNet-1k (Deng et al., 2009) training set using the Inception-style (Szegedy", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 461, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 506, + 474 + ], + "score": 1.0, + "content": "et al., 2015) pre-processing without extra training data or strong augmentation. For all models, we", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 472, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 505, + 484 + ], + "score": 1.0, + "content": "search for the best learning rate and weight decay for vanilla training, and then use the same values", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 483, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 401, + 495 + ], + "score": 1.0, + "content": "for the experiments with SAM and GSAM. For ResNets, we search for", + "type": "text" + }, + { + "bbox": [ + 401, + 485, + 408, + 495 + ], + "score": 0.8, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 483, + 505, + 495 + ], + "score": 1.0, + "content": "from 0.01 to 0.05 with", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 494, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 325, + 506 + ], + "score": 1.0, + "content": "a stepsize 0.01. For ViTs and Mixers, we search for", + "type": "text" + }, + { + "bbox": [ + 325, + 496, + 333, + 505 + ], + "score": 0.8, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 494, + 505, + 506 + ], + "score": 1.0, + "content": "from 0.05 to 0.6 with a stepsize 0.05. In", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 505, + 506, + 517 + ], + "spans": [ + { + "bbox": [ + 106, + 505, + 200, + 517 + ], + "score": 1.0, + "content": "GSAM, we search for", + "type": "text" + }, + { + "bbox": [ + 201, + 506, + 209, + 515 + ], + "score": 0.76, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 505, + 222, + 517 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 223, + 505, + 295, + 517 + ], + "score": 0.89, + "content": "\\{ 0 . 0 1 , 0 . 0 2 , 0 . 0 3 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 505, + 367, + 517 + ], + "score": 1.0, + "content": "for ResNets and", + "type": "text" + }, + { + "bbox": [ + 368, + 507, + 376, + 515 + ], + "score": 0.73, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 505, + 389, + 517 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 390, + 505, + 448, + 517 + ], + "score": 0.89, + "content": "\\{ 0 . 1 , 0 . 2 , 0 . 3 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 505, + 506, + 517 + ], + "score": 1.0, + "content": "for ViTs and", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 515, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 506, + 529 + ], + "score": 1.0, + "content": "Mixers. Considering that each step in SAM and GSAM requires twice the computation of vanilla", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 527, + 506, + 540 + ], + "spans": [ + { + "bbox": [ + 104, + 527, + 506, + 540 + ], + "score": 1.0, + "content": "training, we experiment with the vanilla training for twice the epochs of SAM and GSAM, but we", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 538, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 505, + 550 + ], + "score": 1.0, + "content": "observe no significant improvements from the longer training (Table 5 in appendix). We summarize", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 549, + 338, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 549, + 338, + 561 + ], + "score": 1.0, + "content": "the best hyper-parameters for each model in Appendix B.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 27.5, + "bbox_fs": [ + 104, + 426, + 506, + 561 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 565, + 505, + 654 + ], + "lines": [ + { + "bbox": [ + 105, + 564, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 506, + 578 + ], + "score": 1.0, + "content": "We report the performances on ImageNet (Deng et al., 2009), ImageNet-v2 (Recht et al., 2019) and", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 577, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 505, + 589 + ], + "score": 1.0, + "content": "ImageNet-Real (Beyer et al., 2020) in Table 1. 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We empirically validate", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 347, + 722 + ], + "score": 1.0, + "content": "this in Fig. 4. As shown in the left subfigure, for some fixed", + "type": "text" + }, + { + "bbox": [ + 347, + 712, + 353, + 721 + ], + "score": 0.76, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 709, + 400, + 722 + ], + "score": 1.0, + "content": ", increasing", + "type": "text" + }, + { + "bbox": [ + 401, + 712, + 408, + 720 + ], + "score": 0.72, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "decreases the dominant", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 721, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 732 + ], + "score": 1.0, + "content": "value and improves generalization (test accuracy). In the middle subfigure, we plot the dominant", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 326, + 506, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 281, + 340 + ], + "score": 1.0, + "content": "eigenvalues estimated by the surrogate gap,", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 281, + 326, + 343, + 339 + ], + "score": 0.93, + "content": "\\sigma _ { m a x } \\approx 2 h / \\rho ^ { 2 }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 344, + 326, + 506, + 340 + ], + "score": 1.0, + "content": "(Lemma 3.3). In the right subfigure, we", + "type": "text", + "cross_page": true + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 337, + 506, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 337, + 506, + 351 + ], + "score": 1.0, + "content": "directly calculate the dominant eigenvalues using the power-iteration (Mises & Pollaczek-Geiringer,", + "type": "text", + "cross_page": true + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 348, + 505, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 440, + 362 + ], + "score": 1.0, + "content": "1929). The estimated dominant eigenvalues (middle) match the real eigenvalues", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 440, + 350, + 464, + 360 + ], + "score": 0.88, + "content": "\\sigma _ { m a x }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 464, + 348, + 505, + 362 + ], + "score": 1.0, + "content": "(right) in", + "type": "text", + "cross_page": true + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 360, + 506, + 373 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 198, + 373 + ], + "score": 1.0, + "content": "terms of the trend that", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 198, + 361, + 222, + 371 + ], + "score": 0.91, + "content": "\\sigma _ { m a x }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 222, + 360, + 284, + 373 + ], + "score": 1.0, + "content": "decreases with", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 285, + 361, + 293, + 370 + ], + "score": 0.78, + "content": "\\alpha", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 293, + 360, + 311, + 373 + ], + "score": 1.0, + "content": "and", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 311, + 361, + 318, + 371 + ], + "score": 0.79, + "content": "\\rho", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 318, + 360, + 434, + 373 + ], + "score": 1.0, + "content": ". 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These", + "type": "text", + "cross_page": true + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 382, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 505, + 394 + ], + "score": 1.0, + "content": "results show that the ascent step in GSAM minimizes the dominant eigenvalue by minimizing the", + "type": "text", + "cross_page": true + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 393, + 248, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 208, + 405 + ], + "score": 1.0, + "content": "surrogate loss, validating", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 209, + 393, + 244, + 403 + ], + "score": 0.29, + "content": "\\mathrm { T h m } 5 . 3", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 244, + 393, + 248, + 405 + ], + "score": 1.0, + "content": ".", + "type": "text", + "cross_page": true + } + ], + "index": 23 + } + ], + "index": 44.5, + "bbox_fs": [ + 105, + 687, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 111, + 124, + 499, + 497 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 95, + 503, + 118 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 95, + 505, + 108 + ], + "spans": [ + { + "bbox": [ + 106, + 95, + 207, + 108 + ], + "score": 1.0, + "content": "Table 1: Top-1 Accuracy", + "type": "text" + }, + { + "bbox": [ + 208, + 96, + 222, + 107 + ], + "score": 0.73, + "content": "( \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 95, + 505, + 108 + ], + "score": 1.0, + "content": "on ImageNet datasets for ResNets, ViTs and MLP-Mixers trained with", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 106, + 330, + 120 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 330, + 120 + ], + "score": 1.0, + "content": "Vanilla SGD or AdamW, SAM, and GSAM optimizers.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table_body", + "bbox": [ + 111, + 124, + 499, + 497 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 111, + 124, + 499, + 497 + ], + "spans": [ + { + "bbox": [ + 111, + 124, + 499, + 497 + ], + "score": 0.983, + "html": "
ModelTrainingImageNet-v1ImageNet-RealImageNet-V2ImageNet-RImageNet-C
ResNet
ResNet50Vanilla (SGD)76.082.463.622.244.6
SAM76.983.364.423.846.5
GSAM77.283.964.623.647.6
ResNet101Vanilla (SGD)77.883.965.324.448.5
SAM78.684.866.725.951.3
GSAM78.985.267.326.351.8
ResNet152Vanilla (SGD)78.584.266.325.350.0
SAM79.384.967.325.752.2
GSAM80.085.968.627.354.1
ViT-S/32Vanilla (AdamW)68.4Vision Transformer 75.254.3
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GSAM73.880.460.422.546.2 48.2
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SAM78.184.165.624.753.0
GSAM79.585.367.325.353.3
ViT-B/32Vanilla (AdamW)71.477.557.523.444.0
SAM73.680.360.024.050.7
GSAM76.882.763.025.151.7
ViT-B/16Vanilla (AdamW)74.679.861.320.146.6
SAM79.985.267.526.456.5
GSAM81.086.569.227.155.7
Mixer-S/32Vanilla (AdamW)63.9MLP-Mixer 70.349.516.935.2
SAM66.773.852.418.639.3
GSAM68.675.855.022.644.6
Mixer-S/16Vanilla (AdamW)68.875.154.815.935.6
SAM72.979.858.920.142.0
GSAM75.081.761.923.748.5
Mixer-S/8Vanilla (AdamW)70.276.256.115.434.6
SAM75.982.562.320.542.4
GSAM76.883.464.024.647.8
Mixer-B/32Vanilla (AdamW)62.568.147.614.633.8
SAM72.479.058.022.846.2
GSAM73.680.259.927.952.1
Mixer-B/16Vanilla (AdamW)66.472.150.814.533.8
SAM77.483.563.924.748.8
GSAM77.884.064.928.354.4
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Left: Top-1 accuracy on ImageNet. Middle:", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 667, + 506, + 683 + ], + "spans": [ + { + "bbox": [ + 104, + 667, + 365, + 683 + ], + "score": 1.0, + "content": "Estimation of the dominant eigenvalues from the surrogate gap,", + "type": "text" + }, + { + "bbox": [ + 365, + 668, + 427, + 680 + ], + "score": 0.93, + "content": "\\sigma _ { m a x } \\approx 2 h / \\rho ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 667, + 506, + 683 + ], + "score": 1.0, + "content": ". Right: Dominant", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 680, + 505, + 692 + ], + "spans": [ + { + "bbox": [ + 105, + 680, + 505, + 692 + ], + "score": 1.0, + "content": "eigenvalues of the Hessian calculated via the power iteration. Middle and right figures match in", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 690, + 506, + 704 + ], + "spans": [ + { + "bbox": [ + 105, + 690, + 506, + 704 + ], + "score": 1.0, + "content": "the trend of curves, validating that the surrogate gap can be viewed as a proxy of the dominant", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 702, + 199, + 714 + ], + "spans": [ + { + "bbox": [ + 105, + 702, + 199, + 714 + ], + "score": 1.0, + "content": "eigenvalue of Hessian.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10.5 + } + ], + "index": 8.25 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 105, + 25, + 294, + 39 + ], + "spans": [ + { + "bbox": [ + 105, + 25, + 294, + 39 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 300, + 750, + 309, + 761 + ], + "spans": [ + { + "bbox": [ + 300, + 750, + 309, + 761 + ], + "score": 1.0, + "content": "8", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 111, + 124, + 499, + 497 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 95, + 503, + 118 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 95, + 505, + 108 + ], + "spans": [ + { + "bbox": [ + 106, + 95, + 207, + 108 + ], + "score": 1.0, + "content": "Table 1: Top-1 Accuracy", + "type": "text" + }, + { + "bbox": [ + 208, + 96, + 222, + 107 + ], + "score": 0.73, + "content": "( \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 95, + 505, + 108 + ], + "score": 1.0, + "content": "on ImageNet datasets for ResNets, ViTs and MLP-Mixers trained with", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 106, + 330, + 120 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 330, + 120 + ], + "score": 1.0, + "content": "Vanilla SGD or AdamW, SAM, and GSAM optimizers.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table_body", + "bbox": [ + 111, + 124, + 499, + 497 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 111, + 124, + 499, + 497 + ], + "spans": [ + { + "bbox": [ + 111, + 124, + 499, + 497 + ], + "score": 0.983, + "html": "
ModelTrainingImageNet-v1ImageNet-RealImageNet-V2ImageNet-RImageNet-C
ResNet
ResNet50Vanilla (SGD)76.082.463.622.244.6
SAM76.983.364.423.846.5
GSAM77.283.964.623.647.6
ResNet101Vanilla (SGD)77.883.965.324.448.5
SAM78.684.866.725.951.3
GSAM78.985.267.326.351.8
ResNet152Vanilla (SGD)78.584.266.325.350.0
SAM79.384.967.325.752.2
GSAM80.085.968.627.354.1
ViT-S/32Vanilla (AdamW)68.4Vision Transformer 75.254.3
SAM70.577.556.919.0 21.443.3
GSAM73.880.460.422.546.2 48.2
ViT-S/16Vanilla (AdamW)74.480.461.720.046.5
SAM78.184.165.624.753.0
GSAM79.585.367.325.353.3
ViT-B/32Vanilla (AdamW)71.477.557.523.444.0
SAM73.680.360.024.050.7
GSAM76.882.763.025.151.7
ViT-B/16Vanilla (AdamW)74.679.861.320.146.6
SAM79.985.267.526.456.5
GSAM81.086.569.227.155.7
Mixer-S/32Vanilla (AdamW)63.9MLP-Mixer 70.349.516.935.2
SAM66.773.852.418.639.3
GSAM68.675.855.022.644.6
Mixer-S/16Vanilla (AdamW)68.875.154.815.935.6
SAM72.979.858.920.142.0
GSAM75.081.761.923.748.5
Mixer-S/8Vanilla (AdamW)70.276.256.115.434.6
SAM75.982.562.320.542.4
GSAM76.883.464.024.647.8
Mixer-B/32Vanilla (AdamW)62.568.147.614.633.8
SAM72.479.058.022.846.2
GSAM73.680.259.927.952.1
Mixer-B/16Vanilla (AdamW)66.472.150.814.533.8
SAM77.483.563.924.748.8
GSAM77.884.064.928.354.4
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Datasetmin(fp+Xh)GSAM
ImageNet75.476.8
ImageNet-Real81.182.7
ImageNet-v260.963.0
ImageNet-R23.925.1
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ViT-B/16ViT-S/16
VanillaSAMGSAMVanillaSAMGSAM
Cifar1098.198.698.897.698.298.4
Cifar10087.689.189.785.787.688.1
Flowers88.591.891.286.491.590.3
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Datasetmin(fp+Xh)GSAM
ImageNet75.476.8
ImageNet-Real81.182.7
ImageNet-v260.963.0
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ViT-B/16ViT-S/16
VanillaSAMGSAMVanillaSAMGSAM
Cifar1098.198.698.897.698.298.4
Cifar10087.689.189.785.787.688.1
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We report the best accuracy of each method. Top-1 accuracy in", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 632, + 404, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 632, + 404, + 645 + ], + "score": 1.0, + "content": "Table 2 show the superior performance of GSAM, validating our analysis.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 38.5, + "bbox_fs": [ + 104, + 577, + 506, + 645 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 648, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 106, + 648, + 506, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 648, + 151, + 661 + ], + "score": 0.91, + "content": "\\operatorname* { m i n } ( f _ { p } , h )", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 648, + 166, + 663 + ], + "score": 1.0, + "content": "vs.", + "type": "text" + }, + { + "bbox": [ + 167, + 648, + 208, + 661 + ], + "score": 0.88, + "content": "\\operatorname* { m i n } ( f , h )", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 648, + 267, + 663 + ], + "score": 1.0, + "content": "GSAM solves", + "type": "text" + }, + { + "bbox": [ + 267, + 649, + 312, + 661 + ], + "score": 0.92, + "content": "\\operatorname* { m i n } ( f _ { p } , h )", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 648, + 368, + 663 + ], + "score": 1.0, + "content": "by descent in", + "type": "text" + }, + { + "bbox": [ + 368, + 649, + 387, + 661 + ], + "score": 0.9, + "content": "\\nabla f _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 648, + 446, + 663 + ], + "score": 1.0, + "content": ", decomposing", + "type": "text" + }, + { + "bbox": [ + 446, + 649, + 461, + 660 + ], + "score": 0.88, + "content": "\\nabla f", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 648, + 482, + 663 + ], + "score": 1.0, + "content": "onto", + "type": "text" + }, + { + "bbox": [ + 483, + 649, + 501, + 661 + ], + "score": 0.9, + "content": "\\nabla f _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 648, + 506, + 663 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 658, + 506, + 674 + ], + "spans": [ + { + "bbox": [ + 105, + 658, + 338, + 674 + ], + "score": 1.0, + "content": "and an ascent step in the orthogonal direction to increase", + "type": "text" + }, + { + "bbox": [ + 338, + 660, + 345, + 672 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 658, + 393, + 674 + ], + "score": 1.0, + "content": "while keep", + "type": "text" + }, + { + "bbox": [ + 393, + 660, + 404, + 672 + ], + "score": 0.88, + "content": "f _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 658, + 506, + 674 + ], + "score": 1.0, + "content": "intact. 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We choose", + "type": "text" + }, + { + "bbox": [ + 265, + 693, + 309, + 705 + ], + "score": 0.94, + "content": "\\operatorname* { m i n } ( f _ { p } , h )", + "type": "inline_equation" + }, + { + "bbox": [ + 310, + 691, + 494, + 706 + ], + "score": 1.0, + "content": "mainly to make the minimal change to SAM.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 44, + "bbox_fs": [ + 105, + 648, + 506, + 706 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "GSAM benefits transfer learning Using weights trained on ImageNet-1k, we finetune models with", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "SGD on downstream tasks including the CIFAR10/CIFAR100 (Krizhevsky et al., 2009), Oxford-", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 234, + 504, + 247 + ], + "spans": [ + { + "bbox": [ + 106, + 234, + 504, + 247 + ], + "score": 1.0, + "content": "flowers (Nilsback & Zisserman, 2008) and Oxford-IITPets (Parkhi et al., 2012). 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We propose the GSAM method, which improves the generalization over SAM", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 404, + 506, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 506, + 417 + ], + "score": 1.0, + "content": "at negligible computation cost. We show the convergence and provably better generalization of", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 415, + 489, + 428 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 489, + 428 + ], + "score": 1.0, + "content": "GSAM compared to SAM, and validate the superior performance of GSAM on various models.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18.5 + }, + { + "type": "title", + "bbox": [ + 108, + 444, + 219, + 455 + ], + "lines": [ + { + "bbox": [ + 107, + 444, + 220, + 457 + ], + "spans": [ + { + "bbox": [ + 107, + 444, + 220, + 457 + ], + "score": 1.0, + "content": "ACKNOWLEDGEMENT", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 468, + 504, + 501 + ], + "lines": [ + { + "bbox": [ + 106, + 468, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 505, + 480 + ], + "score": 1.0, + "content": "We would like to thank Xiangning Chen (UCLA) and Hossein Mobahi (Google) for discussions, Yi", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 478, + 505, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 505, + 492 + ], + "score": 1.0, + "content": "Tay (Google) for help with datasets, and Yeqing Li, Xianzhi Du, and Shawn Wang (Google) for help", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 490, + 244, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 490, + 244, + 503 + ], + "score": 1.0, + "content": "with TensorFlow implementation.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23 + }, + { + "type": "title", + "bbox": [ + 108, + 518, + 210, + 531 + ], + "lines": [ + { + "bbox": [ + 106, + 518, + 213, + 532 + ], + "spans": [ + { + "bbox": [ + 106, + 518, + 213, + 532 + ], + "score": 1.0, + "content": "ETHICS STATEMENT", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 543, + 505, + 588 + ], + "lines": [ + { + "bbox": [ + 106, + 543, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 106, + 543, + 505, + 555 + ], + "score": 1.0, + "content": "This paper focuses on the development of optimization methodologies and can be applied to the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 555, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 505, + 567 + ], + "score": 1.0, + "content": "training of different deep neural networks for a wide range of applications. Therefore, the ethical", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 565, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 505, + 578 + ], + "score": 1.0, + "content": "impact of our work would primarily be determined by the specific models that are trained using our", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 576, + 214, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 214, + 590 + ], + "score": 1.0, + "content": "new optimization strategy.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27.5 + }, + { + "type": "title", + "bbox": [ + 108, + 604, + 267, + 617 + ], + "lines": [ + { + "bbox": [ + 106, + 603, + 269, + 619 + ], + "spans": [ + { + "bbox": [ + 106, + 603, + 269, + 619 + ], + "score": 1.0, + "content": "REPRODUCIBILITY STATEMENT", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 629, + 505, + 674 + ], + "lines": [ + { + "bbox": [ + 106, + 629, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 106, + 629, + 505, + 641 + ], + "score": 1.0, + "content": "We provide the detailed proof of theoretical results in Appendix A and provide the data pre-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 640, + 505, + 654 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 505, + 654 + ], + "score": 1.0, + "content": "processing and hyper-parameter settings in Appendix B. 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Normalize", + "type": "text" + }, + { + "bbox": [ + 478, + 589, + 483, + 597 + ], + "score": 0.82, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 587, + 505, + 600 + ], + "score": 1.0, + "content": "such", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 597, + 234, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 124, + 612 + ], + "score": 1.0, + "content": "that", + "type": "text" + }, + { + "bbox": [ + 124, + 598, + 164, + 610 + ], + "score": 0.93, + "content": "| | \\delta | | _ { 2 } = \\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 597, + 234, + 612 + ], + "score": 1.0, + "content": ", Hence we have:", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5 + }, + { + "type": "interline_equation", + "bbox": [ + 146, + 617, + 465, + 642 + ], + "lines": [ + { + "bbox": [ + 146, + 617, + 465, + 642 + ], + "spans": [ + { + "bbox": [ + 146, + 617, + 465, + 642 + ], + "score": 0.9, + "content": "h ( w ) = f _ { p } ( w ) - f ( w ) = \\operatorname* { m a x } _ { | | \\delta | | _ { 2 } \\leq \\rho } f ( w + \\delta ) - f ( w ) = \\frac { 1 } { 2 } \\sigma _ { m a x } \\rho ^ { 2 } + O ( \\rho ^ { 3 } )", + "type": "interline_equation", + "image_path": "266770658db8ee578fb42a9532ffea10b591945e46e2429298703fe2af9dc6a3.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 146, + 617, + 465, + 642 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 648, + 506, + 672 + ], + "lines": [ + { + "bbox": [ + 106, + 648, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 106, + 648, + 134, + 661 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 135, + 651, + 158, + 660 + ], + "score": 0.88, + "content": "\\sigma _ { m a x }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 648, + 335, + 661 + ], + "score": 1.0, + "content": "is the dominate eigenvalue of the hessian", + "type": "text" + }, + { + "bbox": [ + 336, + 649, + 346, + 659 + ], + "score": 0.85, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 648, + 505, + 661 + ], + "score": 1.0, + "content": ", and first order term is 0 because the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 659, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 305, + 673 + ], + "score": 1.0, + "content": "gradient is 0 at local minima. 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Normalize", + "type": "text" + }, + { + "bbox": [ + 478, + 589, + 483, + 597 + ], + "score": 0.82, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 587, + 505, + 600 + ], + "score": 1.0, + "content": "such", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 597, + 234, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 124, + 612 + ], + "score": 1.0, + "content": "that", + "type": "text" + }, + { + "bbox": [ + 124, + 598, + 164, + 610 + ], + "score": 0.93, + "content": "| | \\delta | | _ { 2 } = \\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 597, + 234, + 612 + ], + "score": 1.0, + "content": ", Hence we have:", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 587, + 505, + 612 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 146, + 617, + 465, + 642 + ], + "lines": [ + { + "bbox": [ + 146, + 617, + 465, + 642 + ], + "spans": [ + { + "bbox": [ + 146, + 617, + 465, + 642 + ], + "score": 0.9, + "content": "h ( w ) = f _ { p } ( w ) - f ( w ) = \\operatorname* { m a x } _ { | | \\delta | | _ { 2 } \\leq \\rho } f ( w + \\delta ) - f ( w ) = \\frac { 1 } { 2 } \\sigma _ { m a x } \\rho ^ { 2 } + O ( \\rho ^ { 3 } )", + "type": "interline_equation", + "image_path": "266770658db8ee578fb42a9532ffea10b591945e46e2429298703fe2af9dc6a3.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 146, + 617, + 465, + 642 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 648, + 506, + 672 + ], + "lines": [ + { + "bbox": [ + 106, + 648, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 106, + 648, + 134, + 661 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 135, + 651, + 158, + 660 + ], + "score": 0.88, + "content": "\\sigma _ { m a x }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 648, + 335, + 661 + ], + "score": 1.0, + "content": "is the dominate eigenvalue of the hessian", + "type": "text" + }, + { + "bbox": [ + 336, + 649, + 346, + 659 + ], + "score": 0.85, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 648, + 505, + 661 + ], + "score": 1.0, + "content": ", and first order term is 0 because the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 659, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 305, + 673 + ], + "score": 1.0, + "content": "gradient is 0 at local minima. 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\\delta _ { t } \\rangle \\leq L ( 1 + \\alpha ) G \\rho _ { t } \\eta _ { t }", + "type": "interline_equation", + "image_path": "1537a5c50c1dbc3ef55e80028d160facf36f2e371f8f6300f550c2d55ddfa66b.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 216, + 293, + 395, + 308 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 309, + 182, + 321 + ], + "lines": [ + { + "bbox": [ + 106, + 307, + 183, + 322 + ], + "spans": [ + { + "bbox": [ + 106, + 307, + 183, + 322 + ], + "score": 1.0, + "content": "Similarly, we have", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "interline_equation", + "bbox": [ + 192, + 323, + 416, + 386 + ], + "lines": [ + { + "bbox": [ + 192, + 323, + 416, + 386 + ], + "spans": [ + { + "bbox": [ + 192, + 323, + 416, + 386 + ], + "score": 0.93, + "content": "\\begin{array} { r l } & { \\mathbb { E } \\Big \\lvert \\Big \\lvert \\delta _ { t + 1 } - \\delta _ { t } \\Big \\rvert \\Big \\rvert ^ { 2 } \\leq \\rho _ { t } ^ { 2 } \\mathbb { E } \\Big \\lvert \\Big \\lvert \\frac { g ^ { ( t ) } } { \\lvert | g ^ { ( t ) } \\rvert \\rvert + \\epsilon } - 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By Lemma.1 in Laurent", + "type": "text" + }, + { + "bbox": [ + 380, + 651, + 389, + 660 + ], + "score": 0.4, + "content": "\\&", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 650, + 491, + 663 + ], + "score": 1.0, + "content": "Massart (2000), we have", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5 + }, + { + "type": "interline_equation", + "bbox": [ + 212, + 665, + 399, + 682 + ], + "lines": [ + { + "bbox": [ + 212, + 665, + 399, + 682 + ], + "spans": [ + { + "bbox": [ + 212, + 665, + 399, + 682 + ], + "score": 0.92, + "content": "\\mathbb { P } \\big ( | | \\delta | | _ { 2 } ^ { 2 } - k b ^ { 2 } \\geq 2 b ^ { 2 } \\sqrt { k t } + 2 t b ^ { 2 } \\big ) \\leq e x p ( - t )", + "type": "interline_equation", + "image_path": "9a80a6dc4791072d1c9581bc88f146f071e5194bae61c7457c7879f3b159b60f.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 212, + 665, + 399, + 682 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 685, + 310, + 698 + ], + "lines": [ + { + "bbox": [ + 106, + 684, + 311, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 684, + 229, + 699 + ], + "score": 1.0, + "content": "hence with probability at least", + "type": "text" + }, + { + "bbox": [ + 230, + 685, + 271, + 698 + ], + "score": 0.92, + "content": "1 - 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\\nabla f ( w _ { t } ) , \\alpha \\eta _ { t } \\nabla f _ { \\perp } ( w _ { t } ) \\rangle = - \\alpha \\eta _ { t } \\big \\lvert \\big \\rvert \\nabla f _ { \\perp } ( w _ { t } ) \\big \\rvert \\big \\rvert _ { 2 } ^ { 2 } < 0 } \\end{array}", + "type": "interline_equation", + "image_path": "66412a0182b9ce0f9b598bc60dfeec285823de2b3ebd5c0497afcee9ca5d6d06.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 172, + 364, + 439, + 382 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 386, + 505, + 446 + ], + "lines": [ + { + "bbox": [ + 105, + 385, + 506, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 506, + 399 + ], + "score": 1.0, + "content": "Above results demonstrate that ascent step decreases the loss gap, while descent step might increase", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 396, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 338, + 410 + ], + "score": 1.0, + "content": "the loss gap. A sufficient (but not necessary) condition for", + "type": "text" + }, + { + "bbox": [ + 338, + 397, + 417, + 409 + ], + "score": 0.93, + "content": "\\mathbb { E } \\langle \\nabla h ( w _ { t } ) , d t \\rangle \\leq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 396, + 452, + 410 + ], + "score": 1.0, + "content": "requires", + "type": "text" + }, + { + "bbox": [ + 453, + 399, + 460, + 407 + ], + "score": 0.8, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 396, + 506, + 410 + ], + "score": 1.0, + "content": "to be large", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 407, + 506, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 118, + 423 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 118, + 408, + 256, + 423 + ], + "score": 0.92, + "content": "| \\big | \\nabla f ( w _ { t } ) \\big | \\big | _ { 2 } \\cos \\theta _ { t } \\leq \\big | \\big | \\nabla f _ { p } ( w _ { t } ) \\big | \\big |", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 407, + 416, + 423 + ], + "score": 1.0, + "content": ". In practice, the perturbation amplitude", + "type": "text" + }, + { + "bbox": [ + 416, + 411, + 423, + 421 + ], + "score": 0.81, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 407, + 506, + 423 + ], + "score": 1.0, + "content": "is small and we can", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 420, + 506, + 437 + ], + "spans": [ + { + "bbox": [ + 104, + 420, + 138, + 437 + ], + "score": 1.0, + "content": "assume", + "type": "text" + }, + { + "bbox": [ + 138, + 423, + 148, + 434 + ], + "score": 0.87, + "content": "\\theta _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 420, + 218, + 437 + ], + "score": 1.0, + "content": "is close to 0 and", + "type": "text" + }, + { + "bbox": [ + 218, + 422, + 268, + 436 + ], + "score": 0.92, + "content": "\\big | \\big | \\nabla f _ { p } ( w _ { t } ) \\big | \\big |", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 420, + 313, + 437 + ], + "score": 1.0, + "content": "is close to", + "type": "text" + }, + { + "bbox": [ + 313, + 421, + 360, + 436 + ], + "score": 0.94, + "content": "\\left| \\left| \\nabla f ( w _ { t } ) \\right| \\right|", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 420, + 486, + 437 + ], + "score": 1.0, + "content": ", we can also set the parameter", + "type": "text" + }, + { + "bbox": [ + 486, + 425, + 493, + 433 + ], + "score": 0.77, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 420, + 506, + 437 + ], + "score": 1.0, + "content": "to", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 432, + 273, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 273, + 447 + ], + "score": 1.0, + "content": "be large in order to decrease the loss gap.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19, + "bbox_fs": [ + 104, + 385, + 506, + 447 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 457, + 486, + 480 + ], + "lines": [ + { + "bbox": [ + 106, + 457, + 486, + 469 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 486, + 469 + ], + "score": 1.0, + "content": "STEP 2: UPPER AND LOWER BOUND OF DECREASE IN LOSS GAP (BY THE ASCENT STEP IN", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 469, + 362, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 469, + 362, + 480 + ], + "score": 1.0, + "content": "ORTHOGONAL GRADIENT DIRECTION) COMPARED TO SAM.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5, + "bbox_fs": [ + 106, + 457, + 486, + 480 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 488, + 505, + 534 + ], + "lines": [ + { + "bbox": [ + 104, + 487, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 104, + 489, + 288, + 502 + ], + "score": 1.0, + "content": "Next we give an estimate of the decrease in", + "type": "text" + }, + { + "bbox": [ + 288, + 487, + 296, + 500 + ], + "score": 0.84, + "content": "\\widehat { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 489, + 506, + 502 + ], + "score": 1.0, + "content": "caused by our ascent step. We refer to Eq. 69 and", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 500, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 104, + 500, + 505, + 514 + ], + "score": 1.0, + "content": "Eq. 70 to analyze the change in loss gap caused by the descent and ascent (orthogonally) respectively.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 511, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 525 + ], + "score": 1.0, + "content": "It can be seen that gradient descent step might not decrease loss gap, in fact they often increase loss", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 522, + 411, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 411, + 537 + ], + "score": 1.0, + "content": "gap in practice; while the ascent step is guaranteed to decrease the loss gap.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25.5, + "bbox_fs": [ + 104, + 487, + 506, + 537 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 539, + 218, + 551 + ], + "lines": [ + { + "bbox": [ + 105, + 537, + 218, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 218, + 552 + ], + "score": 1.0, + "content": "The decrease in loss gap is:", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 537, + 218, + 552 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 154, + 556, + 456, + 595 + ], + "lines": [ + { + "bbox": [ + 154, + 556, + 456, + 595 + ], + "spans": [ + { + "bbox": [ + 154, + 556, + 456, + 595 + ], + "score": 0.92, + "content": "\\begin{array} { r l r } & { } & { \\Delta \\widehat { h } _ { t } = - 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\\nabla \\widehat { f } ( w _ { t } ) | | } { | | \\widehat { f } ( w _ { t } ) | | } = \\frac { | | \\rho _ { t } \\widehat { H } ( w _ { t } ) + O ( \\rho _ { t } ^ { 2 } ) | | } { | | \\nabla \\widehat { f } ( w _ { t } ) | | } \\geq \\frac { \\rho _ { t } | \\sigma | _ { m i n } } { G }", + "type": "interline_equation", + "image_path": "042d990fe9c2542b4f7cac44cb240740149d52dcd6da21645ab887890c29e706.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 145, + 228, + 465, + 238.66666666666666 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 145, + 238.66666666666666, + 465, + 249.33333333333331 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 145, + 249.33333333333331, + 465, + 260.0 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 263, + 503, + 296 + ], + "lines": [ + { + "bbox": [ + 106, + 262, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 262, + 133, + 276 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 263, + 142, + 273 + ], + "score": 0.85, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 262, + 305, + 276 + ], + "score": 1.0, + "content": "is the upper-bound on norm of gradient,", + "type": "text" + }, + { + "bbox": [ + 305, + 263, + 333, + 275 + ], + "score": 0.92, + "content": "| \\sigma | _ { m i n }", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 262, + 505, + 276 + ], + "score": 1.0, + "content": "is the minimum absolute eigenvalue of the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 274, + 504, + 285 + ], + "spans": [ + { + "bbox": [ + 106, + 274, + 408, + 285 + ], + "score": 1.0, + "content": "Hessian. The intuition is that as perturbation amplitude decreases, the angle", + "type": "text" + }, + { + "bbox": [ + 408, + 274, + 417, + 285 + ], + "score": 0.87, + "content": "\\theta _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 274, + 504, + 285 + ], + "score": 1.0, + "content": "decreases at a similar", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 285, + 369, + 297 + ], + "spans": [ + { + "bbox": [ + 106, + 285, + 369, + 297 + ], + "score": 1.0, + "content": "rate, though the scale constant might be different. Hence we have", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13 + }, + { + "type": "interline_equation", + "bbox": [ + 203, + 299, + 407, + 429 + ], + "lines": [ + { + "bbox": [ + 203, + 299, + 407, + 429 + ], + "spans": [ + { + "bbox": [ + 203, + 299, + 407, + 429 + ], + "score": 0.95, + "content": "\\begin{array} { r l } { { \\sum _ { t = 1 } ^ { T } \\Delta \\hat { h } _ { t } = \\sum _ { t = 1 } ^ { T } \\alpha \\eta _ { t } \\vert \\big \\vert \\nabla \\widehat { f } ( w _ { t } ) \\big \\vert \\big \\vert _ { 2 } ^ { 2 } \\vert \\tan \\theta _ { t } \\vert ^ { 2 } + O ( \\theta _ { t } ^ { 4 } ) } \\ ~ } \\\\ & { \\geq \\sum _ { t = 1 } ^ { T } \\alpha \\eta _ { t } c ^ { 2 } \\Big ( \\frac { \\rho _ { t } \\vert \\sigma \\vert _ { m i n } } { G } \\Big ) ^ { 2 } } \\\\ & { = \\frac { \\alpha c ^ { 2 } \\rho _ { 0 } ^ { 2 } \\eta _ { 0 } \\vert \\sigma \\vert _ { m i n } ^ { 2 } } { G ^ { 2 } } \\sum _ { t = 1 } ^ { T } \\frac { 1 } { t ^ { 3 / 2 } } } \\\\ & { \\geq \\frac { \\alpha c ^ { 2 } \\rho _ { 0 } ^ { 2 } \\eta _ { 0 } \\vert \\sigma \\vert _ { m i n } ^ { 2 } } { G ^ { 2 } } } \\end{array}", + "type": "interline_equation", + "image_path": "696775696d71ffbfb2af20956be032a8254a55afd9eb75d5eb8751a605040322.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 203, + 299, + 407, + 313.44444444444446 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 203, + 313.44444444444446, + 407, + 327.8888888888889 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 203, + 327.8888888888889, + 407, + 342.33333333333337 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 203, + 342.33333333333337, + 407, + 356.7777777777778 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 203, + 356.7777777777778, + 407, + 371.2222222222223 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 203, + 371.2222222222223, + 407, + 385.66666666666674 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 203, + 385.66666666666674, + 407, + 400.1111111111112 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 203, + 400.1111111111112, + 407, + 414.55555555555566 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 203, + 414.55555555555566, + 407, + 429.0000000000001 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 433, + 505, + 468 + ], + "lines": [ + { + "bbox": [ + 105, + 432, + 506, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 134, + 447 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 433, + 144, + 444 + ], + "score": 0.86, + "content": "c ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 432, + 237, + 447 + ], + "score": 1.0, + "content": "is the lower bound of", + "type": "text" + }, + { + "bbox": [ + 237, + 432, + 268, + 446 + ], + "score": 0.93, + "content": "| | \\nabla \\widehat { f } | | ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 432, + 506, + 447 + ], + "score": 1.0, + "content": "(e.g. due to noise in data and gradient observation). Re-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 444, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 505, + 457 + ], + "score": 1.0, + "content": "sults above indicate that the decrease in loss gap caused by the ascent step is non-trivial, hence our", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 455, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 407, + 468 + ], + "score": 1.0, + "content": "proposed method efficiently improves generalization compared with SAM.", + "type": "text" + }, + { + "bbox": [ + 493, + 456, + 505, + 467 + ], + "score": 1.0, + "content": "\u0003", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25 + }, + { + "type": "title", + "bbox": [ + 108, + 479, + 282, + 491 + ], + "lines": [ + { + "bbox": [ + 106, + 479, + 284, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 284, + 493 + ], + "score": 1.0, + "content": "A.7 DISCUSSION ON COROLLARY 5.2.1", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 500, + 506, + 568 + ], + "lines": [ + { + "bbox": [ + 105, + 500, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 505, + 513 + ], + "score": 1.0, + "content": "The comment “‘The corollary gives a bound on the risk in terms of the perturbed training loss", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 171, + 524 + ], + "score": 1.0, + "content": "if one removes", + "type": "text" + }, + { + "bbox": [ + 171, + 512, + 180, + 522 + ], + "score": 0.81, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "from both sides”’ is correct. But there is a misunderstanding in the statement", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 519, + 504, + 538 + ], + "spans": [ + { + "bbox": [ + 104, + 519, + 477, + 538 + ], + "score": 1.0, + "content": "“‘the perturbed training loss is small then the model has a small risk”’: it’s only true when", + "type": "text" + }, + { + "bbox": [ + 478, + 524, + 504, + 535 + ], + "score": 0.45, + "content": "\\rho _ { t r a i n }", + "type": "inline_equation" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 533, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 241, + 546 + ], + "score": 1.0, + "content": "for training equals its real value", + "type": "text" + }, + { + "bbox": [ + 241, + 535, + 264, + 545 + ], + "score": 0.87, + "content": "\\rho _ { t r u e }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 533, + 506, + 546 + ], + "score": 1.0, + "content": "determined by the data distribution; in practice, we never", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 544, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 131, + 559 + ], + "score": 1.0, + "content": "know", + "type": "text" + }, + { + "bbox": [ + 131, + 546, + 153, + 556 + ], + "score": 0.87, + "content": "\\rho _ { t r u e }", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 544, + 383, + 559 + ], + "score": 1.0, + "content": ". In the following we show that the minimization of both", + "type": "text" + }, + { + "bbox": [ + 383, + 545, + 390, + 555 + ], + "score": 0.82, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 544, + 408, + 559 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 409, + 545, + 419, + 557 + ], + "score": 0.89, + "content": "f _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 544, + 505, + 559 + ], + "score": 1.0, + "content": "is better than simply", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 554, + 256, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 155, + 570 + ], + "score": 1.0, + "content": "minimizing", + "type": "text" + }, + { + "bbox": [ + 155, + 556, + 165, + 568 + ], + "score": 0.89, + "content": "f _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 554, + 192, + 570 + ], + "score": 1.0, + "content": "when", + "type": "text" + }, + { + "bbox": [ + 192, + 556, + 252, + 568 + ], + "score": 0.9, + "content": "\\rho _ { t r u e } \\neq \\rho _ { t r a i n }", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 554, + 256, + 570 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30.5 + }, + { + "type": "text", + "bbox": [ + 108, + 572, + 333, + 583 + ], + "lines": [ + { + "bbox": [ + 106, + 570, + 334, + 586 + ], + "spans": [ + { + "bbox": [ + 106, + 570, + 334, + 586 + ], + "score": 1.0, + "content": "1. First, we re-write the conclusion of Corollary 5.2.1 as", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "interline_equation", + "bbox": [ + 167, + 586, + 444, + 622 + ], + "lines": [ + { + "bbox": [ + 167, + 586, + 444, + 622 + ], + "spans": [ + { + "bbox": [ + 167, + 586, + 444, + 622 + ], + "score": 0.64, + "content": "\\begin{array} { r } { \\mathbb { E } _ { w } \\mathbb { E } _ { x } f ( w , x ) \\le f _ { p } + R = C + \\widehat { h } + R = C + \\rho ^ { 2 } \\sigma / 2 + R + O ( \\rho ^ { 3 } ) } \\\\ { w i t h p r o b a b i l i t y \\left( 1 - a \\right) [ 1 - e ^ { - ( \\frac { \\rho } { \\sqrt { 2 } b } - \\sqrt { k } ) ^ { 2 } } ] } \\end{array}", + "type": "interline_equation", + "image_path": "cdaf2ab0abf25e025e63bc79a18bd9a65ace8a902a290b154e1abfb27914e5e9.jpg" + } + ] + } + ], + "index": 36, + "virtual_lines": [ + { + "bbox": [ + 167, + 586, + 444, + 598.0 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 167, + 598.0, + 444, + 610.0 + ], + "spans": [], + "index": 36 + }, + { + "bbox": [ + 167, + 610.0, + 444, + 622.0 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 624, + 503, + 657 + ], + "lines": [ + { + "bbox": [ + 105, + 623, + 505, + 637 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 133, + 637 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 625, + 142, + 634 + ], + "score": 0.83, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 623, + 249, + 637 + ], + "score": 1.0, + "content": "is the regularization term,", + "type": "text" + }, + { + "bbox": [ + 250, + 625, + 259, + 634 + ], + "score": 0.83, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 623, + 339, + 637 + ], + "score": 1.0, + "content": "is the training loss,", + "type": "text" + }, + { + "bbox": [ + 339, + 627, + 346, + 635 + ], + "score": 0.77, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 623, + 505, + 637 + ], + "score": 1.0, + "content": "is the dominant eigenvalue of Hessian.", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 635, + 505, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 435, + 648 + ], + "score": 1.0, + "content": "As in lemma 3.3, we perform Taylor-expansion and can ignore the high-order term", + "type": "text" + }, + { + "bbox": [ + 435, + 635, + 461, + 648 + ], + "score": 0.92, + "content": "O ( \\rho ^ { 3 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 635, + 505, + 648 + ], + "score": 1.0, + "content": ". We focus", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 648, + 120, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 120, + 658 + ], + "score": 1.0, + "content": "on", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39 + }, + { + "type": "interline_equation", + "bbox": [ + 250, + 654, + 361, + 670 + ], + "lines": [ + { + "bbox": [ + 250, + 654, + 361, + 670 + ], + "spans": [ + { + "bbox": [ + 250, + 654, + 361, + 670 + ], + "score": 0.92, + "content": "f _ { p } = C + \\widehat { h } = C + \\rho ^ { 2 } \\sigma / 2", + "type": "interline_equation", + "image_path": "a4c4871c27e9cd5aff90d0bbfaebf1c51736abe97c45a1184e05d9ae4fff9a26.jpg" + } + ] + } + ], + "index": 41, + "virtual_lines": [ + { + "bbox": [ + 250, + 654, + 361, + 670 + ], + "spans": [], + "index": 41 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 676, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 676, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 145, + 691 + ], + "score": 1.0, + "content": "2. When", + "type": "text" + }, + { + "bbox": [ + 146, + 677, + 206, + 689 + ], + "score": 0.89, + "content": "\\rho _ { t r u e } \\neq \\rho _ { t r a i n }", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 676, + 261, + 691 + ], + "score": 1.0, + "content": ", minimizing", + "type": "text" + }, + { + "bbox": [ + 261, + 677, + 268, + 687 + ], + "score": 0.79, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 676, + 453, + 691 + ], + "score": 1.0, + "content": "achieves a lower risk than only minimizing", + "type": "text" + }, + { + "bbox": [ + 454, + 677, + 464, + 689 + ], + "score": 0.86, + "content": "f _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 676, + 506, + 691 + ], + "score": 1.0, + "content": ". 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The intuition is that as perturbation amplitude decreases, the angle", + "type": "text" + }, + { + "bbox": [ + 408, + 274, + 417, + 285 + ], + "score": 0.87, + "content": "\\theta _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 274, + 504, + 285 + ], + "score": 1.0, + "content": "decreases at a similar", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 285, + 369, + 297 + ], + "spans": [ + { + "bbox": [ + 106, + 285, + 369, + 297 + ], + "score": 1.0, + "content": "rate, though the scale constant might be different. Hence we have", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13, + "bbox_fs": [ + 106, + 262, + 505, + 297 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 203, + 299, + 407, + 429 + ], + "lines": [ + { + "bbox": [ + 203, + 299, + 407, + 429 + ], + "spans": [ + { + "bbox": [ + 203, + 299, + 407, + 429 + ], + "score": 0.95, + "content": "\\begin{array} { r l } { { \\sum _ { t = 1 } ^ { T } \\Delta \\hat { h } _ { t } = \\sum _ { t = 1 } ^ { T } \\alpha \\eta _ { t } \\vert \\big \\vert \\nabla \\widehat { f } ( w _ { t } ) \\big \\vert \\big \\vert _ { 2 } ^ { 2 } \\vert \\tan \\theta _ { t } \\vert ^ { 2 } + O ( \\theta _ { t } ^ { 4 } ) } \\ ~ } \\\\ & { \\geq \\sum _ { t = 1 } ^ { T } \\alpha \\eta _ { t } c ^ { 2 } \\Big ( \\frac { \\rho _ { t } \\vert \\sigma \\vert _ { m i n } } { G } \\Big ) ^ { 2 } } \\\\ & { = \\frac { \\alpha c ^ { 2 } \\rho _ { 0 } ^ { 2 } \\eta _ { 0 } \\vert \\sigma \\vert _ { m i n } ^ { 2 } } { G ^ { 2 } } \\sum _ { t = 1 } ^ { T } \\frac { 1 } { t ^ { 3 / 2 } } } \\\\ & { \\geq \\frac { \\alpha c ^ { 2 } \\rho _ { 0 } ^ { 2 } \\eta _ { 0 } \\vert \\sigma \\vert _ { m i n } ^ { 2 } } { G ^ { 2 } } } \\end{array}", + "type": "interline_equation", + "image_path": "696775696d71ffbfb2af20956be032a8254a55afd9eb75d5eb8751a605040322.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 203, + 299, + 407, + 313.44444444444446 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 203, + 313.44444444444446, + 407, + 327.8888888888889 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 203, + 327.8888888888889, + 407, + 342.33333333333337 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 203, + 342.33333333333337, + 407, + 356.7777777777778 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 203, + 356.7777777777778, + 407, + 371.2222222222223 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 203, + 371.2222222222223, + 407, + 385.66666666666674 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 203, + 385.66666666666674, + 407, + 400.1111111111112 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 203, + 400.1111111111112, + 407, + 414.55555555555566 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 203, + 414.55555555555566, + 407, + 429.0000000000001 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 433, + 505, + 468 + ], + "lines": [ + { + "bbox": [ + 105, + 432, + 506, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 134, + 447 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 433, + 144, + 444 + ], + "score": 0.86, + "content": "c ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 432, + 237, + 447 + ], + "score": 1.0, + "content": "is the lower bound of", + "type": "text" + }, + { + "bbox": [ + 237, + 432, + 268, + 446 + ], + "score": 0.93, + "content": "| | \\nabla \\widehat { f } | | ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 432, + 506, + 447 + ], + "score": 1.0, + "content": "(e.g. due to noise in data and gradient observation). Re-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 444, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 505, + 457 + ], + "score": 1.0, + "content": "sults above indicate that the decrease in loss gap caused by the ascent step is non-trivial, hence our", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 455, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 407, + 468 + ], + "score": 1.0, + "content": "proposed method efficiently improves generalization compared with SAM.", + "type": "text" + }, + { + "bbox": [ + 493, + 456, + 505, + 467 + ], + "score": 1.0, + "content": "\u0003", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 432, + 506, + 468 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 479, + 282, + 491 + ], + "lines": [ + { + "bbox": [ + 106, + 479, + 284, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 284, + 493 + ], + "score": 1.0, + "content": "A.7 DISCUSSION ON COROLLARY 5.2.1", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 500, + 506, + 568 + ], + "lines": [ + { + "bbox": [ + 105, + 500, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 505, + 513 + ], + "score": 1.0, + "content": "The comment “‘The corollary gives a bound on the risk in terms of the perturbed training loss", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 171, + 524 + ], + "score": 1.0, + "content": "if one removes", + "type": "text" + }, + { + "bbox": [ + 171, + 512, + 180, + 522 + ], + "score": 0.81, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "from both sides”’ is correct. But there is a misunderstanding in the statement", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 519, + 504, + 538 + ], + "spans": [ + { + "bbox": [ + 104, + 519, + 477, + 538 + ], + "score": 1.0, + "content": "“‘the perturbed training loss is small then the model has a small risk”’: it’s only true when", + "type": "text" + }, + { + "bbox": [ + 478, + 524, + 504, + 535 + ], + "score": 0.45, + "content": "\\rho _ { t r a i n }", + "type": "inline_equation" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 533, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 241, + 546 + ], + "score": 1.0, + "content": "for training equals its real value", + "type": "text" + }, + { + "bbox": [ + 241, + 535, + 264, + 545 + ], + "score": 0.87, + "content": "\\rho _ { t r u e }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 533, + 506, + 546 + ], + "score": 1.0, + "content": "determined by the data distribution; in practice, we never", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 544, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 131, + 559 + ], + "score": 1.0, + "content": "know", + "type": "text" + }, + { + "bbox": [ + 131, + 546, + 153, + 556 + ], + "score": 0.87, + "content": "\\rho _ { t r u e }", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 544, + 383, + 559 + ], + "score": 1.0, + "content": ". In the following we show that the minimization of both", + "type": "text" + }, + { + "bbox": [ + 383, + 545, + 390, + 555 + ], + "score": 0.82, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 544, + 408, + 559 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 409, + 545, + 419, + 557 + ], + "score": 0.89, + "content": "f _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 544, + 505, + 559 + ], + "score": 1.0, + "content": "is better than simply", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 554, + 256, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 155, + 570 + ], + "score": 1.0, + "content": "minimizing", + "type": "text" + }, + { + "bbox": [ + 155, + 556, + 165, + 568 + ], + "score": 0.89, + "content": "f _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 554, + 192, + 570 + ], + "score": 1.0, + "content": "when", + "type": "text" + }, + { + "bbox": [ + 192, + 556, + 252, + 568 + ], + "score": 0.9, + "content": "\\rho _ { t r u e } \\neq \\rho _ { t r a i n }", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 554, + 256, + 570 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30.5, + "bbox_fs": [ + 104, + 500, + 506, + 570 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 572, + 333, + 583 + ], + "lines": [ + { + "bbox": [ + 106, + 570, + 334, + 586 + ], + "spans": [ + { + "bbox": [ + 106, + 570, + 334, + 586 + ], + "score": 1.0, + "content": "1. First, we re-write the conclusion of Corollary 5.2.1 as", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34, + "bbox_fs": [ + 106, + 570, + 334, + 586 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 167, + 586, + 444, + 622 + ], + "lines": [ + { + "bbox": [ + 167, + 586, + 444, + 622 + ], + "spans": [ + { + "bbox": [ + 167, + 586, + 444, + 622 + ], + "score": 0.64, + "content": "\\begin{array} { r } { \\mathbb { E } _ { w } \\mathbb { E } _ { x } f ( w , x ) \\le f _ { p } + R = C + \\widehat { h } + R = C + \\rho ^ { 2 } \\sigma / 2 + R + O ( \\rho ^ { 3 } ) } \\\\ { w i t h p r o b a b i l i t y \\left( 1 - a \\right) [ 1 - e ^ { - ( \\frac { \\rho } { \\sqrt { 2 } b } - \\sqrt { k } ) ^ { 2 } } ] } \\end{array}", + "type": "interline_equation", + "image_path": "cdaf2ab0abf25e025e63bc79a18bd9a65ace8a902a290b154e1abfb27914e5e9.jpg" + } + ] + } + ], + "index": 36, + "virtual_lines": [ + { + "bbox": [ + 167, + 586, + 444, + 598.0 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 167, + 598.0, + 444, + 610.0 + ], + "spans": [], + "index": 36 + }, + { + "bbox": [ + 167, + 610.0, + 444, + 622.0 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 624, + 503, + 657 + ], + "lines": [ + { + "bbox": [ + 105, + 623, + 505, + 637 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 133, + 637 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 625, + 142, + 634 + ], + "score": 0.83, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 623, + 249, + 637 + ], + "score": 1.0, + "content": "is the regularization term,", + "type": "text" + }, + { + "bbox": [ + 250, + 625, + 259, + 634 + ], + "score": 0.83, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 623, + 339, + 637 + ], + "score": 1.0, + "content": "is the training loss,", + "type": "text" + }, + { + "bbox": [ + 339, + 627, + 346, + 635 + ], + "score": 0.77, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 623, + 505, + 637 + ], + "score": 1.0, + "content": "is the dominant eigenvalue of Hessian.", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 635, + 505, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 435, + 648 + ], + "score": 1.0, + "content": "As in lemma 3.3, we perform Taylor-expansion and can ignore the high-order term", + "type": "text" + }, + { + "bbox": [ + 435, + 635, + 461, + 648 + ], + "score": 0.92, + "content": "O ( \\rho ^ { 3 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 635, + 505, + 648 + ], + "score": 1.0, + "content": ". We focus", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 648, + 120, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 120, + 658 + ], + "score": 1.0, + "content": "on", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39, + "bbox_fs": [ + 105, + 623, + 505, + 658 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 250, + 654, + 361, + 670 + ], + "lines": [ + { + "bbox": [ + 250, + 654, + 361, + 670 + ], + "spans": [ + { + "bbox": [ + 250, + 654, + 361, + 670 + ], + "score": 0.92, + "content": "f _ { p } = C + \\widehat { h } = C + \\rho ^ { 2 } \\sigma / 2", + "type": "interline_equation", + "image_path": "a4c4871c27e9cd5aff90d0bbfaebf1c51736abe97c45a1184e05d9ae4fff9a26.jpg" + } + ] + } + ], + "index": 41, + "virtual_lines": [ + { + "bbox": [ + 250, + 654, + 361, + 670 + ], + "spans": [], + "index": 41 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 676, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 676, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 145, + 691 + ], + "score": 1.0, + "content": "2. When", + "type": "text" + }, + { + "bbox": [ + 146, + 677, + 206, + 689 + ], + "score": 0.89, + "content": "\\rho _ { t r u e } \\neq \\rho _ { t r a i n }", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 676, + 261, + 691 + ], + "score": 1.0, + "content": ", minimizing", + "type": "text" + }, + { + "bbox": [ + 261, + 677, + 268, + 687 + ], + "score": 0.79, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 676, + 453, + 691 + ], + "score": 1.0, + "content": "achieves a lower risk than only minimizing", + "type": "text" + }, + { + "bbox": [ + 454, + 677, + 464, + 689 + ], + "score": 0.86, + "content": "f _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 676, + 506, + 691 + ], + "score": 1.0, + "content": ". (1) Note", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 686, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 180, + 700 + ], + "score": 1.0, + "content": "that after training,", + "type": "text" + }, + { + "bbox": [ + 180, + 689, + 189, + 698 + ], + "score": 0.81, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 686, + 301, + 700 + ], + "score": 1.0, + "content": "(training loss) is fixed, but", + "type": "text" + }, + { + "bbox": [ + 302, + 688, + 308, + 698 + ], + "score": 0.8, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 686, + 378, + 700 + ], + "score": 1.0, + "content": "could vary with", + "type": "text" + }, + { + "bbox": [ + 378, + 690, + 384, + 699 + ], + "score": 0.77, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 686, + 505, + 700 + ], + "score": 1.0, + "content": "(e.g. when training on dataset", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "A and testing on an unrelated dataset B, the training loss remains unchanged, but the risk would", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 708, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 190, + 723 + ], + "score": 1.0, + "content": "be huge and a large", + "type": "text" + }, + { + "bbox": [ + 190, + 712, + 197, + 721 + ], + "score": 0.8, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 708, + 483, + 723 + ], + "score": 1.0, + "content": "is required for a valid bound). (2) With an example, we show a low", + "type": "text" + }, + { + "bbox": [ + 483, + 710, + 494, + 722 + ], + "score": 0.88, + "content": "f _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 708, + 506, + 723 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 719, + 333, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 270, + 734 + ], + "score": 1.0, + "content": "insufficient for generalization, and a low", + "type": "text" + }, + { + "bbox": [ + 271, + 723, + 277, + 730 + ], + "score": 0.77, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 719, + 333, + 734 + ], + "score": 1.0, + "content": "is necessary:", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 44, + "bbox_fs": [ + 105, + 676, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 128, + 81, + 506, + 230 + ], + "lines": [ + { + "bbox": [ + 129, + 81, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 129, + 81, + 211, + 96 + ], + "score": 1.0, + "content": "A Suppose we use", + "type": "text" + }, + { + "bbox": [ + 212, + 85, + 237, + 94 + ], + "score": 0.86, + "content": "\\rho _ { t r a i n }", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 81, + 426, + 96 + ], + "score": 1.0, + "content": "for training, and consider two solutions with", + "type": "text" + }, + { + "bbox": [ + 426, + 83, + 454, + 94 + ], + "score": 0.91, + "content": "C _ { 1 } , \\sigma _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 81, + 506, + 96 + ], + "score": 1.0, + "content": "(SAM) and", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 142, + 93, + 479, + 107 + ], + "spans": [ + { + "bbox": [ + 142, + 94, + 169, + 105 + ], + "score": 0.9, + "content": "C _ { 2 } , \\sigma _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 170, + 93, + 326, + 107 + ], + "score": 1.0, + "content": "(GSAM). Suppose they have the same", + "type": "text" + }, + { + "bbox": [ + 326, + 94, + 336, + 106 + ], + "score": 0.89, + "content": "f _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 93, + 437, + 107 + ], + "score": 1.0, + "content": "during training for some", + "type": "text" + }, + { + "bbox": [ + 438, + 96, + 463, + 105 + ], + "score": 0.62, + "content": "\\rho _ { t r a i n }", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 93, + 479, + 107 + ], + "score": 1.0, + "content": ", so", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 208, + 110, + 439, + 126 + ], + "spans": [ + { + "bbox": [ + 208, + 110, + 439, + 126 + ], + "score": 0.89, + "content": "f _ { p 1 } = C _ { 1 } + \\sigma _ { 1 } / 2 \\times \\rho _ { t r a i n } ^ { 2 } = C _ { 2 } + \\sigma _ { 2 } / 2 \\times \\rho _ { t r a i n } ^ { 2 } = f _ { p 2 }", + "type": "inline_equation", + "image_path": "4db8b113636d6520040de09b22820b5243b1ea3a6833aaba864da604befe1ded.jpg" + } + ], + "index": 2 + }, + { + "bbox": [ + 141, + 129, + 268, + 145 + ], + "spans": [ + { + "bbox": [ + 141, + 129, + 178, + 145 + ], + "score": 1.0, + "content": "Suppose", + "type": "text" + }, + { + "bbox": [ + 178, + 131, + 216, + 142 + ], + "score": 0.91, + "content": "C _ { 1 } < C _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 129, + 228, + 145 + ], + "score": 1.0, + "content": "so", + "type": "text" + }, + { + "bbox": [ + 229, + 132, + 263, + 142 + ], + "score": 0.89, + "content": "\\sigma _ { 1 } > \\sigma _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 129, + 268, + 145 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 129, + 145, + 268, + 159 + ], + "spans": [ + { + "bbox": [ + 129, + 145, + 168, + 159 + ], + "score": 1.0, + "content": "B When", + "type": "text" + }, + { + "bbox": [ + 169, + 147, + 228, + 158 + ], + "score": 0.9, + "content": "\\rho _ { t r u e } > \\rho _ { t r a i n }", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 145, + 268, + 159 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 141, + 162, + 505, + 177 + ], + "spans": [ + { + "bbox": [ + 141, + 163, + 201, + 177 + ], + "score": 1.0, + "content": "risk bound", + "type": "text" + }, + { + "bbox": [ + 201, + 162, + 505, + 176 + ], + "score": 0.82, + "content": "\\begin{array} { r } { - 1 = C _ { 1 } + \\sigma _ { 1 } / 2 \\times \\rho _ { t r u e } ^ { 2 } + R > \\mathrm { r i s k . b o u n d . } 2 = C _ { 2 } + \\sigma _ { 2 } / 2 \\times \\rho _ { t r u e } ^ { 2 } + R } \\end{array}", + "type": "inline_equation", + "image_path": "47b03d8f7d3cf477315f317bc020420e920f222a2bbf5a602954754d729f4a5f.jpg" + } + ], + "index": 5 + }, + { + "bbox": [ + 141, + 180, + 504, + 195 + ], + "spans": [ + { + "bbox": [ + 141, + 180, + 245, + 195 + ], + "score": 1.0, + "content": "This implies that a small", + "type": "text" + }, + { + "bbox": [ + 245, + 183, + 253, + 192 + ], + "score": 0.77, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 180, + 404, + 195 + ], + "score": 1.0, + "content": "helps generalization, but only a low", + "type": "text" + }, + { + "bbox": [ + 405, + 182, + 419, + 194 + ], + "score": 0.9, + "content": "f _ { p 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 180, + 491, + 195 + ], + "score": 1.0, + "content": "(caused by a low", + "type": "text" + }, + { + "bbox": [ + 491, + 182, + 504, + 193 + ], + "score": 0.88, + "content": "C _ { 1 }", + "type": "inline_equation" + } + ], + "index": 6 + }, + { + "bbox": [ + 142, + 193, + 356, + 205 + ], + "spans": [ + { + "bbox": [ + 142, + 193, + 179, + 205 + ], + "score": 1.0, + "content": "and high", + "type": "text" + }, + { + "bbox": [ + 179, + 194, + 191, + 204 + ], + "score": 0.82, + "content": "\\sigma _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 193, + 356, + 205 + ], + "score": 1.0, + "content": ") is insufficient for a good generalization.", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 129, + 205, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 129, + 205, + 183, + 222 + ], + "score": 1.0, + "content": "C Note that", + "type": "text" + }, + { + "bbox": [ + 183, + 209, + 209, + 219 + ], + "score": 0.89, + "content": "\\rho _ { t r a i n }", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 205, + 371, + 222 + ], + "score": 1.0, + "content": "is fixed during training, so minimizing", + "type": "text" + }, + { + "bbox": [ + 371, + 208, + 398, + 219 + ], + "score": 0.91, + "content": "h _ { t r a i n }", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 205, + 505, + 222 + ], + "score": 1.0, + "content": "during training is equiva-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 142, + 218, + 285, + 231 + ], + "spans": [ + { + "bbox": [ + 142, + 218, + 216, + 231 + ], + "score": 1.0, + "content": "lently minimizing", + "type": "text" + }, + { + "bbox": [ + 216, + 221, + 223, + 228 + ], + "score": 0.76, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 218, + 285, + 231 + ], + "score": 1.0, + "content": "by Lemma 3.3", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 106, + 240, + 505, + 318 + ], + "lines": [ + { + "bbox": [ + 104, + 239, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 104, + 239, + 274, + 254 + ], + "score": 1.0, + "content": "3. Why we are often unlucky to have", + "type": "text" + }, + { + "bbox": [ + 274, + 242, + 339, + 252 + ], + "score": 0.85, + "content": "\\rho _ { t r u e } ~ > ~ \\rho _ { t r a i n }", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 239, + 505, + 254 + ], + "score": 1.0, + "content": "(1) First, the test sets are almost surely", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 252, + 505, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 505, + 264 + ], + "score": 1.0, + "content": "outside the convex hull of the training set because “‘interpolation almost surely never occurs in", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 261, + 506, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 261, + 181, + 276 + ], + "score": 1.0, + "content": "high-dimensional", + "type": "text" + }, + { + "bbox": [ + 181, + 263, + 214, + 273 + ], + "score": 0.81, + "content": "( > ~ 1 0 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 261, + 506, + 276 + ], + "score": 1.0, + "content": "cases”’ Balestriero et al. (2021). As a result, the variability of (train", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 273, + 506, + 285 + ], + "spans": [ + { + "bbox": [ + 106, + 275, + 114, + 284 + ], + "score": 0.69, + "content": "^ +", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 273, + 415, + 285 + ], + "score": 1.0, + "content": "test) sets is almost surely larger than the variability of (train) set. 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For", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 284, + 506, + 297 + ], + "spans": [ + { + "bbox": [ + 106, + 284, + 506, + 297 + ], + "score": 1.0, + "content": "vanilla training we report results for training 300 epochs and 600 epochs, for GSAM we report the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 295, + 199, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 199, + 307 + ], + "score": 1.0, + "content": "results for 300 epochs.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5 + }, + { + "type": "table_body", + "bbox": [ + 136, + 315, + 474, + 363 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 136, + 315, + 474, + 363 + ], + "spans": [ + { + "bbox": [ + 136, + 315, + 474, + 363 + ], + "score": 0.975, + "html": "
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As", + "type": "text" + }, + { + "bbox": [ + 342, + 253, + 349, + 261 + ], + "score": 0.77, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 250, + 506, + 264 + ], + "score": 1.0, + "content": "increases, the surrogate gap decreases,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 262, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 106, + 262, + 505, + 275 + ], + "score": 1.0, + "content": "validating that the ascent step in GSAM efficiently minimizes the surrogate gap. Furthermore, the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 273, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 330, + 286 + ], + "score": 1.0, + "content": "surrogate gap increases with training steps for any fixed", + "type": "text" + }, + { + "bbox": [ + 330, + 274, + 338, + 283 + ], + "score": 0.77, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 273, + 505, + 286 + ], + "score": 1.0, + "content": ", indicating that the training process grad-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 284, + 382, + 296 + ], + "spans": [ + { + "bbox": [ + 106, + 284, + 382, + 296 + ], + "score": 1.0, + "content": "ually falls into local minimum in order to minimize the training loss.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13.5 + }, + { + "type": "title", + "bbox": [ + 107, + 311, + 217, + 323 + ], + "lines": [ + { + "bbox": [ + 105, + 310, + 218, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 310, + 218, + 325 + ], + "score": 1.0, + "content": "D RELATED WORKS", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 336, + 505, + 424 + ], + "lines": [ + { + "bbox": [ + 105, + 335, + 505, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 505, + 349 + ], + "score": 1.0, + "content": "Besides SAM and ASAM, other methods were proposed in the literature to improve generalization:", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 348, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 505, + 360 + ], + "score": 1.0, + "content": "Lin et al. (2020) proposed extrapolation of gradient, Xie et al. (2021) proposed to manipulate the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 358, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 505, + 370 + ], + "score": 1.0, + "content": "noise in gradient, and Damian et al. (2021) proved label noise improves generalization, Yue et al.", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 370, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 370, + 505, + 381 + ], + "score": 1.0, + "content": "(2020) proposed to adjust learning rate according to sharpness, and Zheng et al. (2021) proposed", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 380, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 505, + 393 + ], + "score": 1.0, + "content": "model perturbation with similar idea to SAM. Izmailov et al. (2018) proposed averaging weights", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 391, + 504, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 504, + 403 + ], + "score": 1.0, + "content": "to improve generalization, and Heo et al. (2020) restricted the norm of updated weights to improve", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 402, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 505, + 415 + ], + "score": 1.0, + "content": "generalization. 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In Fig. 8 the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 211, + 505, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 211, + 298, + 225 + ], + "score": 1.0, + "content": "cosine value is always above 0.9, indicating that", + "type": "text" + }, + { + "bbox": [ + 299, + 212, + 332, + 224 + ], + "score": 0.93, + "content": "\\nabla f ( w _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 211, + 349, + 225 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 350, + 212, + 387, + 224 + ], + "score": 0.94, + "content": "\\nabla f _ { p } ( w _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 211, + 505, + 225 + ], + "score": 1.0, + "content": "point to very close directions", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 223, + 506, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 484, + 236 + ], + "score": 1.0, + "content": "considering the high dimension of parameters. This empirically validates our assumption that", + "type": "text" + }, + { + "bbox": [ + 485, + 223, + 494, + 234 + ], + "score": 0.87, + "content": "\\theta _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 223, + 506, + 236 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 233, + 195, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 233, + 195, + 248 + ], + "score": 1.0, + "content": "small during training.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 8, + "bbox_fs": [ + 105, + 167, + 506, + 248 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 250, + 505, + 295 + ], + "lines": [ + { + "bbox": [ + 105, + 250, + 506, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 341, + 264 + ], + "score": 1.0, + "content": "We also plot the surrogate gap during training in Fig. 9. As", + "type": "text" + }, + { + "bbox": [ + 342, + 253, + 349, + 261 + ], + "score": 0.77, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 250, + 506, + 264 + ], + "score": 1.0, + "content": "increases, the surrogate gap decreases,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 262, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 106, + 262, + 505, + 275 + ], + "score": 1.0, + "content": "validating that the ascent step in GSAM efficiently minimizes the surrogate gap. Furthermore, the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 273, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 330, + 286 + ], + "score": 1.0, + "content": "surrogate gap increases with training steps for any fixed", + "type": "text" + }, + { + "bbox": [ + 330, + 274, + 338, + 283 + ], + "score": 0.77, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 273, + 505, + 286 + ], + "score": 1.0, + "content": ", indicating that the training process grad-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 284, + 382, + 296 + ], + "spans": [ + { + "bbox": [ + 106, + 284, + 382, + 296 + ], + "score": 1.0, + "content": "ually falls into local minimum in order to minimize the training loss.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 250, + 506, + 296 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 311, + 217, + 323 + ], + "lines": [ + { + "bbox": [ + 105, + 310, + 218, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 310, + 218, + 325 + ], + "score": 1.0, + "content": "D RELATED WORKS", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 336, + 505, + 424 + ], + "lines": [ + { + "bbox": [ + 105, + 335, + 505, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 505, + 349 + ], + "score": 1.0, + "content": "Besides SAM and ASAM, other methods were proposed in the literature to improve generalization:", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 348, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 505, + 360 + ], + "score": 1.0, + "content": "Lin et al. (2020) proposed extrapolation of gradient, Xie et al. (2021) proposed to manipulate the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 358, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 505, + 370 + ], + "score": 1.0, + "content": "noise in gradient, and Damian et al. (2021) proved label noise improves generalization, Yue et al.", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 370, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 370, + 505, + 381 + ], + "score": 1.0, + "content": "(2020) proposed to adjust learning rate according to sharpness, and Zheng et al. (2021) proposed", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 380, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 505, + 393 + ], + "score": 1.0, + "content": "model perturbation with similar idea to SAM. Izmailov et al. (2018) proposed averaging weights", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 391, + 504, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 504, + 403 + ], + "score": 1.0, + "content": "to improve generalization, and Heo et al. (2020) restricted the norm of updated weights to improve", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 402, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 505, + 415 + ], + "score": 1.0, + "content": "generalization. Many of aforementioned methods can be combined with GSAM to further improve", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 413, + 168, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 168, + 425 + ], + "score": 1.0, + "content": "generalization.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 335, + 505, + 425 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 430, + 505, + 561 + ], + "lines": [ + { + "bbox": [ + 105, + 430, + 504, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 504, + 442 + ], + "score": 1.0, + "content": "Besides modified training schemes, there are other two types of techniques to improve generaliza-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 441, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 505, + 453 + ], + "score": 1.0, + "content": "tion: data augmentation and model regularization. Data augmentation typically generates new data", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 451, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 505, + 466 + ], + "score": 1.0, + "content": "from training samples; besides standard data augmentation such as flipping or rotation of images,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 462, + 505, + 476 + ], + "spans": [ + { + "bbox": [ + 104, + 462, + 505, + 476 + ], + "score": 1.0, + "content": "recent data augmentations include label smoothing (Muller et al., 2019) and mixup (M ¨ uller et al., ¨", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 473, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 505, + 486 + ], + "score": 1.0, + "content": "2019) which trains on convex combinations of both inputs and labels, automatically learned augmen-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 485, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 505, + 497 + ], + "score": 1.0, + "content": "tation (Cubuk et al., 2018), and cutout (DeVries & Taylor, 2017) which randomly masks out parts", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 496, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 505, + 508 + ], + "score": 1.0, + "content": "of an image. Model regularization typically applies auxiliary losses besides the training loss such as", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 506, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 505, + 519 + ], + "score": 1.0, + "content": "weight decay (Loshchilov & Hutter, 2017), other methods randomly modify the model architecture", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 518, + 505, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 505, + 530 + ], + "score": 1.0, + "content": "during training, such as dropout (Srivastava et al., 2014) and shake-shake regularization (Gastaldi,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 527, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 505, + 542 + ], + "score": 1.0, + "content": "2017). Note that the data augmentation and model regularization literature mentioned here typically", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 540, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 505, + 552 + ], + "score": 1.0, + "content": "train with the standard back-propagation (Rumelhart et al., 1985) and first-order gradient optimizers,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 550, + 313, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 313, + 563 + ], + "score": 1.0, + "content": "and both techniques can be combined with GSAM.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 30.5, + "bbox_fs": [ + 104, + 430, + 505, + 563 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 567, + 503, + 600 + ], + "lines": [ + { + "bbox": [ + 106, + 567, + 504, + 579 + ], + "spans": [ + { + "bbox": [ + 106, + 567, + 504, + 579 + ], + "score": 1.0, + "content": "Besides SGD, Adam and AdaBelief, GSAM can be combined with other first-order gradient opti-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 577, + 506, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 506, + 591 + ], + "score": 1.0, + "content": "mizers, such as AdaBound (Luo et al., 2019), RAdam (Liu et al., 2019), Yogi (Zaheer et al., 2018),", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 589, + 470, + 601 + ], + "spans": [ + { + "bbox": [ + 106, + 589, + 470, + 601 + ], + "score": 1.0, + "content": "AdaGrad (Duchi et al., 2011), AMSGrad (Reddi et al., 2019) and AdaDelta (Zeiler, 2012).", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38, + "bbox_fs": [ + 105, + 567, + 506, + 601 + ] + } + ] + } + ], + "_backend": "pipeline", + "_version_name": "2.2.2" +} \ No newline at end of file diff --git a/parse/dev/edONMAnhLu-/edONMAnhLu-_model.json b/parse/dev/edONMAnhLu-/edONMAnhLu-_model.json new file mode 100644 index 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ModelTrainingImageNet-v1ImageNet-RealImageNet-V2ImageNet-RImageNet-C
ResNet
ResNet50Vanilla (SGD)76.082.463.622.244.6
SAM76.983.364.423.846.5
GSAM77.283.964.623.647.6
ResNet101Vanilla (SGD)77.883.965.324.448.5
SAM78.684.866.725.951.3
GSAM78.985.267.326.351.8
ResNet152Vanilla (SGD)78.584.266.325.350.0
SAM79.384.967.325.752.2
GSAM80.085.968.627.354.1
ViT-S/32Vanilla (AdamW)68.4Vision Transformer 75.254.3
SAM70.577.556.919.0 21.443.3
GSAM73.880.460.422.546.2 48.2
ViT-S/16Vanilla (AdamW)74.480.461.720.046.5
SAM78.184.165.624.753.0
GSAM79.585.367.325.353.3
ViT-B/32Vanilla (AdamW)71.477.557.523.444.0
SAM73.680.360.024.050.7
GSAM76.882.763.025.151.7
ViT-B/16Vanilla (AdamW)74.679.861.320.146.6
SAM79.985.267.526.456.5
GSAM81.086.569.227.155.7
Mixer-S/32Vanilla (AdamW)63.9MLP-Mixer 70.349.516.935.2
SAM66.773.852.418.639.3
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GSAM73.680.259.927.952.1
Mixer-B/16Vanilla (AdamW)66.472.150.814.533.8
SAM77.483.563.924.748.8
GSAM77.884.064.928.354.4
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ViT-B/16ViT-S/16
VanillaSAMGSAMVanillaSAMGSAM
Cifar1098.198.698.897.698.298.4
Cifar10087.689.189.785.787.688.1
Flowers88.591.891.286.491.590.3
Pets91.993.194.490.492.993.5
mean91.593.293.590.092.692.6
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Datasetmin(fp+Xh)GSAM
ImageNet75.476.8
ImageNet-Real81.182.7
ImageNet-v260.963.0
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ViT-B/320.60.10.63e-33e-50.3AdamW30010kLinear
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Mixer-S/320.50.00.23e-33e-50.3AdamW30010kLinear
Mixer-S/160.50.00.63e-33e-50.3AdamW30010kLinear
Mixer-S/80.50.10.13e-33e-50.3AdamW30010kLinear
Mixer-B/320.70.20.053e-33e-50.3AdamW30010kLinear
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MethodEpochsImageNetImageNet-RealImageNet-v2ImageNet-R
Vanilla30071.477.557.523.4
60072.078.257.923.6
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WITH ALGORITHM DISTILLATION + +Michael Laskin∗†, Luyu Wang∗, Junhyuk Oh, Emilio Parisotto, Stephen Spencer, Richie Steigerwald, DJ Strouse, Steven Hansen, Angelos Filos, Ethan Brooks, Maxime Gazeau, Himanshu Sahni, Satinder Singh, Volodymyr Mnih† DeepMind + +# ABSTRACT + +We propose Algorithm Distillation (AD), a method for distilling reinforcement learning (RL) algorithms into neural networks by modeling their training histories with a causal sequence model. Algorithm Distillation treats learning to reinforcement learn as an across-episode sequential prediction problem. A dataset of learning histories is generated by a source RL algorithm, and then a causal transformer is trained by autoregressively predicting actions given their preceding learning histories as context. Unlike sequential policy prediction architectures that distill post-learning or expert sequences, AD is able to improve its policy entirely in-context without updating its network parameters. We demonstrate that AD can reinforcement learn in-context in a variety of environments with sparse rewards, combinatorial task structure, and pixel-based observations, and find that AD learns a more data-efficient RL algorithm than the one that generated the source data. + +![](images/679c16b167e631679ba20610706e8543cd9d762d7dab80268efc9673f356c188.jpg) +Figure 1: Algorithm Distillation (AD) has two steps – (i) a dataset of learning histories is collected from individual single-task RL algorithms solving different tasks; (ii) a causal transformer predicts actions from these histories using across-episodic contexts. Since the RL policy improves throughout the learning histories, by predicting actions accurately AD learns to output an improved policy relative to the one seen in its context. AD models state-action-reward tokens, and does not condition on returns. + +# 1 INTRODUCTION + +Transformers have emerged as powerful neural network architectures for sequence modeling (Vaswani et al., 2017). A striking property of pre-trained transformers is their ability to adapt to downstream tasks through prompt conditioning or in-context learning. After pre-training on large offline datasets, large transformers have been shown to generalize to downstream tasks in text completion (Brown et al., 2020), language understanding (Devlin et al., 2018), and image generation (Yu et al., 2022). + +Recent work demonstrated that transformers can also learn policies from offline data by treating offline Reinforcement Learning (RL) as a sequential prediction problem. While Chen et al. (2021) showed that transformers can learn single-task policies from offline RL data via imitation learning, subsequent work showed that transformers can also extract multi-task policies in both same-domain (Lee et al., 2022) and cross-domain settings (Reed et al., 2022). These works suggest a promising paradigm for extracting generalist multi-task policies – first collect a large and diverse dataset of environment interactions, then extract a policy from the data via sequential modeling. We refer to the family of approaches that learns policies from offline RL data via imitation learning as Offline Policy Distillation, or simply Policy Distillation1 (PD). + +Despite its simplicity and scalability, a substantial drawback of PD is that the resulting policy does not improve incrementally from additional interaction with the environment. For instance, the MultiGame Decision Transformer (MGDT, Lee et al., 2022) learns a return-conditioned policy that plays many Atari games while Gato (Reed et al., 2022) learns a policy that solves tasks across diverse environments by inferring tasks through context, but neither method can improve its policy in-context through trial and error. MGDT adapts the transformer to new tasks by finetuning the model weights while Gato requires prompting with an expert demonstration to adapt to a new task. In short, Policy Distillation methods learn policies but not Reinforcement Learning algorithms. + +We hypothesize that the reason Policy Distillation does not improve through trial and error is that it trains on data that does not show learning progress. Current methods either learn policies from data that contains no learning (e.g. by distilling fixed expert policies) or data with learning (e.g. the replay buffer of an RL agent) but with a context size that is too small to capture policy improvement. + +Our key observation is that the sequential nature of learning within RL algorithm training could, in principle, make it possible to model the process of reinforcement learning itself as a causal sequence prediction problem. Specifically, if a transformer’s context is long enough to include policy improvement due to learning updates it should be able to represent not only a fixed policy but a policy improvement operator by attending to states, actions and rewards from previous episodes. This opens the possibility that any RL algorithm can be distilled into a sufficiently powerful sequence model such as a transformer via imitation learning, converting it into an in-context RL algorithm. By in-context RL we mean that the transformer should improve its policy through trial and error within the environment by attending to its context, without updating its parameters. + +We present Algorithm Distillation (AD), a method that learns an in-context policy improvement operator by optimizing a causal sequence prediction loss on the learning histories of an RL algorithm. AD has two components. First, a large multi-task dataset is generated by saving the training histories of an RL algorithm on many individual tasks. Next, a transformer models actions causally using the preceding learning history as its context. Since the policy improves throughout the course of training of the source RL algorithm, AD is forced to learn the improvement operator in order to accurately model the actions at any given point in the training history. Crucially, the transformer context size must be sufficiently large (i.e. across-episodic) to capture improvement in the training data. The full method is shown in Fig. 1. + +We show that by imitating gradient-based RL algorithms using a causal transformer with sufficiently large contexts, AD can reinforcement learn new tasks entirely in-context. We evaluate AD across a number of partially observed environments that require exploration, including the pixel-based Watermaze (Morris, 1981) from DMLab (Beattie et al., 2016). We show that AD is capable of in-context exploration, temporal credit assignment, and generalization. We also show that AD learns a more data-efficient algorithm than the one that generated the source data for transformer training. To the best of our knowledge, AD is the first method to demonstrate in-context reinforcement learning via sequential modeling of offline data with an imitation loss. + +# 2 BACKGROUND + +Partially Observable Markov Decision Processes: A Markov Decision Process (MDP) consists of states $s \in S$ , actions $a \in { \mathcal { A } }$ , rewards $r \in \mathcal { R }$ , a discount factor $\gamma$ , and a transition probability function $p ( s _ { t + 1 } | s _ { t } , a _ { t } )$ , where $t$ is an integer denoting the timestep and $( \cal { S } , \cal { A } )$ are state and action spaces. In environments described by an MDP, at each timestep $t$ the agent observes the state $s _ { t }$ , selects an action $a _ { t } \sim \pi ( \cdot | s _ { t } )$ from its policy, and then observes the next state $s _ { t + 1 } \sim p \big ( \cdot | s _ { t } , a _ { t } \big )$ sampled from the transition dynamics of the environment. In this work, we operate in the Partially Observable Markov Decision Process (POMDP) setting where instead of states $s \in S$ the agent receives observations $o \in \mathcal { O }$ that only have partial information about the true state of the environment. Full state information may be incomplete due to missing information about the goal in the environment, which the agent must instead infer through rewards with memory, or because the observations are pixel-based, or both. + +Online and Offline Reinforcement Learning: Reinforcement Learning algorithms aim to maximize the return, defined as the cumulative sum of rewards $\textstyle \sum _ { t } \gamma ^ { t } r _ { t }$ , throughout an agent’s lifetime or episode of training. RL algorithms broadly fall into two categories: on-policy algorithms (Williams, 1992) where the agent directly maximizes a Monte-Carlo estimate of the total returns or off-policy (Mnih et al., 2013; 2015) where an agent learns and maximizes a value function that approximates the total future return. Most RL algorithms maximize returns through trial-and-error by directly interacting with the environment. However, offline RL (Levine et al., 2020) has recently emerged as an alternate and often complementary paradigm for RL where an agent aims to extract return maximizing policies from offline data gathered by another agent. The offline dataset consists of $( s , a , r )$ tuples which are often used to train an off-policy agent, though other algorithms for extracting return maximizing policies from offline data are also possible. + +Self-Attention and Transformers The self-attention (Vaswani et al., 2017) operation begins by projecting input data $X$ with three separate matrices onto $D$ -dimensional vectors called queries $Q$ , keys $K$ , and values $V$ . These vectors are then passed through the attention function: + +$$ +\mathrm { A t t e n t i o n } ( Q , K , V ) = \mathrm { s o f t m a x } ( Q K ^ { T } / \sqrt { D } ) V . +$$ + +The $Q K ^ { T }$ term computes an inner product between two projections of the input data $X$ . The inner product is then normalized and projected back to a $D$ -dimensional vector with the scaling term $V$ . Transformers (Vaswani et al., 2017; Devlin et al., 2018; Brown et al., 2020) utilize self-attention as a core part of the architecture to process sequential data such as text sequences. Transformers are usually pre-trained with a self-supervised objective that predicts tokens within the sequential data. Common prediction tasks include predicting randomly masked out tokens (Devlin et al., 2018) or applying a causal mask and predicting the next token (Radford et al., 2018). + +Offline Policy Distillation: We refer to the family of methods that treat offline Reinforcement Learning as a sequential prediction problem as Offline Policy Distillation, or Policy Distillation (PD) for brevity. Rather than learning a value function from offline data, PD extracts policies by predicting actions in the offline data (i.e. behavior cloning) with a sequence model and either return conditioning (Chen et al., 2021; Lee et al., 2022) or filtering out suboptimal data (Reed et al., 2022). Initially proposed to learn single-task policies (Chen et al., 2021; Janner et al., 2021), PD was recently extended to learn multi-task policies from diverse offline data (Lee et al., 2022; Reed et al., 2022). + +In-Context Learning: In-context learning refers to the ability to infer tasks from context. For example, large language models like GPT-3 (Brown et al., 2020) or Gopher (Rae et al., 2021) can be directed at solving tasks such as text completion, code generation, and text summarization by specifying the task through language as a prompt. This ability to infer the task from prompt is often called in-context learning. We use the terms ‘in-weights learning’ and ‘in-context learning’ from prior work on sequence models (Brown et al., 2020; Chan et al., 2022) to distinguish between gradient-based learning with parameter updates and gradient-free learning from context, respectively. + +# 3 METHOD + +Over the course of its lifetime a capable reinforcement learning (RL) agent will exhibit complex behaviours, such as exploration, temporal credit assignment, and planning. Our key insight is that an agent’s actions, regardless of the environment it inhabits, its internal structure, and implementation, can be viewed as a function of its past experience, which we refer to as its history. Formally, we write: + +$$ +\mathcal { H } \ni h _ { t } : = \left( o _ { 0 } , a _ { 0 } , r _ { 0 } , \ldots , o _ { t - 1 } , a _ { t - 1 } , r _ { t - 1 } , o _ { t } , a _ { t } , r _ { t } \right) = \left( o _ { \leq t } , r _ { \leq t } , a _ { \leq t } \right) +$$ + +and we refer to a $l o n g ^ { 2 }$ history-conditioned policy as an algorithm: + +$$ +P : \mathcal { H } \cup \mathcal { O } \Delta ( \mathcal { A } ) , +$$ + +where $\Delta ( \mathcal { A } )$ denotes the space of probability distributions over the action space $\mathcal { A }$ . Eqn. (3) suggests that, similar to a policy, an algorithm can be unrolled in an environment to generate sequences of observations, rewards, and actions. For brevity, we denote the algorithm as $P$ and environment (i.e. task) as $\mathcal { M }$ , such that the history of learning for any given task $\mathcal { M }$ is generated by the algorithm $P _ { \mathcal { M } }$ . + +$$ +( O _ { 0 } , A _ { 0 } , R _ { 0 } , \ldots , O _ { T } , A _ { T } , R _ { T } ) \sim P _ { \mathcal { M } } . +$$ + +Here, we’re denoting random variables with uppercase Latin letters, e.g. $O , A , R$ , and their values with lowercase Latin letters, $e . g . \ o , a , r$ . By viewing algorithms as long history-conditioned policies, we hypothesize that any algorithm that generated a set of learning histories can be distilled into a neural network by performing behavioral cloning over actions. Next, we present a method that, provided agents’ lifetimes, learns a sequence model with behavioral cloning to map long histories to distributions over actions. + +# 3.1 ALGORITHM DISTILLATION + +Suppose the agents’ lifetimes, which we also call learning histories, are generated by the source algorithm $P ^ { \mathrm { s o u r c e } }$ for many individual tasks $\{ \mathcal { M } _ { n } \} _ { n = 1 } ^ { N }$ , producing the dataset $\mathcal { D }$ : + +$$ +\begin{array} { r } { \mathcal { D } : = \left\{ \left( o _ { 0 } ^ { ( n ) } , a _ { 0 } ^ { ( n ) } , r _ { 0 } ^ { ( n ) } , \ldots , o _ { T } ^ { ( n ) } , a _ { T } ^ { ( n ) } , r _ { T } ^ { ( n ) } \right) \sim P _ { \mathcal { M } _ { n } } ^ { \mathrm { s o u r c e } } \right\} _ { n = 1 } ^ { N } . } \end{array} +$$ + +Then we distill the source algorithm’s behaviour into a sequence model that maps long histories to probabilities over actions with a negative log likelihood (NLL) loss and refer to this process as algorithm distillation (AD). In this work, we consider neural network models $P _ { \theta }$ with parameters $\theta$ which we train by minimizing the following loss function: + +$$ +\mathcal { L } ( \theta ) : = - \sum _ { n = 1 } ^ { N } \sum _ { t = 1 } ^ { T - 1 } \log P _ { \theta } ( A = a _ { t } ^ { ( n ) } | h _ { t - 1 } ^ { ( n ) } , o _ { t } ^ { ( n ) } ) . +$$ + +Intuitively, a sequence model with fixed parameters that is trained with AD should amortise the source RL algorithm $P ^ { \mathrm { s o u r c e } }$ and by doing so exhibit similarly complex behaviours, such as exploration and temporal credit assignment. Since the RL policy improves throughout the learning history of the source algorithm, accurate action prediction requires the sequence model to not only infer the current policy from the preceding context but also infer the improved policy, therefore distilling the policy improvement operator. + +# Algorithm 1 Algorithm Distillation + +Require: Train $\{ \mathcal { M } ^ { \mathrm { t r a i n } } \}$ and test $\{ \mathcal { M } ^ { \mathrm { t e s t } } \}$ tasks, observations $o \in \mathcal { O }$ , actions $a \in { \mathcal { A } }$ , and rewards $r \in \mathcal { R }$ . Require: Network parameters $\phi _ { i }$ for $i = 1 , \ldots , N$ source RL algorithms. +Require: Network parameters $\theta$ for a causal transformer $P _ { \theta }$ that predicts actions. +Require: An empty buffer to store data $\mathcal { D }$ . +1: for $i = 1 \dots N$ do ▷ Part 1: Dataset Generation 2: Sample a task $\mathcal { M } _ { i } ^ { \mathrm { t r a i n } }$ randomly from the train task distribution. +3: Train the source RL algorithm $\phi _ { i }$ until it converges to the optimal policy. +4: Save the learning history $h _ { T } ^ { ( i ) } = ( o _ { 0 } , a _ { 0 } , r _ { 0 } , \ldots , o _ { T } , a _ { T } , r _ { T } ) _ { i }$ to the dataset $\mathcal { D } \mathcal { D } \cup h _ { T } ^ { ( i ) }$ . 5: end for +6: while $P _ { \theta }$ not converged do ▷ Part 2: Algorithm Distillation 7: Randomly sample a multi-episodic subsequence $\bar { h } _ { j } ^ { ( i ) } = \left( o _ { j } , a _ { j } , r _ { j } , \dotsc , o _ { j + c } , a _ { j + c } , r _ { j + c } \right) _ { i }$ of length $c$ 8: Autoregressively predict the actions with $P _ { \theta }$ and compute the NLL loss in Eq. 6. +9: Backpropagate to update the transformer parameters. +10: end while +11: for $k = 1 \dots M _ { \mathrm { s e e d s } }$ do ▷ Part 3: Autoregressive Evaluation 12: Sample a task $\mathcal { M } _ { k } ^ { \mathrm { t e s t } }$ randomly from the test task distribution. Initialize empty context queue $C$ . 13: Unroll the transformer $P _ { \theta } ( \cdot | C )$ in the environment storing sequential transitions (i.e. histories) in $C$ . 14: Measure the return accumulated by the agent for each episode of evaluation. +15: end for + +# 3.2 PRACTICAL IMPLEMENTATION + +In practice, we implement AD as a two-step procedure. First, a dataset of learning histories is collected by running an individual gradient-based RL algorithm on many different tasks. Next, a sequence model with multi-episodic context is trained to predict actions from the histories. We describe these two steps below and detail the full practical implementation in Algorithm 1. + +Dataset Generation: A dataset of learning histories is collected by training $N$ individual single-task gradient-based RL algorithms. To prevent overfitting to any specific task during sequence model training, a task $\mathcal { M }$ is sampled randomly from a task distribution for each RL run. The data generation step is RL algorithm agnostic - any RL algorithm can be distilled. We show results distilling UCB exploration (Lai & Robbins, 1985), an on-policy actor-critic (Mnih et al., 2016), and an off-policy DQN (Mnih et al., 2013), in both distributed and single-stream settings. We denote the dataset of learning histories as $\mathcal { D }$ in Eq. 5. + +Training the Sequence Model: Once a dataset of learning histories $\mathcal { D }$ is collected, a sequential prediction model is trained to predict actions given the preceding histories. We utilize the GPT (Radford et al., 2018) causal transformer model for sequential action prediction, but AD is compatible with any sequence model including RNNs (Williams & Zipser, 1989). For instance, we show in Appendix $\mathrm { L }$ that AD can also be achieved with an LSTM (Hochreiter & Schmidhuber, 1997), though less effectively than AD with causal transformers. Since causal transformer training and inference are quadratic in the sequence length, we sample across-episodic subsequences $\bar { h _ { j } } ^ { - } = \left( o _ { j } , r _ { j } , a _ { j } \ldots , o _ { j + c } , r _ { j + c } , a _ { j + c } \right)$ of length $c < T$ from $\mathcal { D }$ rather than training full histories. + +# 4 EXPERIMENTAL SETUP + +# 4.1 ENVIRONMENTS + +To investigate the in-context RL capabilities of AD and the baselines (see next section), we focus on environments that cannot be solved through zero-shot generalization after pre-training. Specifically, we require that each environment supports many tasks, that the tasks cannot be inferred easily from the observation, and that episodes are short enough to feasibly train across-episodic causal transformers - for more details regarding environments see Appendix B. We list the evaluation environments below: + +Adversarial Bandit: a multi-armed bandit with 10 arms and 100 trials similar to the environment considered in $\mathrm { { R L ^ { 2 } } }$ (Duan et al., 2016). However, during evaluation the reward is out of distribution. Reward is more likely distributed under odd arms $9 5 \%$ of the time during training. At evaluation, the opposite happens - reward appears more often under even arms $9 5 \%$ of the time. + +![](images/a3da0f2d56a2dea7e0ac773cbc4072974da4d3b19ed7b19d036e66b3977c8b9e.jpg) +Figure 2: Agent view from the DMLab Watermaze environment. The task is to find a hidden platform that elevates once found. + +Dark Room: a 2D discrete POMDP where an agent spawns in a room and must find a goal location. The agent only knows its own $( x , y )$ coordinates but does not know the goal location and must infer it from the reward. The room size is $9 \times 9$ , the possible actions are one step left, right, up, down, and no-op, the episode length is 20, and the agent resets at the center of the map. We test two environment variants – Dark Room where the agent receives $r = 1$ every time the goal is reached and Dark Room Hard, a hard exploration variant with a $1 7 \times 1 7$ size and sparse reward $\mathbf { \boldsymbol { r } } = 1$ exactly once for reaching the goal). When not $r = 1$ , then $r = 0$ . + +Dark Key-to-Door: similar to Dark Room but this environment requires an agent to first find an invisible key upon which it receives a reward of $r = 1$ once and then open an invisible door upon which it receives a reward of $r = 1$ once again. Otherwise, the reward is $r = 0$ . The room size is $9 \times 9$ making the task space combinatorial with $8 1 ^ { 2 } = 6 5 6 1$ possible tasks. This environment is similar to the one considered in Chen et al. (2021) except the key and door are invisible and the reward is semisparse $\boldsymbol { r } = 1$ for both key and the door). The agent is randomly reset. The episode length is 50 steps. + +DMLab Watermaze: a partially observable 3D visual DMLab environment based on the classic Morris Watermaze (Morris, 1981). The task is to navigate the water maze to find a randomly spawned trap-door. The maze walls have color patterns that can be used to remember the goal location. Observations are pixel images of size $7 2 \times 9 6 \times 3$ . There are 8 possible actions in total, including going forward, backward, left, or right, rotating left or right, and rotating left or right while going forward. The episode length is 50, and the agent resets at the center of the map. Similar to Dark + +![](images/d5537f516f3e701ce2d1f385b3b62e3325664799bf1950f1bd449d7fede0b517.jpg) +Figure 3: Adversarial Bandit (Section 5): AD, $\mathrm { { R L } ^ { 2 } }$ , and ED evaluated on a 10-arm bandit with 100 trials. The source data for AD comes from learning histories from UCB (Lai & Robbins, 1985). During training, the reward is distributed under odd arms $9 5 \%$ of the time and under even arms $9 5 \%$ of the time during evaluation. Both AD and $\mathrm { { R L } ^ { 2 } }$ can in-context learn in-distribution tasks, but AD generalizes better out of distribution. Running $\mathrm { { R L } ^ { 2 } }$ with a transformer generally doesn’t offer an advantage over the original LSTM variant. ED performs poorly both in and out of distribution relative to AD and $\mathrm { { R L } ^ { 2 } }$ . Scores are normalized relative to UCB. + +Room, the agent cannot see the location of the goal from the observations and must infer it through the reward of $r = 1$ if reached and $r = 0$ otherwise; however, the goal space is continuous and therefore there are an infinite number of goals. + +# 4.2 BASELINES + +The main aim of this work is to investigate to what extent AD reinforcement learns in-context relative to prior related work. AD is mostly closely related to Policy Distillation, where a policy is learned with a sequential model from offline interaction data. In-context online meta-RL is also related though not directly comparable to AD, since AD is an in-context offline meta-RL method. Still, we consider both types of baselines to better contextualize our work. For a more detailed discussion of these baseline choices we refer the reader to Appendix C. Our baselines include: + +Expert Distillation $( E D )$ : this baseline is exactly the same as AD but the source data consists of expert trajectories only, rather than learning histories. ED is most similar to Gato (Reed et al., 2022) except ED models state-action-reward sequences like AD, while Gato models state-action sequences. + +Source Algorithm: we compare AD to the gradient-based source RL algorithm that generates the training data for distillation. We include running the source algorithm from scratch as a baseline to compare the data-efficiency of in-context RL to the in-weights source algorithm. + +$R L ^ { 2 }$ (Duan et al., 2016): an online meta-RL algorithm where exploration and fast in-context adaptation are learned jointly by maximizing a multi-episodic value function. $\mathrm { { R L } ^ { 2 } }$ is not directly comparable to AD for similar reasons to why online and offline RL algorithms are not directly comparable – RL2 gets to interact with the environment during training while AD does not. We use $\mathrm { { R L } ^ { 2 } }$ asymptotic performance as an approximate upper bound for AD. + +# 4.3 EVALUATION + +After pre-training, the AD transformer $P _ { \theta }$ can reinforcement learn in-context. Evaluation is exactly the same as with an in-weights RL algorithm except the learning happens entirely in-context without updating the transformer network parameters. Given an MDP (or POMDP), the transformer interacts with the environment and populates its own context (i.e. without demonstrations), where the context is a queue containing the last $c$ transitions. The transformer’s performance is then evaluated in terms of its ability to maximize return. For all evaluation runs, we average results across 5 training seeds with 20 evaluation seeds each for a total of 100 seeds. A task $\mathcal { M }$ is sampled uniformly from the test task distribution and fixed for each evaluation seed. The aggregate statistics reported reflect multi-task performance. We evaluate for 1000 and 160 episodes for the Dark and Watermaze environments respectively and plot performance as a function of total environment steps at test-time. + +# 5 EXPERIMENTS + +The main research question of this work is whether an in-weights RL algorithm can be amortized into an in-context one via Algorithm Distillation. The in-context RL algorithm should behave in a similar way as the in-weights one and exhibit exploration, credit assignment, and generalization capabilities. We begin our analysis in a clean and simple experimental setting where all three properties are required to solve the task - the Adversarial Bandit described in Sec. 4. + +![](images/9ee4c7c970fbe6d5f893cf896ce23072080991bcd6e0b43221766e63d5068738.jpg) +Figure 4: Main results: we evaluate AD, $\mathrm { { R L } ^ { 2 } }$ , ED, and the source algorithm on environments that require memory and exploration. In these environments, an agent must reach a goal location that can only be inferred through a binary reward. AD is consistently able to in-context reinforcement learn across all environments and is more data-efficient than the A3C (“Dark” environments) (Mnih et al., 2016) or DQN (Watermaze) (Mnih et al., 2013) source algorithm it distilled. We report the mean return $\pm \nobreakspace 1 \nobreakspace$ standard deviation over 5 training seeds with 20 test seeds each. + +To generate the source data, we sample a set of training tasks Bound algorithm (Lai & Robbins, 1985), and save its learning $\{ \mathcal { M } _ { j } \} _ { j = 1 } ^ { N }$ , run the Upper Confidence We then train a transformer to predict actions as described in Alg. 1. We evaluate AD, ED, and $\mathrm { { R L } ^ { 2 } }$ and normalize their scores relative to UCB and a random policy $( r - r _ { r a n d . } ) / ( r _ { U C B } - r _ { r a n d . } )$ . The results are shown in Fig. 3. We find that both AD and $\mathtt { R L } ^ { 2 }$ can reliably in-context learn tasks sampled from the training distribution while ED cannot, though ED does do better than random guessing when evaluated in-distribution. However, AD can also in-context learn to solve out of distribution tasks whereas the other methods cannot. This experiment shows that AD can explore the bandit arms, can assign credit by exploiting an arm once reached, and can generalize well out of distribution nearly as well as UCB. We now move beyond the bandit setting and investigate similar research questions in more challenging RL environments and present our results as answers to a series of research questions. + +Does Algorithm Distillation exhibit in-context reinforcement learning? To answer this question, we first generate source data for Algorithm Distillation. In the Dark Room and Dark Key-to-Door environments we collect 2000 learning histories with an Asynchronous Advantage Actor-Critic (A3C) (Mnih et al., 2016) with 100 actors, while in DMLab Watermaze we collect 4000 learning histories with a distributed DQN with 16 parallel actors (see Appendix F for asymptotic learning curves of the source algorithm and Appendix O for hyperparameters). Shown in Fig. 4, AD in-context reinforcement learns in all of the environments. In contrast, ED fails to explore and learn in-context in most settings. We use $\mathrm { { R L } ^ { 2 } }$ trained for 1 billion environment steps as a proxy for the upper bound of performance for a meta-RL method. $\mathrm { { R L ^ { 2 } } }$ achieves a near-optimal asymptotic score in all the environments except for Dark Room (Hard). Despite learning from offline data, AD matches asymptotic $\mathtt { R L } ^ { 2 }$ on the Dark environments and approaches it (within $1 3 \%$ ) on Watermaze. + +Credit-assignment: In Dark Room, the agent receives $r = 1$ each time it visits the goal location. Even though AD is trained to condition only on single timestep reward and not episodic return tokens, it is still able to maximize the reward, which suggests that AD has learned to do credit assignment. + +Exploration: Dark Room (Hard) tests the agents exploration capability. Since the reward is sparse $r = 1$ exactly once), most of the learning history has reward values of $r = 0$ . Nevertheless, AD infers the goal from previous episodes in its context which means it has learned to explore and exploit. + +Generalization: Dark Key-to-Door tests in-distribution generalization with a combinatorial task space. While the environment has a total of $\sim 6 . 5 \mathrm { k }$ tasks, less than 2k were seen during training. During evaluation, AD both generalizes and achieves near-optimal performance on mostly unseen tasks. + +Can Algorithm Distillation learn from pixel-based observations? DMLab Watermaze is a pixelbased environment that is larger than the Dark environments with tasks sampled from a continuous uniform distribution. The environment is partially observable in two ways - the goal is invisible until the agent has reached it and the first-person view limits the agent’s field of vision. Shown in Fig. 4, AD maximizes the episodic return with in-context RL while ED does not learn. + +Can AD learn a more data-efficient RL algorithm than the one that produced the source data? In Fig. 4, AD is significantly more data-efficient than the source algorithm. This gain is a byproduct of distilling a multi-stream algorithm into a single-stream one. The source algorithms (A3C and DQN) are distributed, which means they run many actors in parallel to achieve good performance.3 A distributed RL algorithm may not be very data-efficient in aggregate but each individual actor can be data-efficient. Since the learning history for each actor is saved separately, + +![](images/e72730fa811ddedf85713313c74350731b168f42cfdbd85e088031cd6898d726.jpg) +Figure 5: $A D$ and $E D$ conditioned on partial demonstrations: We compare the performance of AD and ED when prompted with a demonstration from the source algorithm’s training history on Dark Room (semi-dense). While ED slightly improves and then maintains performance from the input policy, AD is able to improve it in-context until the policy is optimal or nearly optimal. + +AD achieves similar performance to the multi-stream distributed RL algorithm, but is more data-efficient as a single-stream method. + +These data-efficiency gains are also evident for distilling single-stream algorithms. In Fig 6 we show that by subsampling every $k$ -th episode (where $k = 1 0$ ) from a single stream A3C learning history, AD can still learn a more data-efficient in-context RL algorithm (for more detail, see Appendix J). Therefore, AD can be more data-efficient than both a multi and single-stream source RL algorithm. + +![](images/c652f1346777c272cd0157cbac3dddfaa6bac8bab0398dde13e052045c590f1e.jpg) +Figure 6: Single-stream Algorithm Distillation: AD trained on the learning history from an A3C agent with only one actor (i.e. single-stream). By training on subsampled learning histories (see Sec. 5), AD learns are more data-efficient in-context RL algorithm. + +While AD is more data-efficient, the source algorithm achieves slightly higher asymptotic performance (see Appendix F). However, the source algorithm produces many single-task agents with a unique set of weights $\phi _ { n }$ per task $\mathcal { M } _ { n }$ , while AD produces a single generalist agent with weights $\theta$ that are fixed across all tasks. + +Is it possible to accelerate AD by prompting it with demonstrations? Although AD can reinforcement learn without relying on demonstrations, it has the added benefit that, unlike the source algorithm, it can be conditioned on or prompted with external data. To answer the research question, we sample policies from the hold-out test-set data along different points of the source algorithm history - from a near-random policy to a near-expert policy. We then pre-fill the context for both AD and ED with this policy data, and run both methods in the environment in Dark Room (Fig. 5). While ED maintains the performance of the input policy, AD improves every policy in-context until it is near-optimal. Importantly, the more optimal the input policy the faster AD improves it until it is optimal. + +![](images/326a444f7a476152beb4d779e7e9c87e058d5d0329f877516a605551d10515aa.jpg) +Figure 7: Context size: AD in Dark Key-toDoor with different context sizes. In-context RL only emerges once the context size is large enough and across-episodic. + +What context size is required for in-context RL to emerge? We’ve hypothesized that AD requires sufficiently long (i.e. across-episodic) contexts to in-context reinforcement learn. We test this hypothesis by training several AD variants with different context sizes on the Dark Room environment. We plot the learning curves of these different variants in Fig. 7 and find that multi-episodic contexts of 2-4 episodes are necessary to learn a near-optimal in-context RL algorithm. Initial signs of in-context RL begin to emerge when the context size is roughly the length of an episode. The reason for this is likely that the context is large enough to retrain across-episodic information – e.g., at the start of a new episode, the context will be filled with transitions from most of the previous episode. + +# 6 RELATED WORK + +Offline Policy Distillation: Most closely related to our work are the recent advances in learning policies from offline environment interaction data with transformers, which we have been referring to as Policy Distillation (PD). Initial PD architectures such as Decision Transformer (DT) (Chen et al., 2021) and Trajectory Transformer (Janner et al., 2021) showed that transformers can learn single-task policies from offline data. Subsequently the Multi-Game Decision Transformer (MGDT) (Lee et al., 2022) and Gato (Reed et al., 2022) showed that PD architectures can also learn multi-task same domain and cross-domain policies, respectively. Importantly, these prior methods use contexts substantially smaller than an episode length, which is likely the reason in-context RL was not observed in these works. Instead, they rely on alternate ways to adapt to new tasks - MGDT finetunes the model parameters while Gato gets prompted with expert demonstrations to adapt to downstream tasks. AD adapts in-context without finetuning and does not rely on demonstrations. A number of recent works have explored more generalized PD architectures (Furuta et al., 2021), prompt conditioning (Xu et al., 2022), and online gradient-based RL (Zheng et al., 2022). Some PD architectures such as DT and MGDT are instantiations of Upside Down RL (UDRL) Schmidhuber (2019); Srivastava et al. (2019) where rather than learning a value function, a policy is conditioned directly on the desired return. However, AD is not explicitly doing UDRL since it is not conditioned on returns. In fact, return maximization is not specified anywhere in the AD objective but rather emerges implicitly by modeling the learning histories of an RL algorithm. + +Meta Reinforcement Learning: AD falls into the category of methods that learn to reinforcement learn, also known as meta-RL. Specifically, AD is an in-context offline meta-RL method. This general idea of learning the policy improvement process has a long history in reinforcement learning, but has been limited to meta-learning hyper-parameters4 until recently (Ishii et al., 2002). In-context deep meta-RL methods introduced by Wang et al. (2016) and Duan et al. (2016) are usually trained in the online setting by maximizing multi-episodic value functions with memory-based architectures through environment interactions (Ni et al., 2022). Meta-RL through multi-episodic value functions has been done in both on-policy (Duan et al., 2016) and off-policy (Rakelly et al., 2019; Fakoor et al., 2020) settings. Another common approach to online meta-RL includes optimization-based methods that find good network parameter initializations for meta-RL (Hochreiter et al., 2001; Finn et al., 2017; Nichol et al., 2018) and adapt by taking additional gradient steps. Like other in-context meta-RL approaches, AD is gradient-free - it adapts to downstream tasks without updating its network parameters. Recent works have proposed learning to reinforcement learn from offline datasets, or offline meta-RL, using Bayesian RL (Dorfman et al., 2021) and optimization-based meta-RL (Mitchell et al., 2021). Given the difficulty of offline meta-RL, Pong et al. (2022) proposed a hybrid offline-online strategy for meta-RL. + +In-Context Learning with Transformers: In this work, we make the distinction between in-context learning and incremental or in-context learning. In-context learning involves learning from a provided prompt or demonstration while incremental in-context learning involves learning from one’s own behavior through trial and error. While many recent works have demonstrated the former, it is much less common to see methods that exhibit the latter. Arguably, the most impressive demonstrations of in-context learning to date have been shown in the text completion setting (Radford et al., 2018; Chen et al., 2020; Brown et al., 2020) through prompt conditioning. Similar methodology was recently extended to show powerful composability properties in text-conditioned image generation (Yu et al., 2022). Recent work showed that transformers can also learn simple algorithm classes, such as linear regression, in-context in a small-scale setting (Garg et al., 2022). Like prior in-context learning methods, Garg et al. (2022) required initializing the transformer prompt with expert examples. While the aforementioned approaches were examples of in-context learning, a recent work (Chen et al., 2022) demonstrated incremental in-context learning for hyperparameter optimization by treating hyperparameter optimization as a sequential prediction problem with a score function. + +# 7 CONCLUSION + +We have demonstrated that can distill RL algorithms by modeling their learning histories causally with imitation learning and that AD can learn more data-efficient algorithms than those that generated the source data. We hope that AD inspires further investigation into in-context reinforcement learning from the research community. + +# REFERENCES + +Charles Beattie, Joel Z. 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Online decision transformer. arXiv preprint arXiv:2202.05607, 2022. + +# A DISCUSSION AND LIMITATIONS + +Discussion: While AD is more data-efficient than the source algorithm, we found that the source algorithm achieves a slightly higher asymptotic score in harder environments. This presents a tradeoff between data-efficiency and asymptotic optimality when using AD that should be taken into consideration when applying AD to a specific problem. In terms of long-term consequences, AD presents a path for converting narrow single-task RL agents into multi-task generalist ones. To date, deep RL research has mostly focused on powerful single-task agents Mnih et al. (2015); Schrittwieser et al. (2019); Hessel et al. (2021). These algorithms have produced powerful but data-inefficient agents, which has limited their applicability beyond narrow domains. AD offers a path toward training substantially more data-efficient, though perhaps less optimal, generalist agents by distilling narrow RL algorithms into sequence models like transformers. + +Limitations: While AD is a general method, in this work we’ve shown that AD generalizes to new tasks within the same domain. Showing the ability to distill cross-domain RL algorithms and generalize to new domains would be an interesting line if inquiry for future work. AD also requires storing many learning histories which could take up significant memory, though this may not be too much of an issue since the learning histories can be stored on disk rather than RAM. Perhaps the main limitation of AD is that most RL environments of interest have long episodes and modeling multiepisodic context requires more powerful long-horizon sequential models than the ones considered in this work. We believe this is a promising direction for future research. + +# B ENVIRONMENT CONSIDERATIONS + +In this work, we consider environments where zero-shot generalization is difficult, so the agent must learn through trial and error. We also want environments where overfitting to any particular task is difficult to ensure our method is general. A final practical consideration is that we consider environments wher across-episodic histories can be feasibly modeled with a causal transformer. Given these considerations, our evaluation environments need to satisfy three criteria: + +1. Supports many tasks: The environments must be multi-task to ensure that our agent and the baselines do not overfit to any single task and instead is able to in-context reinforcement learn across many tasks within a given domain. +2. Task must be hard to infer: To ensure that the downstream tasks are hard to generalize to in zero-shot, we use environments that require exploration. Namely, we require environments where either the task can only be inferred from the reward and not the observation, or tasks that are partially observable. +3. Supports multi-episodic contexts: Lastly, we impose a practical constraint - the environment episodes must be short enough such that a normal GPT-like transformer can fit multiple episodes in its context. Since this work introduces AD as a method, we wish to investigate it in the cleanest possible setting using a canonical architecture. We leave investigating AD with more complex architectures that scale to longer sequences for future work. + +Prior related works (Chen et al., 2021; Lee et al., 2022; Janner et al., 2021; Reed et al., 2022) evaluated on Atari, OpenAI gym, and as well as other environments. However, Atari and OpenAI gym don’t satisfy at least one of the above criteria. Atari and OpenAI gym episodes are often long and can contain thousands or more transitions per episode, so it’s technically challenging to populate a causal transformer’s context with across-episode histories. Indeed, the prior related works only considered within-episode context lengths. Additionally, it is often easy to infer the task from either the observation or the dense reward alone in both Atari and OpenAI gym, which reduces the need for exploration. For these reasons, we evaluate in environments that satisfy all three criteria instead. + +# C CLOSELY RELATED PRIOR METHODS + +In our main results we use Expert Distillation (ED) as a baseline. Here, we discuss how the most closely related methods differ from AD and why ED is sufficient to support the paper’s claims. + +Expert Distillation (ED): ED is most similar to Gato (Reed et al., 2022), which models expert sequences from a converged RL policy using a causal transformer. ED also trains a causal transformer to predict actions using expert policy data. There are two key differences between ED and Gato. First, unlike Gato which utilizes small (relative to an episode length) within-episode contexts, ED is trained on the same across-episode contexts as AD, so the architectures used by ED and AD are the same. The benefits of AD cannot therefore be attributed to across-episode contexts alone but also learning progress in the offline data used to train AD. Second, ED models state-action-reward sequences while Gato models only state-action sequences. The main difference between ED and AD is that AD is trained on full multi-task learning histories rather than expert policy data. + +Decision Transformer (DT) (Chen et al., 2021) and Multi-Game Decision Transformer (Lee et al., 2022): DTs learn return-conditioned policies from single-task offline data collected by an RL agent. While the training data itself (an RL agent’s replay buffer) contains learning, the context sizes used in DT are too small to capture any learning progress or identify the task using across-episode information. For instance, the Atari experiments use a context of length $3 0 - 5 0$ tokens, or $1 0 - 1 7$ transitions. Atari games can have hundreds or thousands of transitions in a single episode, which means these contexts capture mostly within-episode information. Additionally, very little learning progress happens in the underlying replay buffer data within that many transitions. + +Another difference between DT and AD / ED is that DT learns a return-conditioned model whereas AD / ED are both reward-conditioned. In our setting return-conditioning alone cannot yield an optimal policy since the agent does not know the task until after it explores the environment and can identify it using across-episode contexts. Since (i) DT uses small within-episodic contexts and (ii) return-conditioning would not help in the environments considered, this baseline is similar to ED with a small within-episode context which is strictly weaker than the long across-episode context variant of ED we consider. + +Trajectory Transformer (TTO) (Janner et al., 2021): Like AD, TTO also models state-action-reward tokens but in addition to predicting actions it also learns a world model by predicting states and rewards. To maximize return, TTO then uses beam search to select high-reward actions. However, in our setting, TTO will run into the same problem as DT. To model rewards accurately it will need longer across-episodic contexts since one environment supports many tasks. Similar to DT, MGDT, and Gato, TTO uses smaller within-episode contexts. For this reason, TTO will fare no better than DT, MGDT, or ED in the settings we consider. We also note that in contrast to TTO, AD is model-free. In AD, actions are sampled from the transformer history-conditioned predictions and return maximization emerges from modeling the learning histories of an RL algorithm. + +To summarize, AD differs from prior methods mainly because its context is across-episodic and hence large enough to capture learning progress and task information. AD could further be augmented by learning world models like TTO or conditioning on returns like DT, but these investigations would be well suited for future work since they are tangential to the main research question addressed in this work – whether in-context RL can emerge by imitating the learning histories of an RL algorithm with long across-episodic contexts. + +AD is also closely related to prior work in in-context meta-RL. While both AD and in-context meta-RL model across-episodic histories with memory-based architectures, prior in-context meta-RL algorithms, such as RL2 (Duan et al., 2016) are trained online and rely on learning multi-episodic value functions with TD learning while AD is trained offline and uses a supervised imitation learning objective. + +# D EXPERT DISTILLATION MAIN RESULTS + +We elaborate further on the main results in Fig. 4 and provide intuition regarding the behaviors of the ED baseline. In Dark Room, Dark Room (Hard), and Watermaze, ED performance is either flat or it degrades. The reason for this is that ED only saw expert trajectories during training, but during evaluation it needs to first explore (i.e. perform non-expert behavior) to identify the task. This required behavior is out-of-distribution for ED and for this reason it does not reinforcement learning in-context. In Dark Key-to-Door the agent is reset randomly at the beginning of each episode, whereas in all of the environments the agent’s starting position is fixed. Due to random resets, the ED agent is sometimes reset by the first goal in Dark Key-to-Door which allows it to occasionally identify the first goal of the task, which is why it shows slight improvement. + +# E MODEL SIZE + +We investigate how transformer capacity affects performance in Fig. 8. While in-context RL emerges across all model sizes investigated, we find that increasing the model depth, the model width in terms of embedding dimension, and (to a lesser extent) the number of attention heads improves performance on Dark Key-to-Door. + +![](images/7268b332c41895dedc19c08ef9b6099f4da74ea1f1754cc2dd5c84977a7dc825.jpg) +Figure 8: Model size investigations: We investigate how increasing model capacity affects AD. While in-context RL with AD emerges regardless of the model capacity, increasing the model depth and width helps improve AD until it achieves near-optimal performance. + +![](images/5225296573b7d46d8a1d9f73f0c47aa7acab27725bdc5f8cd9d76937d0d127b6.jpg) +Figure 9: Asymptotic performance of the A3C (Mnih et al., 2016) and a $\mathbf { Q - } \lambda$ variant of the DQN (Mnih et al., 2013) RL algorithms used to produce learning histories for the Dark and Watermaze environments. These curves show the learning histories AD is trained on. The source algorithms plotted in Fig. 4 are the same as in these plots. + +# G LABEL SMOOTHING ABLATION + +For the harder exploration task of Dark Room (Hard), we found that adding label smoothing regularization (Szegedy et al., 2016; Muller et al. ¨ , 2019) improved the in-context learning ability of AD . In Figure 10 we ablate the benefit of using label smoothing for 3 different $\alpha$ values as well as with it turned off. Each curve in the figure denotes average performance over 5 training seeds. We can see that adding label smoothing up to a point improves the in-context learning ability of Algorithm Distillation, with performance continually increasing with the number of evaluation episodes. + +![](images/e07003b625836d03f8abae3f9b6de0dbf38604250cecc84a2937ab5a8c642300.jpg) +Figure 10: AD trained with different amounts of label smoothing on Dark Room (Hard). + +# H $\mathtt { R L } ^ { 2 }$ NETWORK ARCHITECTURE: TRANSFORMER VS LSTM + +We compared using a transformer as the architecture for $\mathrm { { R L ^ { 2 } } }$ instead of an LSTM. In Figure 11, we ran both transformer and LSTM $\mathrm { { R L ^ { 2 } } }$ agents over the Dark Room environment. The curves shown are the best from a sweep over learning rate and unroll length hyperparameters. The transformer architecture is 4-layers with a model size of 256 and pre-norm layer normalization placement. While both agents reached a similar level of final performance, all $\mathtt { R L } ^ { \dot { 2 } }$ transformer models trained tended to be more unstable with the average return not as consistent as with an LSTM architecture. Given the poor performance of the transformer-based $\mathrm { { R L ^ { 2 } } }$ on the simpler Dark Room setting, our other experimental settings used the LSTM. + +![](images/d84a5680152d9b0d5a45f824e697e85bb345213b6f295d3072548835b0a69b86.jpg) +LSTM v.s. Transformer +Figure 11: Comparison of LSTM and Transformer architecture for $\mathtt { R L } ^ { 2 }$ agent on Dark Room. Each curve is averaged over 5 training seeds with the shaded area representing the standard error. + +# I NUMBER OF TRAINING TASKS IN SOURCE DATA + +![](images/3f6801f52bcd71f0fcf1231b4856e8cbd97a9ff344198d24cc0710b9d3353f73.jpg) +Figure 12: Algorithm distillation trained on different numbers of training tasks on Dark Key-to-Door evaluated on a fixed set of test tasks for 300 episodes of evaluation. + +One interesting question is how many tasks AD needs to be trained on to learn an algorithm that generalizes to held out tasks. We trained AD on different numbers of Dark Key-to-Door training tasks and evaluated the resulting models on the same set of test tasks. Figure 12 shows the incontext learning plots for the resulting AD models on the set of test tasks. As a reminder, there are $8 1 ^ { 2 } = 6 5 6 1$ unique Dark Key-to-Door tasks. Models trained on 1, 9 or 18 training tasks did not show any in-context learning on test tasks. While models trained on 37, 75 and 151 tasks did not achieve good performance overall, they did exhibit some in-context learning over the course of 300 episodes. The best models were trained on 1212 and 2424 tasks which corresponds to roughly $1 8 \%$ and $3 7 \%$ of the total number of tasks in the Dark Key-to-Door domain. + +# J SINGLE-STREAM ALGORITHM DISTILLATION + +We provide more details around the experimental setup for the single-stream result shown in Fig. 6. We showed in Fig. 4 that when AD is trained on data from a subset of the actors of a distributed source RL algorithm, the resulting model is more data efficient than the source algorithm. Here we confirm that AD can produce a faster algorithm than the one it was trained on in the single-stream setting. For this experiment we trained A3C on 2048 Dark Key-to-Door tasks for 2000 episodes each. We then trained AD on the resulting data while subsampling the learning histories by a factor of 10. More concretely, we took every 10th episode from each of the learning history, which resulted in a 200 episode compressed learning trajectory for each task. Figure 6 compares the resulting AD model evaluated on a set of test tasks to the performance of the source algorithm on these tasks. The model learned by AD learns much faster than the source algorithm confirming that Algorithm Distillation can turn a slow gradient-based algorithm into a much more data efficient in-context learning algorithm. + +# K RANDOM MASK + +![](images/1038f6e9da6d542b87e80cfa6740b77180b4083883611e385d6a893aeadde962.jpg) +Figure 13: Downstream performance of Algorithm Distillation with different values of random masking during training in $9 \mathrm { { x 9 1 } }$ goal gridworld. + +During training, input tokens were randomly masked to avoid overfitting to training data. This plot shows the downstreams results on a $9 \mathrm { x } 9$ Dark Key-to-Door domain with different values of this random masking. Values of $0 . 3 - 0 . 5$ perform the best with the value of 0.3 chosen for all experiments. + +# L AD NETWORK ARCHITECTURE: TRANSFORMER VS LSTM + +Here we consider the importance of the Transformer architecture to the success of algorithm distillation (AD) by comparing to AD based off of an LSTM (Hochreiter & Schmidhuber, 1997). Specifically, the LSTM receives the concatenated embeddings of $\left( o _ { i } , a _ { i } , r _ { i } \right)$ triplets up to the most recent time step $t - 1$ . The output of the LSTM is then concatenated with the current observation $o _ { t }$ embedding and both are then fed through a multi-layer perceptron (MLP) policy torso to produce a distribution over the present action $a _ { t }$ . The LSTM hidden size (512), MLP depth (2), and MLP width (256) were swept and tuned by grid search based on downstream reward attainment. + +Comparing Transformer AD and LSTM AD on the Dark Key-to-Door task (Figure 14), we find that both agents are capable of in-context learning, demonstrating that the success of AD is not tied to the underlying network architecture. However, we also find that the Transformer variant consistently outperforms the LSTM variant, which is why all other experiments in this paper employ the Transformer variant. This finding is consistent with the recent wider success of Transformer-based architectures over recurrent neural network (RNN)-based architectures in sequence prediction tasks. + +# M DISTILLING ACTIONS VS PROBABILITIES + +In the rest of the paper, we use (one-hot) actions taken by the source policy as the prediction target for AD. Here, we compare that choice to predicting the source policy probabilities from which that action was sampled. In other words, is it better to distill actions or probabilities? + +Figure 15 compares all combinations of: 1) distilling actions vs probabilities, and 2) conditioning those predictions on past actions, probabilities, or both. The plot labels represent various input/output combinations, e.g. $\mathtt { S a r - } > \mathtt { a }$ indicates observing states, actions, and rewards while predicting actions (i.e. the main variant of AD in the rest of the paper), sar $- > \mathrm { p }$ represents using the same observations but instead predicting probabilities, and $\tt S a p r - > a$ represents observing states, actions, probabilities, and rewards while predicting actions, etc. + +Two conclusions stand out from this plot. First, the original action distillation variant of AD $( s \mathsf { a r } - \mathsf { > a } )$ performs best, followed closely by probability distillation $( \mathsf { s a r - > p } )$ . In the paragraphs and experiments below, we explore why action distillation outperforms probability distillation. Second, all variants that condition on past probabilities catastrophically fail. We speculate on two somewhat contradictory reasons why this might occur: 1) The AD prediction task involves a combination of inferring the current policy, as well as predicting when policy updates will occur. Since observing policy probabilities provides more information about the current policy than actions do, it is possible that this leads to AD fully focusing on inferring the current policy and ignoring the prediction of policy updates. In other words, observing only actions taken may act as a useful information bottleneck. 2) On the other hand, observing probabilities may inadvertently leak information about when policy updates occur. If the transformer context includes multiple visits to the same state, AD could learn to compare the policy probabilities to infer whether a policy update occurred in between the two visits. If AD learned during training to rely on this information, then during autoregressive evaluation, it may be “waiting“ for a policy update than never occurs. If this is the source of the issue, then a potential solution would be retraining with explicit policy update tokens and including them between evaluation episodes, however we leave this investigation to future work. + +![](images/454e8467c64103db58787ec5b2c5c859eefc5a040f070dda6d155b0f1e59d8e6.jpg) +Figure 14: Comparison between algorithm distillation with a Transformer and LSTM architecture on Dark Key-to-Door. Mean $\pm \nobreakspace 1 \nobreakspace$ standard deviation over 5 training seeds and 20 evaluation seeds. 300 episodes corresponds to $1 5 \mathrm { k }$ environment steps. + +![](images/a807c9a242becc876d7e1138b0bdc3096c70672568caf1845fcca6c88bec2352.jpg) +Figure 15: Comparison between algorithm distillation with various input/output combinations on Dark Key-to-Door. Mean $\pm \nobreakspace 1 \nobreakspace$ standard deviation over 5 training seeds and 20 evaluation seeds. 1000 episodes corresponds to $5 0 \mathrm { k }$ environment steps. + +Now we return to the question of why action distillation $\left( s \mathsf { a r } - \mathsf { > a } \right)$ outperforms probability distillation $( s \mathsf { a r } \mathrm { - } \mathsf { > p } )$ . Intuitively, distilling one-hot actions will result in a more deterministic policy than distilling probabilities when training on finite data. Indeed, the red curves in the top two plots of Figure 16 show that probability distillation (top left) converges to a higher entropy ${ \sim } 1 . 2$ bits) policy than does action distillation (top right, ${ \sim } 1$ bit). Noting this discrepancy, we speculated that encouraging probability distillation towards a more deterministic policy might lead to increased performance. To do so, we experimented with an entropy penalty regularizer added to the AD loss from equation 6, leading to the modified objective: + +$$ +\tilde { \mathcal { L } } ( \theta , \alpha ) : = \sum _ { n = 1 } ^ { N } \sum _ { t = 1 } ^ { T - 1 } - \log P _ { \theta } ( A = a _ { t } ^ { ( n ) } | h _ { t - 1 } ^ { ( n ) } , o _ { t } ^ { ( n ) } ) + \alpha H \Big [ P _ { \theta } ( A | h _ { t - 1 } ^ { ( n ) } , o _ { t } ^ { ( n ) } ) \Big ] , +$$ + +where $H [ \cdot ]$ is the Shannon entropy in bits, and $\alpha$ is a regularization weight. The top two panels of Figure 16 show the effect of increasing $\alpha$ on decreasing policy entropy for action distillation (top right) and probability distillation (top left), throughout training. The bottom two panels show the corresponding changes in evaluation returns. Notably, entropy penalization indeed improves performance for probability distillation (bottom left), with the best performing regularization value $\alpha = 0 . 1$ ) achieving a similar return to the original unregularized action distillation (bottom right, red). Interestingly, this is the amount of regularization that leads probability distillation to have the most similar entropy to the unregularized action distillation variant $_ { \sim 1 }$ bit) as well. Further entropy penalization for action distillation, on the other hand, does not lead to increased performance (bottom right). Together, these results suggest that action distillation naturally leads to the “optimal“ amount of entropy regularization on its own, at least in the environments we study. + +![](images/1193b4fb8b446f1e1eb9a0060b68dfb52c35743a1269cf2b5dabdd292afbbd8e.jpg) +Figure 16: Policy entropy during training and reward during evaluation for the action distillation $( s \mathsf { a r } - \mathsf { > a } )$ and probability distillation $( \mathsf { s a r - > p } )$ variants of AD on Dark Key-to-Door. Colors indicate the strength of the weight on the entropy penalty regularizer from equation 7. Mean $\pm \nobreakspace 1 \nobreakspace$ standard deviation over 5 training seeds and 20 evaluation seeds. 1000 episodes corresponds to $5 0 \mathrm { k }$ environment steps. + +In addition to closing gap between probability and action distillation with an entropy penalty, we also explored retuning various hyperparameters for $\mathtt { S a r - > p }$ rather than reusing those tuned for $\mathtt { S a r - } > \mathtt { a }$ . We found that many hyperparameters, such as the dropout rate, attention dropout rate, and sequence mask probability, had similar optimal values in the two cases, and so retuning them did not help. However, one hyperparameter that did have an effect was the size of the transformer context window (Figure 17). While action distillation performance increased only up to a context window size of 200 steps (4 episodes) and then plateaued (Figure 17, right), probability distillation performance continued to increase up to a context window size of 300 steps (6 episodes) before plateauing (Figure 17, left). Thus, while a comparison at our default context window size of 200 steps favored action distillation, for larger context window sizes, action and probability distillation performed similarly. We leave any further explanation of why probability distillation requires larger context windows than action distillation to future work. + +![](images/5dfce0c0db48409f890ffe00174597da457e40c2ba659bd24f1e455836d5c726.jpg) +Figure 17: Context window size dependence for the action distillation $( s \mathsf { a r } - > \mathsf { a } )$ and probability distillation $( \mathsf { s a r - > p } )$ variants of AD on Dark Key-to-Door. Mean $\pm \nobreakspace 1 \nobreakspace$ standard deviation over 5 training seeds and 20 evaluation seeds. 1000 episodes corresponds to $5 0 \mathrm { k }$ environment steps. + +N ALGORITHM DISTILLATION HYPERPARAMETERS + +
HyperparameterDark RoomDark Room (Hard)Dark Key-to-DoorWatermaze
Embedding Dim.64
Number of Layers
Number of Heads4
Feedforward Dim.2048
Position EncodingsAbsolute
LayerNorm PlacementPost Norm
Dropout Rate0.1
Context Window600 tokens (200 timesteps)
Attention Dropout Rate00.50.5
Sequence Mask Prob0.5 0.30.50.30.3
Label Smoothing α00.200
+ +Table 1: Algorithm Distillation Architecture Hyperparameters. + +
HyperparameterValue
Batch Size128
OptimizerAdam
β0.9
0.99
Gradient Clip Norm Threshold1
Learning Rate ScheduleCosine Decay
Initial Value2e-6
Peak Value3e-4
+ +Table 2: Algorithm Distillation Optimization Hyperparameters. +Table 3: Watermaze Image Encoder Hyperparameters. + +
LayerHyperparameterValue
Conv Block
ConvChannel128
Kernel5
Stride2
BatchNormDecay Rate0.999
Activationeps1e-5
-ReLU
Max PoolingKernel2
Stride2
DropoutRate0.2
Network
Conv Blocks=3
FinalLinearLayerUnits256
+ +# O SOURCE RL ALGORITHM HYPERPARAMETERS + +# O.1 DARK ENVIRONMENTS + +Table 4: Source A3C Algorithm Hyperparameters for Dark Environments. + +
Hyperparameter Value
Batch Size (Num.Actors)100
入0.95
Agent Discount0.99
Entropy Bonus Weight0.01
MLPLayers3
MLPHidden Dim128
OptimizerAdam
β0.9
β20.999
E1e-6
Learning Rate1e-4
+ +# O.2 DMLAB WATERMAZE + +Table 5: Source $\mathrm { D Q N } ( \mathrm { Q } - \lambda )$ Algorithm Hyperparameters for Watermaze. + +
HyperparameterValue
Batch Size8
Rollout Length40
Rollout Overlap31
Number of Actors16
Reply Buffer Capacity1e5
Offline Data Fraction0.7
入0.75
E0.01
Agent Discount0.9
Target Update Period50
ResNet Channels[32, 64, 64]
ResNetKernels[3,3,3]
ResNet Strides[1, 1, 1]
Pool Kernels[3,3,3]
Pool Strides[2,2,2]
OptimizerAdam
β0.9
阳0.999
E1e-6
Gradient Clip Norm Threshold10
Learning Rate1e-4
+ +# P RL2 HYPERPARAMETERS + +Table 6: RL2 Hyperparameters used in “Dark” Environments. + +
HyperparameterValue
RL AlgorithmA3C
Learning Rate3e-4
Batch Size256
Unroll Length20
LSTM Hidden Dim. LSTMNumber ofLayers256
Episodes Per Trial2 10
+ +
HyperparameterValue
RL AlgorithmDQN(Q-入)
Learning Rate1e-4
Batch Size96
Unroll Length40
LSTM Hidden Dim.256
LSTMNumber of Layers1
Episodes Per Trial30
+ +Table 7: RL2 Hyperparameters used in the Watermaze Environment. + +![](images/ea1fc762f193fd44d789c64e6914ec2d901af77612e21bb3dcbce6fbfe7da485.jpg) +0 1700Figure 18: Attention maps for AD from five separate seeds. White and gray colors correspond to low and high attention. Red and blue colors indicate that those transitions correspond to an episode restart and a positive reward token, respectively. The left column plots attention for an AD transformer after 200 time-steps of evaluation (when the context is initially filled). The right column plots attention after 1900 steps (38 episodes) of evaluation. Each episode has a length of 50 steps. From these patterns, it is evident that AD attends to tokens across several episodes to predict its next action. \ No newline at end of file diff --git a/parse/dev/hy0a5MMPUv/hy0a5MMPUv_content_list.json b/parse/dev/hy0a5MMPUv/hy0a5MMPUv_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..334994107bab8681a27890220da1176f345a5063 --- /dev/null +++ b/parse/dev/hy0a5MMPUv/hy0a5MMPUv_content_list.json @@ -0,0 +1,2346 @@ +[ + { + "type": "text", + "text": "IN-CONTEXT REINFORCEMENT LEARNING WITH ALGORITHM DISTILLATION ", + "text_level": 1, + "bbox": [ + 173, + 98, + 684, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Michael Laskin∗†, Luyu Wang∗, Junhyuk Oh, Emilio Parisotto, Stephen Spencer, Richie Steigerwald, DJ Strouse, Steven Hansen, Angelos Filos, Ethan Brooks, Maxime Gazeau, Himanshu Sahni, Satinder Singh, Volodymyr Mnih† DeepMind ", + "bbox": [ + 184, + 169, + 748, + 227 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 263, + 544, + 279 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We propose Algorithm Distillation (AD), a method for distilling reinforcement learning (RL) algorithms into neural networks by modeling their training histories with a causal sequence model. Algorithm Distillation treats learning to reinforcement learn as an across-episode sequential prediction problem. A dataset of learning histories is generated by a source RL algorithm, and then a causal transformer is trained by autoregressively predicting actions given their preceding learning histories as context. Unlike sequential policy prediction architectures that distill post-learning or expert sequences, AD is able to improve its policy entirely in-context without updating its network parameters. We demonstrate that AD can reinforcement learn in-context in a variety of environments with sparse rewards, combinatorial task structure, and pixel-based observations, and find that AD learns a more data-efficient RL algorithm than the one that generated the source data. ", + "bbox": [ + 233, + 295, + 766, + 462 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/679c16b167e631679ba20610706e8543cd9d762d7dab80268efc9673f356c188.jpg", + "image_caption": [ + "Figure 1: Algorithm Distillation (AD) has two steps – (i) a dataset of learning histories is collected from individual single-task RL algorithms solving different tasks; (ii) a causal transformer predicts actions from these histories using across-episodic contexts. Since the RL policy improves throughout the learning histories, by predicting actions accurately AD learns to output an improved policy relative to the one seen in its context. AD models state-action-reward tokens, and does not condition on returns. " + ], + "image_footnote": [], + "bbox": [ + 230, + 493, + 766, + 670 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 785, + 336, + 800 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Transformers have emerged as powerful neural network architectures for sequence modeling (Vaswani et al., 2017). A striking property of pre-trained transformers is their ability to adapt to downstream tasks through prompt conditioning or in-context learning. After pre-training on large offline datasets, large transformers have been shown to generalize to downstream tasks in text completion (Brown et al., 2020), language understanding (Devlin et al., 2018), and image generation (Yu et al., 2022). ", + "bbox": [ + 174, + 815, + 825, + 886 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Recent work demonstrated that transformers can also learn policies from offline data by treating offline Reinforcement Learning (RL) as a sequential prediction problem. While Chen et al. (2021) showed that transformers can learn single-task policies from offline RL data via imitation learning, subsequent work showed that transformers can also extract multi-task policies in both same-domain (Lee et al., 2022) and cross-domain settings (Reed et al., 2022). These works suggest a promising paradigm for extracting generalist multi-task policies – first collect a large and diverse dataset of environment interactions, then extract a policy from the data via sequential modeling. We refer to the family of approaches that learns policies from offline RL data via imitation learning as Offline Policy Distillation, or simply Policy Distillation1 (PD). ", + "bbox": [ + 174, + 103, + 825, + 229 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Despite its simplicity and scalability, a substantial drawback of PD is that the resulting policy does not improve incrementally from additional interaction with the environment. For instance, the MultiGame Decision Transformer (MGDT, Lee et al., 2022) learns a return-conditioned policy that plays many Atari games while Gato (Reed et al., 2022) learns a policy that solves tasks across diverse environments by inferring tasks through context, but neither method can improve its policy in-context through trial and error. MGDT adapts the transformer to new tasks by finetuning the model weights while Gato requires prompting with an expert demonstration to adapt to a new task. In short, Policy Distillation methods learn policies but not Reinforcement Learning algorithms. ", + "bbox": [ + 174, + 236, + 825, + 347 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We hypothesize that the reason Policy Distillation does not improve through trial and error is that it trains on data that does not show learning progress. Current methods either learn policies from data that contains no learning (e.g. by distilling fixed expert policies) or data with learning (e.g. the replay buffer of an RL agent) but with a context size that is too small to capture policy improvement. ", + "bbox": [ + 174, + 354, + 825, + 410 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Our key observation is that the sequential nature of learning within RL algorithm training could, in principle, make it possible to model the process of reinforcement learning itself as a causal sequence prediction problem. Specifically, if a transformer’s context is long enough to include policy improvement due to learning updates it should be able to represent not only a fixed policy but a policy improvement operator by attending to states, actions and rewards from previous episodes. This opens the possibility that any RL algorithm can be distilled into a sufficiently powerful sequence model such as a transformer via imitation learning, converting it into an in-context RL algorithm. By in-context RL we mean that the transformer should improve its policy through trial and error within the environment by attending to its context, without updating its parameters. ", + "bbox": [ + 174, + 416, + 825, + 542 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We present Algorithm Distillation (AD), a method that learns an in-context policy improvement operator by optimizing a causal sequence prediction loss on the learning histories of an RL algorithm. AD has two components. First, a large multi-task dataset is generated by saving the training histories of an RL algorithm on many individual tasks. Next, a transformer models actions causally using the preceding learning history as its context. Since the policy improves throughout the course of training of the source RL algorithm, AD is forced to learn the improvement operator in order to accurately model the actions at any given point in the training history. Crucially, the transformer context size must be sufficiently large (i.e. across-episodic) to capture improvement in the training data. The full method is shown in Fig. 1. ", + "bbox": [ + 174, + 549, + 825, + 674 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We show that by imitating gradient-based RL algorithms using a causal transformer with sufficiently large contexts, AD can reinforcement learn new tasks entirely in-context. We evaluate AD across a number of partially observed environments that require exploration, including the pixel-based Watermaze (Morris, 1981) from DMLab (Beattie et al., 2016). We show that AD is capable of in-context exploration, temporal credit assignment, and generalization. We also show that AD learns a more data-efficient algorithm than the one that generated the source data for transformer training. To the best of our knowledge, AD is the first method to demonstrate in-context reinforcement learning via sequential modeling of offline data with an imitation loss. ", + "bbox": [ + 174, + 681, + 825, + 792 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 BACKGROUND ", + "text_level": 1, + "bbox": [ + 176, + 813, + 326, + 829 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Partially Observable Markov Decision Processes: A Markov Decision Process (MDP) consists of states $s \\in S$ , actions $a \\in { \\mathcal { A } }$ , rewards $r \\in \\mathcal { R }$ , a discount factor $\\gamma$ , and a transition probability function $p ( s _ { t + 1 } | s _ { t } , a _ { t } )$ , where $t$ is an integer denoting the timestep and $( \\cal { S } , \\cal { A } )$ are state and action spaces. In environments described by an MDP, at each timestep $t$ the agent observes the state $s _ { t }$ , selects an action $a _ { t } \\sim \\pi ( \\cdot | s _ { t } )$ from its policy, and then observes the next state $s _ { t + 1 } \\sim p \\big ( \\cdot | s _ { t } , a _ { t } \\big )$ sampled from the transition dynamics of the environment. In this work, we operate in the Partially Observable Markov Decision Process (POMDP) setting where instead of states $s \\in S$ the agent receives observations $o \\in \\mathcal { O }$ that only have partial information about the true state of the environment. Full state information may be incomplete due to missing information about the goal in the environment, which the agent must instead infer through rewards with memory, or because the observations are pixel-based, or both. ", + "bbox": [ + 174, + 844, + 825, + 887 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 202 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Online and Offline Reinforcement Learning: Reinforcement Learning algorithms aim to maximize the return, defined as the cumulative sum of rewards $\\textstyle \\sum _ { t } \\gamma ^ { t } r _ { t }$ , throughout an agent’s lifetime or episode of training. RL algorithms broadly fall into two categories: on-policy algorithms (Williams, 1992) where the agent directly maximizes a Monte-Carlo estimate of the total returns or off-policy (Mnih et al., 2013; 2015) where an agent learns and maximizes a value function that approximates the total future return. Most RL algorithms maximize returns through trial-and-error by directly interacting with the environment. However, offline RL (Levine et al., 2020) has recently emerged as an alternate and often complementary paradigm for RL where an agent aims to extract return maximizing policies from offline data gathered by another agent. The offline dataset consists of $( s , a , r )$ tuples which are often used to train an off-policy agent, though other algorithms for extracting return maximizing policies from offline data are also possible. ", + "bbox": [ + 173, + 208, + 825, + 361 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Self-Attention and Transformers The self-attention (Vaswani et al., 2017) operation begins by projecting input data $X$ with three separate matrices onto $D$ -dimensional vectors called queries $Q$ , keys $K$ , and values $V$ . These vectors are then passed through the attention function: ", + "bbox": [ + 174, + 367, + 825, + 410 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/6decb3eacadd4531f7ee314400c746d565a58ae5297223df98af696270cc6c75.jpg", + "text": "$$\n\\mathrm { A t t e n t i o n } ( Q , K , V ) = \\mathrm { s o f t m a x } ( Q K ^ { T } / \\sqrt { D } ) V .\n$$", + "text_format": "latex", + "bbox": [ + 343, + 411, + 651, + 431 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The $Q K ^ { T }$ term computes an inner product between two projections of the input data $X$ . The inner product is then normalized and projected back to a $D$ -dimensional vector with the scaling term $V$ . Transformers (Vaswani et al., 2017; Devlin et al., 2018; Brown et al., 2020) utilize self-attention as a core part of the architecture to process sequential data such as text sequences. Transformers are usually pre-trained with a self-supervised objective that predicts tokens within the sequential data. Common prediction tasks include predicting randomly masked out tokens (Devlin et al., 2018) or applying a causal mask and predicting the next token (Radford et al., 2018). ", + "bbox": [ + 174, + 433, + 825, + 531 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Offline Policy Distillation: We refer to the family of methods that treat offline Reinforcement Learning as a sequential prediction problem as Offline Policy Distillation, or Policy Distillation (PD) for brevity. Rather than learning a value function from offline data, PD extracts policies by predicting actions in the offline data (i.e. behavior cloning) with a sequence model and either return conditioning (Chen et al., 2021; Lee et al., 2022) or filtering out suboptimal data (Reed et al., 2022). Initially proposed to learn single-task policies (Chen et al., 2021; Janner et al., 2021), PD was recently extended to learn multi-task policies from diverse offline data (Lee et al., 2022; Reed et al., 2022). ", + "bbox": [ + 174, + 536, + 825, + 635 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In-Context Learning: In-context learning refers to the ability to infer tasks from context. For example, large language models like GPT-3 (Brown et al., 2020) or Gopher (Rae et al., 2021) can be directed at solving tasks such as text completion, code generation, and text summarization by specifying the task through language as a prompt. This ability to infer the task from prompt is often called in-context learning. We use the terms ‘in-weights learning’ and ‘in-context learning’ from prior work on sequence models (Brown et al., 2020; Chan et al., 2022) to distinguish between gradient-based learning with parameter updates and gradient-free learning from context, respectively. ", + "bbox": [ + 174, + 641, + 825, + 739 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 METHOD ", + "text_level": 1, + "bbox": [ + 176, + 758, + 282, + 775 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Over the course of its lifetime a capable reinforcement learning (RL) agent will exhibit complex behaviours, such as exploration, temporal credit assignment, and planning. Our key insight is that an agent’s actions, regardless of the environment it inhabits, its internal structure, and implementation, can be viewed as a function of its past experience, which we refer to as its history. Formally, we write: ", + "bbox": [ + 174, + 789, + 825, + 844 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/d67bf2227f3dfb62b30ee973ad30f1f001f2ba6e3c101680b45db7d374005255.jpg", + "text": "$$\n\\mathcal { H } \\ni h _ { t } : = \\left( o _ { 0 } , a _ { 0 } , r _ { 0 } , \\ldots , o _ { t - 1 } , a _ { t - 1 } , r _ { t - 1 } , o _ { t } , a _ { t } , r _ { t } \\right) = \\left( o _ { \\leq t } , r _ { \\leq t } , a _ { \\leq t } \\right)\n$$", + "text_format": "latex", + "bbox": [ + 261, + 847, + 736, + 864 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "and we refer to a $l o n g ^ { 2 }$ history-conditioned policy as an algorithm: ", + "bbox": [ + 174, + 868, + 612, + 883 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/8adc0b98f910b4207cf6e035828d2f8c4b84f27f04f82728bd95947853b8e141.jpg", + "text": "$$\nP : \\mathcal { H } \\cup \\mathcal { O } \\Delta ( \\mathcal { A } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 426, + 885, + 570, + 901 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $\\Delta ( \\mathcal { A } )$ denotes the space of probability distributions over the action space $\\mathcal { A }$ . Eqn. (3) suggests that, similar to a policy, an algorithm can be unrolled in an environment to generate sequences of observations, rewards, and actions. For brevity, we denote the algorithm as $P$ and environment (i.e. task) as $\\mathcal { M }$ , such that the history of learning for any given task $\\mathcal { M }$ is generated by the algorithm $P _ { \\mathcal { M } }$ . ", + "bbox": [ + 173, + 103, + 825, + 160 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/6fc3d9f41e0e65739b9ca46cd3d5346ed52f85b5136590eb2509b9a575814e8f.jpg", + "text": "$$\n( O _ { 0 } , A _ { 0 } , R _ { 0 } , \\ldots , O _ { T } , A _ { T } , R _ { T } ) \\sim P _ { \\mathcal { M } } .\n$$", + "text_format": "latex", + "bbox": [ + 367, + 164, + 629, + 180 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Here, we’re denoting random variables with uppercase Latin letters, e.g. $O , A , R$ , and their values with lowercase Latin letters, $e . g . \\ o , a , r$ . By viewing algorithms as long history-conditioned policies, we hypothesize that any algorithm that generated a set of learning histories can be distilled into a neural network by performing behavioral cloning over actions. Next, we present a method that, provided agents’ lifetimes, learns a sequence model with behavioral cloning to map long histories to distributions over actions. ", + "bbox": [ + 174, + 183, + 825, + 267 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.1 ALGORITHM DISTILLATION ", + "text_level": 1, + "bbox": [ + 176, + 284, + 406, + 297 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Suppose the agents’ lifetimes, which we also call learning histories, are generated by the source algorithm $P ^ { \\mathrm { s o u r c e } }$ for many individual tasks $\\{ \\mathcal { M } _ { n } \\} _ { n = 1 } ^ { N }$ , producing the dataset $\\mathcal { D }$ : ", + "bbox": [ + 173, + 309, + 821, + 338 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/d67f7b613e39859f26255bd59838a6e86490dd45678627aa2fd5cbaf14055ebb.jpg", + "text": "$$\n\\begin{array} { r } { \\mathcal { D } : = \\left\\{ \\left( o _ { 0 } ^ { ( n ) } , a _ { 0 } ^ { ( n ) } , r _ { 0 } ^ { ( n ) } , \\ldots , o _ { T } ^ { ( n ) } , a _ { T } ^ { ( n ) } , r _ { T } ^ { ( n ) } \\right) \\sim P _ { \\mathcal { M } _ { n } } ^ { \\mathrm { s o u r c e } } \\right\\} _ { n = 1 } ^ { N } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 294, + 342, + 702, + 372 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Then we distill the source algorithm’s behaviour into a sequence model that maps long histories to probabilities over actions with a negative log likelihood (NLL) loss and refer to this process as algorithm distillation (AD). In this work, we consider neural network models $P _ { \\theta }$ with parameters $\\theta$ which we train by minimizing the following loss function: ", + "bbox": [ + 173, + 375, + 825, + 431 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/7ec3a669edb22323e8b65e2793403150bd27d6e62ff202f3172383124c4f1c03.jpg", + "text": "$$\n\\mathcal { L } ( \\theta ) : = - \\sum _ { n = 1 } ^ { N } \\sum _ { t = 1 } ^ { T - 1 } \\log P _ { \\theta } ( A = a _ { t } ^ { ( n ) } | h _ { t - 1 } ^ { ( n ) } , o _ { t } ^ { ( n ) } ) .\n$$", + "text_format": "latex", + "bbox": [ + 336, + 434, + 661, + 478 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Intuitively, a sequence model with fixed parameters that is trained with AD should amortise the source RL algorithm $P ^ { \\mathrm { s o u r c e } }$ and by doing so exhibit similarly complex behaviours, such as exploration and temporal credit assignment. Since the RL policy improves throughout the learning history of the source algorithm, accurate action prediction requires the sequence model to not only infer the current policy from the preceding context but also infer the improved policy, therefore distilling the policy improvement operator. ", + "bbox": [ + 173, + 479, + 825, + 564 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Algorithm 1 Algorithm Distillation ", + "text_level": 1, + "bbox": [ + 176, + 577, + 475, + 592 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Require: Train $\\{ \\mathcal { M } ^ { \\mathrm { t r a i n } } \\}$ and test $\\{ \\mathcal { M } ^ { \\mathrm { t e s t } } \\}$ tasks, observations $o \\in \\mathcal { O }$ , actions $a \\in { \\mathcal { A } }$ , and rewards $r \\in \\mathcal { R }$ . Require: Network parameters $\\phi _ { i }$ for $i = 1 , \\ldots , N$ source RL algorithms. \nRequire: Network parameters $\\theta$ for a causal transformer $P _ { \\theta }$ that predicts actions. \nRequire: An empty buffer to store data $\\mathcal { D }$ . \n1: for $i = 1 \\dots N$ do ▷ Part 1: Dataset Generation 2: Sample a task $\\mathcal { M } _ { i } ^ { \\mathrm { t r a i n } }$ randomly from the train task distribution. \n3: Train the source RL algorithm $\\phi _ { i }$ until it converges to the optimal policy. \n4: Save the learning history $h _ { T } ^ { ( i ) } = ( o _ { 0 } , a _ { 0 } , r _ { 0 } , \\ldots , o _ { T } , a _ { T } , r _ { T } ) _ { i }$ to the dataset $\\mathcal { D } \\mathcal { D } \\cup h _ { T } ^ { ( i ) }$ . 5: end for \n6: while $P _ { \\theta }$ not converged do ▷ Part 2: Algorithm Distillation 7: Randomly sample a multi-episodic subsequence $\\bar { h } _ { j } ^ { ( i ) } = \\left( o _ { j } , a _ { j } , r _ { j } , \\dotsc , o _ { j + c } , a _ { j + c } , r _ { j + c } \\right) _ { i }$ of length $c$ 8: Autoregressively predict the actions with $P _ { \\theta }$ and compute the NLL loss in Eq. 6. \n9: Backpropagate to update the transformer parameters. \n10: end while \n11: for $k = 1 \\dots M _ { \\mathrm { s e e d s } }$ do ▷ Part 3: Autoregressive Evaluation 12: Sample a task $\\mathcal { M } _ { k } ^ { \\mathrm { t e s t } }$ randomly from the test task distribution. Initialize empty context queue $C$ . 13: Unroll the transformer $P _ { \\theta } ( \\cdot | C )$ in the environment storing sequential transitions (i.e. histories) in $C$ . 14: Measure the return accumulated by the agent for each episode of evaluation. \n15: end for ", + "bbox": [ + 173, + 597, + 826, + 847 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.2 PRACTICAL IMPLEMENTATION ", + "text_level": 1, + "bbox": [ + 176, + 869, + 426, + 883 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In practice, we implement AD as a two-step procedure. First, a dataset of learning histories is collected by running an individual gradient-based RL algorithm on many different tasks. Next, a sequence model with multi-episodic context is trained to predict actions from the histories. We describe these two steps below and detail the full practical implementation in Algorithm 1. ", + "bbox": [ + 173, + 895, + 823, + 924 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 823, + 132 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Dataset Generation: A dataset of learning histories is collected by training $N$ individual single-task gradient-based RL algorithms. To prevent overfitting to any specific task during sequence model training, a task $\\mathcal { M }$ is sampled randomly from a task distribution for each RL run. The data generation step is RL algorithm agnostic - any RL algorithm can be distilled. We show results distilling UCB exploration (Lai & Robbins, 1985), an on-policy actor-critic (Mnih et al., 2016), and an off-policy DQN (Mnih et al., 2013), in both distributed and single-stream settings. We denote the dataset of learning histories as $\\mathcal { D }$ in Eq. 5. ", + "bbox": [ + 174, + 138, + 825, + 237 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Training the Sequence Model: Once a dataset of learning histories $\\mathcal { D }$ is collected, a sequential prediction model is trained to predict actions given the preceding histories. We utilize the GPT (Radford et al., 2018) causal transformer model for sequential action prediction, but AD is compatible with any sequence model including RNNs (Williams & Zipser, 1989). For instance, we show in Appendix $\\mathrm { L }$ that AD can also be achieved with an LSTM (Hochreiter & Schmidhuber, 1997), though less effectively than AD with causal transformers. Since causal transformer training and inference are quadratic in the sequence length, we sample across-episodic subsequences $\\bar { h _ { j } } ^ { - } = \\left( o _ { j } , r _ { j } , a _ { j } \\ldots , o _ { j + c } , r _ { j + c } , a _ { j + c } \\right)$ of length $c < T$ from $\\mathcal { D }$ rather than training full histories. ", + "bbox": [ + 174, + 243, + 826, + 356 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4 EXPERIMENTAL SETUP ", + "text_level": 1, + "bbox": [ + 176, + 376, + 398, + 392 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4.1 ENVIRONMENTS ", + "text_level": 1, + "bbox": [ + 176, + 410, + 328, + 424 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "To investigate the in-context RL capabilities of AD and the baselines (see next section), we focus on environments that cannot be solved through zero-shot generalization after pre-training. Specifically, we require that each environment supports many tasks, that the tasks cannot be inferred easily from the observation, and that episodes are short enough to feasibly train across-episodic causal transformers - for more details regarding environments see Appendix B. We list the evaluation environments below: ", + "bbox": [ + 174, + 436, + 594, + 547 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Adversarial Bandit: a multi-armed bandit with 10 arms and 100 trials similar to the environment considered in $\\mathrm { { R L ^ { 2 } } }$ (Duan et al., 2016). However, during evaluation the reward is out of distribution. Reward is more likely distributed under odd arms $9 5 \\%$ of the time during training. At evaluation, the opposite happens - reward appears more often under even arms $9 5 \\%$ of the time. ", + "bbox": [ + 174, + 554, + 596, + 637 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/a3da0f2d56a2dea7e0ac773cbc4072974da4d3b19ed7b19d036e66b3977c8b9e.jpg", + "image_caption": [ + "Figure 2: Agent view from the DMLab Watermaze environment. The task is to find a hidden platform that elevates once found. " + ], + "image_footnote": [], + "bbox": [ + 627, + 454, + 803, + 556 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Dark Room: a 2D discrete POMDP where an agent spawns in a room and must find a goal location. The agent only knows its own $( x , y )$ coordinates but does not know the goal location and must infer it from the reward. The room size is $9 \\times 9$ , the possible actions are one step left, right, up, down, and no-op, the episode length is 20, and the agent resets at the center of the map. We test two environment variants – Dark Room where the agent receives $r = 1$ every time the goal is reached and Dark Room Hard, a hard exploration variant with a $1 7 \\times 1 7$ size and sparse reward $\\mathbf { \\boldsymbol { r } } = 1$ exactly once for reaching the goal). When not $r = 1$ , then $r = 0$ . ", + "bbox": [ + 174, + 645, + 825, + 742 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Dark Key-to-Door: similar to Dark Room but this environment requires an agent to first find an invisible key upon which it receives a reward of $r = 1$ once and then open an invisible door upon which it receives a reward of $r = 1$ once again. Otherwise, the reward is $r = 0$ . The room size is $9 \\times 9$ making the task space combinatorial with $8 1 ^ { 2 } = 6 5 6 1$ possible tasks. This environment is similar to the one considered in Chen et al. (2021) except the key and door are invisible and the reward is semisparse $\\boldsymbol { r } = 1$ for both key and the door). The agent is randomly reset. The episode length is 50 steps. ", + "bbox": [ + 173, + 750, + 825, + 833 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "DMLab Watermaze: a partially observable 3D visual DMLab environment based on the classic Morris Watermaze (Morris, 1981). The task is to navigate the water maze to find a randomly spawned trap-door. The maze walls have color patterns that can be used to remember the goal location. Observations are pixel images of size $7 2 \\times 9 6 \\times 3$ . There are 8 possible actions in total, including going forward, backward, left, or right, rotating left or right, and rotating left or right while going forward. The episode length is 50, and the agent resets at the center of the map. Similar to Dark ", + "bbox": [ + 174, + 840, + 825, + 924 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/d5537f516f3e701ce2d1f385b3b62e3325664799bf1950f1bd449d7fede0b517.jpg", + "image_caption": [ + "Figure 3: Adversarial Bandit (Section 5): AD, $\\mathrm { { R L } ^ { 2 } }$ , and ED evaluated on a 10-arm bandit with 100 trials. The source data for AD comes from learning histories from UCB (Lai & Robbins, 1985). During training, the reward is distributed under odd arms $9 5 \\%$ of the time and under even arms $9 5 \\%$ of the time during evaluation. Both AD and $\\mathrm { { R L } ^ { 2 } }$ can in-context learn in-distribution tasks, but AD generalizes better out of distribution. Running $\\mathrm { { R L } ^ { 2 } }$ with a transformer generally doesn’t offer an advantage over the original LSTM variant. ED performs poorly both in and out of distribution relative to AD and $\\mathrm { { R L } ^ { 2 } }$ . Scores are normalized relative to UCB. " + ], + "image_footnote": [], + "bbox": [ + 173, + 99, + 821, + 199 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Room, the agent cannot see the location of the goal from the observations and must infer it through the reward of $r = 1$ if reached and $r = 0$ otherwise; however, the goal space is continuous and therefore there are an infinite number of goals. ", + "bbox": [ + 174, + 315, + 823, + 358 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.2 BASELINES", + "text_level": 1, + "bbox": [ + 174, + 376, + 294, + 390 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The main aim of this work is to investigate to what extent AD reinforcement learns in-context relative to prior related work. AD is mostly closely related to Policy Distillation, where a policy is learned with a sequential model from offline interaction data. In-context online meta-RL is also related though not directly comparable to AD, since AD is an in-context offline meta-RL method. Still, we consider both types of baselines to better contextualize our work. For a more detailed discussion of these baseline choices we refer the reader to Appendix C. Our baselines include: ", + "bbox": [ + 174, + 401, + 825, + 486 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Expert Distillation $( E D )$ : this baseline is exactly the same as AD but the source data consists of expert trajectories only, rather than learning histories. ED is most similar to Gato (Reed et al., 2022) except ED models state-action-reward sequences like AD, while Gato models state-action sequences. ", + "bbox": [ + 176, + 492, + 821, + 535 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Source Algorithm: we compare AD to the gradient-based source RL algorithm that generates the training data for distillation. We include running the source algorithm from scratch as a baseline to compare the data-efficiency of in-context RL to the in-weights source algorithm. ", + "bbox": [ + 176, + 541, + 823, + 584 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "$R L ^ { 2 }$ (Duan et al., 2016): an online meta-RL algorithm where exploration and fast in-context adaptation are learned jointly by maximizing a multi-episodic value function. $\\mathrm { { R L } ^ { 2 } }$ is not directly comparable to AD for similar reasons to why online and offline RL algorithms are not directly comparable – RL2 gets to interact with the environment during training while AD does not. We use $\\mathrm { { R L } ^ { 2 } }$ asymptotic performance as an approximate upper bound for AD. ", + "bbox": [ + 174, + 590, + 825, + 660 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.3 EVALUATION ", + "text_level": 1, + "bbox": [ + 174, + 678, + 305, + 693 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "After pre-training, the AD transformer $P _ { \\theta }$ can reinforcement learn in-context. Evaluation is exactly the same as with an in-weights RL algorithm except the learning happens entirely in-context without updating the transformer network parameters. Given an MDP (or POMDP), the transformer interacts with the environment and populates its own context (i.e. without demonstrations), where the context is a queue containing the last $c$ transitions. The transformer’s performance is then evaluated in terms of its ability to maximize return. For all evaluation runs, we average results across 5 training seeds with 20 evaluation seeds each for a total of 100 seeds. A task $\\mathcal { M }$ is sampled uniformly from the test task distribution and fixed for each evaluation seed. The aggregate statistics reported reflect multi-task performance. We evaluate for 1000 and 160 episodes for the Dark and Watermaze environments respectively and plot performance as a function of total environment steps at test-time. ", + "bbox": [ + 174, + 704, + 825, + 843 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 863, + 326, + 880 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The main research question of this work is whether an in-weights RL algorithm can be amortized into an in-context one via Algorithm Distillation. The in-context RL algorithm should behave in a similar way as the in-weights one and exhibit exploration, credit assignment, and generalization capabilities. We begin our analysis in a clean and simple experimental setting where all three properties are required to solve the task - the Adversarial Bandit described in Sec. 4. ", + "bbox": [ + 173, + 895, + 823, + 924 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/9ee4c7c970fbe6d5f893cf896ce23072080991bcd6e0b43221766e63d5068738.jpg", + "image_caption": [ + "Figure 4: Main results: we evaluate AD, $\\mathrm { { R L } ^ { 2 } }$ , ED, and the source algorithm on environments that require memory and exploration. In these environments, an agent must reach a goal location that can only be inferred through a binary reward. AD is consistently able to in-context reinforcement learn across all environments and is more data-efficient than the A3C (“Dark” environments) (Mnih et al., 2016) or DQN (Watermaze) (Mnih et al., 2013) source algorithm it distilled. We report the mean return $\\pm \\nobreakspace 1 \\nobreakspace$ standard deviation over 5 training seeds with 20 test seeds each. " + ], + "image_footnote": [], + "bbox": [ + 174, + 99, + 823, + 184 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 286, + 825, + 328 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "To generate the source data, we sample a set of training tasks Bound algorithm (Lai & Robbins, 1985), and save its learning $\\{ \\mathcal { M } _ { j } \\} _ { j = 1 } ^ { N }$ , run the Upper Confidence We then train a transformer to predict actions as described in Alg. 1. We evaluate AD, ED, and $\\mathrm { { R L } ^ { 2 } }$ and normalize their scores relative to UCB and a random policy $( r - r _ { r a n d . } ) / ( r _ { U C B } - r _ { r a n d . } )$ . The results are shown in Fig. 3. We find that both AD and $\\mathtt { R L } ^ { 2 }$ can reliably in-context learn tasks sampled from the training distribution while ED cannot, though ED does do better than random guessing when evaluated in-distribution. However, AD can also in-context learn to solve out of distribution tasks whereas the other methods cannot. This experiment shows that AD can explore the bandit arms, can assign credit by exploiting an arm once reached, and can generalize well out of distribution nearly as well as UCB. We now move beyond the bandit setting and investigate similar research questions in more challenging RL environments and present our results as answers to a series of research questions. ", + "bbox": [ + 174, + 335, + 825, + 489 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Does Algorithm Distillation exhibit in-context reinforcement learning? To answer this question, we first generate source data for Algorithm Distillation. In the Dark Room and Dark Key-to-Door environments we collect 2000 learning histories with an Asynchronous Advantage Actor-Critic (A3C) (Mnih et al., 2016) with 100 actors, while in DMLab Watermaze we collect 4000 learning histories with a distributed DQN with 16 parallel actors (see Appendix F for asymptotic learning curves of the source algorithm and Appendix O for hyperparameters). Shown in Fig. 4, AD in-context reinforcement learns in all of the environments. In contrast, ED fails to explore and learn in-context in most settings. We use $\\mathrm { { R L } ^ { 2 } }$ trained for 1 billion environment steps as a proxy for the upper bound of performance for a meta-RL method. $\\mathrm { { R L ^ { 2 } } }$ achieves a near-optimal asymptotic score in all the environments except for Dark Room (Hard). Despite learning from offline data, AD matches asymptotic $\\mathtt { R L } ^ { 2 }$ on the Dark environments and approaches it (within $1 3 \\%$ ) on Watermaze. ", + "bbox": [ + 174, + 498, + 825, + 651 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Credit-assignment: In Dark Room, the agent receives $r = 1$ each time it visits the goal location. Even though AD is trained to condition only on single timestep reward and not episodic return tokens, it is still able to maximize the reward, which suggests that AD has learned to do credit assignment. ", + "bbox": [ + 174, + 659, + 825, + 700 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Exploration: Dark Room (Hard) tests the agents exploration capability. Since the reward is sparse $r = 1$ exactly once), most of the learning history has reward values of $r = 0$ . Nevertheless, AD infers the goal from previous episodes in its context which means it has learned to explore and exploit. ", + "bbox": [ + 174, + 708, + 825, + 750 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Generalization: Dark Key-to-Door tests in-distribution generalization with a combinatorial task space. While the environment has a total of $\\sim 6 . 5 \\mathrm { k }$ tasks, less than 2k were seen during training. During evaluation, AD both generalizes and achieves near-optimal performance on mostly unseen tasks. ", + "bbox": [ + 174, + 756, + 825, + 799 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Can Algorithm Distillation learn from pixel-based observations? DMLab Watermaze is a pixelbased environment that is larger than the Dark environments with tasks sampled from a continuous uniform distribution. The environment is partially observable in two ways - the goal is invisible until the agent has reached it and the first-person view limits the agent’s field of vision. Shown in Fig. 4, AD maximizes the episodic return with in-context RL while ED does not learn. ", + "bbox": [ + 174, + 809, + 825, + 878 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Can AD learn a more data-efficient RL algorithm than the one that produced the source data? In Fig. 4, AD is significantly more data-efficient than the source algorithm. This gain is a byproduct of distilling a multi-stream algorithm into a single-stream one. The source algorithms (A3C and DQN) are distributed, which means they run many actors in parallel to achieve good performance.3 A distributed RL algorithm may not be very data-efficient in aggregate but each individual actor can be data-efficient. Since the learning history for each actor is saved separately, ", + "bbox": [ + 176, + 882, + 823, + 924 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/e72730fa811ddedf85713313c74350731b168f42cfdbd85e088031cd6898d726.jpg", + "image_caption": [ + "Figure 5: $A D$ and $E D$ conditioned on partial demonstrations: We compare the performance of AD and ED when prompted with a demonstration from the source algorithm’s training history on Dark Room (semi-dense). While ED slightly improves and then maintains performance from the input policy, AD is able to improve it in-context until the policy is optimal or nearly optimal. " + ], + "image_footnote": [], + "bbox": [ + 215, + 101, + 777, + 207 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 281, + 825, + 324 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "AD achieves similar performance to the multi-stream distributed RL algorithm, but is more data-efficient as a single-stream method. ", + "bbox": [ + 174, + 324, + 549, + 364 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "These data-efficiency gains are also evident for distilling single-stream algorithms. In Fig 6 we show that by subsampling every $k$ -th episode (where $k = 1 0$ ) from a single stream A3C learning history, AD can still learn a more data-efficient in-context RL algorithm (for more detail, see Appendix J). Therefore, AD can be more data-efficient than both a multi and single-stream source RL algorithm. ", + "bbox": [ + 174, + 372, + 549, + 469 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/c652f1346777c272cd0157cbac3dddfaa6bac8bab0398dde13e052045c590f1e.jpg", + "image_caption": [ + "Figure 6: Single-stream Algorithm Distillation: AD trained on the learning history from an A3C agent with only one actor (i.e. single-stream). By training on subsampled learning histories (see Sec. 5), AD learns are more data-efficient in-context RL algorithm. " + ], + "image_footnote": [], + "bbox": [ + 596, + 337, + 789, + 454 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "While AD is more data-efficient, the source algorithm achieves slightly higher asymptotic performance (see Appendix F). However, the source algorithm produces many single-task agents with a unique set of weights $\\phi _ { n }$ per task $\\mathcal { M } _ { n }$ , while AD produces a single generalist agent with weights $\\theta$ that are fixed across all tasks. ", + "bbox": [ + 174, + 477, + 549, + 560 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Is it possible to accelerate AD by prompting it with demonstrations? Although AD can reinforcement learn without relying on demonstrations, it has the added benefit that, unlike the source algorithm, it can be conditioned on or prompted with external data. To answer the research question, we sample policies from the hold-out test-set data along different points of the source algorithm history - from a near-random policy to a near-expert policy. We then pre-fill the context for both AD and ED with this policy data, and run both methods in the environment in Dark Room (Fig. 5). While ED maintains the performance of the input policy, AD improves every policy in-context until it is near-optimal. Importantly, the more optimal the input policy the faster AD improves it until it is optimal. ", + "bbox": [ + 173, + 569, + 549, + 763 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/326a444f7a476152beb4d779e7e9c87e058d5d0329f877516a605551d10515aa.jpg", + "image_caption": [ + "Figure 7: Context size: AD in Dark Key-toDoor with different context sizes. In-context RL only emerges once the context size is large enough and across-episodic. " + ], + "image_footnote": [], + "bbox": [ + 566, + 575, + 818, + 681 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "What context size is required for in-context RL to emerge? We’ve hypothesized that AD requires sufficiently long (i.e. across-episodic) contexts to in-context reinforcement learn. We test this hypothesis by training several AD variants with different context sizes on the Dark Room environment. We plot the learning curves of these different variants in Fig. 7 and find that multi-episodic contexts of 2-4 episodes are necessary to learn a near-optimal in-context RL algorithm. Initial signs of in-context RL begin to emerge when the context size is roughly the length of an episode. The reason for this is likely that the context is large enough to retrain across-episodic information – e.g., at the start of a new episode, the context will be filled with transitions from most of the previous episode. ", + "bbox": [ + 173, + 773, + 825, + 885 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "6 RELATED WORK ", + "text_level": 1, + "bbox": [ + 174, + 102, + 344, + 117 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Offline Policy Distillation: Most closely related to our work are the recent advances in learning policies from offline environment interaction data with transformers, which we have been referring to as Policy Distillation (PD). Initial PD architectures such as Decision Transformer (DT) (Chen et al., 2021) and Trajectory Transformer (Janner et al., 2021) showed that transformers can learn single-task policies from offline data. Subsequently the Multi-Game Decision Transformer (MGDT) (Lee et al., 2022) and Gato (Reed et al., 2022) showed that PD architectures can also learn multi-task same domain and cross-domain policies, respectively. Importantly, these prior methods use contexts substantially smaller than an episode length, which is likely the reason in-context RL was not observed in these works. Instead, they rely on alternate ways to adapt to new tasks - MGDT finetunes the model parameters while Gato gets prompted with expert demonstrations to adapt to downstream tasks. AD adapts in-context without finetuning and does not rely on demonstrations. A number of recent works have explored more generalized PD architectures (Furuta et al., 2021), prompt conditioning (Xu et al., 2022), and online gradient-based RL (Zheng et al., 2022). Some PD architectures such as DT and MGDT are instantiations of Upside Down RL (UDRL) Schmidhuber (2019); Srivastava et al. (2019) where rather than learning a value function, a policy is conditioned directly on the desired return. However, AD is not explicitly doing UDRL since it is not conditioned on returns. In fact, return maximization is not specified anywhere in the AD objective but rather emerges implicitly by modeling the learning histories of an RL algorithm. ", + "bbox": [ + 174, + 123, + 825, + 373 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Meta Reinforcement Learning: AD falls into the category of methods that learn to reinforcement learn, also known as meta-RL. Specifically, AD is an in-context offline meta-RL method. This general idea of learning the policy improvement process has a long history in reinforcement learning, but has been limited to meta-learning hyper-parameters4 until recently (Ishii et al., 2002). In-context deep meta-RL methods introduced by Wang et al. (2016) and Duan et al. (2016) are usually trained in the online setting by maximizing multi-episodic value functions with memory-based architectures through environment interactions (Ni et al., 2022). Meta-RL through multi-episodic value functions has been done in both on-policy (Duan et al., 2016) and off-policy (Rakelly et al., 2019; Fakoor et al., 2020) settings. Another common approach to online meta-RL includes optimization-based methods that find good network parameter initializations for meta-RL (Hochreiter et al., 2001; Finn et al., 2017; Nichol et al., 2018) and adapt by taking additional gradient steps. Like other in-context meta-RL approaches, AD is gradient-free - it adapts to downstream tasks without updating its network parameters. Recent works have proposed learning to reinforcement learn from offline datasets, or offline meta-RL, using Bayesian RL (Dorfman et al., 2021) and optimization-based meta-RL (Mitchell et al., 2021). Given the difficulty of offline meta-RL, Pong et al. (2022) proposed a hybrid offline-online strategy for meta-RL. ", + "bbox": [ + 174, + 381, + 825, + 601 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In-Context Learning with Transformers: In this work, we make the distinction between in-context learning and incremental or in-context learning. In-context learning involves learning from a provided prompt or demonstration while incremental in-context learning involves learning from one’s own behavior through trial and error. While many recent works have demonstrated the former, it is much less common to see methods that exhibit the latter. Arguably, the most impressive demonstrations of in-context learning to date have been shown in the text completion setting (Radford et al., 2018; Chen et al., 2020; Brown et al., 2020) through prompt conditioning. Similar methodology was recently extended to show powerful composability properties in text-conditioned image generation (Yu et al., 2022). Recent work showed that transformers can also learn simple algorithm classes, such as linear regression, in-context in a small-scale setting (Garg et al., 2022). Like prior in-context learning methods, Garg et al. (2022) required initializing the transformer prompt with expert examples. While the aforementioned approaches were examples of in-context learning, a recent work (Chen et al., 2022) demonstrated incremental in-context learning for hyperparameter optimization by treating hyperparameter optimization as a sequential prediction problem with a score function. ", + "bbox": [ + 174, + 609, + 825, + 803 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "7 CONCLUSION ", + "text_level": 1, + "bbox": [ + 176, + 818, + 318, + 833 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We have demonstrated that can distill RL algorithms by modeling their learning histories causally with imitation learning and that AD can learn more data-efficient algorithms than those that generated the source data. We hope that AD inspires further investigation into in-context reinforcement learning from the research community. ", + "bbox": [ + 176, + 843, + 825, + 898 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 176, + 103, + 285, + 117 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Charles Beattie, Joel Z. 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", + "bbox": [ + 173, + 220, + 823, + 248 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A DISCUSSION AND LIMITATIONS ", + "text_level": 1, + "bbox": [ + 178, + 102, + 470, + 117 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Discussion: While AD is more data-efficient than the source algorithm, we found that the source algorithm achieves a slightly higher asymptotic score in harder environments. This presents a tradeoff between data-efficiency and asymptotic optimality when using AD that should be taken into consideration when applying AD to a specific problem. In terms of long-term consequences, AD presents a path for converting narrow single-task RL agents into multi-task generalist ones. To date, deep RL research has mostly focused on powerful single-task agents Mnih et al. (2015); Schrittwieser et al. (2019); Hessel et al. (2021). These algorithms have produced powerful but data-inefficient agents, which has limited their applicability beyond narrow domains. AD offers a path toward training substantially more data-efficient, though perhaps less optimal, generalist agents by distilling narrow RL algorithms into sequence models like transformers. ", + "bbox": [ + 174, + 135, + 825, + 273 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Limitations: While AD is a general method, in this work we’ve shown that AD generalizes to new tasks within the same domain. Showing the ability to distill cross-domain RL algorithms and generalize to new domains would be an interesting line if inquiry for future work. AD also requires storing many learning histories which could take up significant memory, though this may not be too much of an issue since the learning histories can be stored on disk rather than RAM. Perhaps the main limitation of AD is that most RL environments of interest have long episodes and modeling multiepisodic context requires more powerful long-horizon sequential models than the ones considered in this work. We believe this is a promising direction for future research. ", + "bbox": [ + 174, + 281, + 825, + 392 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "B ENVIRONMENT CONSIDERATIONS ", + "text_level": 1, + "bbox": [ + 176, + 415, + 490, + 431 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "In this work, we consider environments where zero-shot generalization is difficult, so the agent must learn through trial and error. We also want environments where overfitting to any particular task is difficult to ensure our method is general. A final practical consideration is that we consider environments wher across-episodic histories can be feasibly modeled with a causal transformer. Given these considerations, our evaluation environments need to satisfy three criteria: ", + "bbox": [ + 174, + 448, + 825, + 517 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "1. Supports many tasks: The environments must be multi-task to ensure that our agent and the baselines do not overfit to any single task and instead is able to in-context reinforcement learn across many tasks within a given domain. \n2. Task must be hard to infer: To ensure that the downstream tasks are hard to generalize to in zero-shot, we use environments that require exploration. Namely, we require environments where either the task can only be inferred from the reward and not the observation, or tasks that are partially observable. \n3. Supports multi-episodic contexts: Lastly, we impose a practical constraint - the environment episodes must be short enough such that a normal GPT-like transformer can fit multiple episodes in its context. Since this work introduces AD as a method, we wish to investigate it in the cleanest possible setting using a canonical architecture. We leave investigating AD with more complex architectures that scale to longer sequences for future work. ", + "bbox": [ + 210, + 532, + 825, + 713 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Prior related works (Chen et al., 2021; Lee et al., 2022; Janner et al., 2021; Reed et al., 2022) evaluated on Atari, OpenAI gym, and as well as other environments. However, Atari and OpenAI gym don’t satisfy at least one of the above criteria. Atari and OpenAI gym episodes are often long and can contain thousands or more transitions per episode, so it’s technically challenging to populate a causal transformer’s context with across-episode histories. Indeed, the prior related works only considered within-episode context lengths. Additionally, it is often easy to infer the task from either the observation or the dense reward alone in both Atari and OpenAI gym, which reduces the need for exploration. For these reasons, we evaluate in environments that satisfy all three criteria instead. ", + "bbox": [ + 174, + 728, + 825, + 839 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "C CLOSELY RELATED PRIOR METHODS ", + "text_level": 1, + "bbox": [ + 174, + 862, + 519, + 878 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "In our main results we use Expert Distillation (ED) as a baseline. Here, we discuss how the most closely related methods differ from AD and why ED is sufficient to support the paper’s claims. ", + "bbox": [ + 174, + 895, + 823, + 924 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Expert Distillation (ED): ED is most similar to Gato (Reed et al., 2022), which models expert sequences from a converged RL policy using a causal transformer. ED also trains a causal transformer to predict actions using expert policy data. There are two key differences between ED and Gato. First, unlike Gato which utilizes small (relative to an episode length) within-episode contexts, ED is trained on the same across-episode contexts as AD, so the architectures used by ED and AD are the same. The benefits of AD cannot therefore be attributed to across-episode contexts alone but also learning progress in the offline data used to train AD. Second, ED models state-action-reward sequences while Gato models only state-action sequences. The main difference between ED and AD is that AD is trained on full multi-task learning histories rather than expert policy data. ", + "bbox": [ + 174, + 103, + 825, + 229 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Decision Transformer (DT) (Chen et al., 2021) and Multi-Game Decision Transformer (Lee et al., 2022): DTs learn return-conditioned policies from single-task offline data collected by an RL agent. While the training data itself (an RL agent’s replay buffer) contains learning, the context sizes used in DT are too small to capture any learning progress or identify the task using across-episode information. For instance, the Atari experiments use a context of length $3 0 - 5 0$ tokens, or $1 0 - 1 7$ transitions. Atari games can have hundreds or thousands of transitions in a single episode, which means these contexts capture mostly within-episode information. Additionally, very little learning progress happens in the underlying replay buffer data within that many transitions. ", + "bbox": [ + 174, + 236, + 825, + 348 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Another difference between DT and AD / ED is that DT learns a return-conditioned model whereas AD / ED are both reward-conditioned. In our setting return-conditioning alone cannot yield an optimal policy since the agent does not know the task until after it explores the environment and can identify it using across-episode contexts. Since (i) DT uses small within-episodic contexts and (ii) return-conditioning would not help in the environments considered, this baseline is similar to ED with a small within-episode context which is strictly weaker than the long across-episode context variant of ED we consider. ", + "bbox": [ + 174, + 353, + 825, + 450 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Trajectory Transformer (TTO) (Janner et al., 2021): Like AD, TTO also models state-action-reward tokens but in addition to predicting actions it also learns a world model by predicting states and rewards. To maximize return, TTO then uses beam search to select high-reward actions. However, in our setting, TTO will run into the same problem as DT. To model rewards accurately it will need longer across-episodic contexts since one environment supports many tasks. Similar to DT, MGDT, and Gato, TTO uses smaller within-episode contexts. For this reason, TTO will fare no better than DT, MGDT, or ED in the settings we consider. We also note that in contrast to TTO, AD is model-free. In AD, actions are sampled from the transformer history-conditioned predictions and return maximization emerges from modeling the learning histories of an RL algorithm. ", + "bbox": [ + 174, + 458, + 825, + 583 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "To summarize, AD differs from prior methods mainly because its context is across-episodic and hence large enough to capture learning progress and task information. AD could further be augmented by learning world models like TTO or conditioning on returns like DT, but these investigations would be well suited for future work since they are tangential to the main research question addressed in this work – whether in-context RL can emerge by imitating the learning histories of an RL algorithm with long across-episodic contexts. ", + "bbox": [ + 174, + 590, + 825, + 674 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "AD is also closely related to prior work in in-context meta-RL. While both AD and in-context meta-RL model across-episodic histories with memory-based architectures, prior in-context meta-RL algorithms, such as RL2 (Duan et al., 2016) are trained online and rely on learning multi-episodic value functions with TD learning while AD is trained offline and uses a supervised imitation learning objective. ", + "bbox": [ + 174, + 681, + 823, + 751 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "D EXPERT DISTILLATION MAIN RESULTS ", + "text_level": 1, + "bbox": [ + 174, + 102, + 537, + 118 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "We elaborate further on the main results in Fig. 4 and provide intuition regarding the behaviors of the ED baseline. In Dark Room, Dark Room (Hard), and Watermaze, ED performance is either flat or it degrades. The reason for this is that ED only saw expert trajectories during training, but during evaluation it needs to first explore (i.e. perform non-expert behavior) to identify the task. This required behavior is out-of-distribution for ED and for this reason it does not reinforcement learning in-context. In Dark Key-to-Door the agent is reset randomly at the beginning of each episode, whereas in all of the environments the agent’s starting position is fixed. Due to random resets, the ED agent is sometimes reset by the first goal in Dark Key-to-Door which allows it to occasionally identify the first goal of the task, which is why it shows slight improvement. ", + "bbox": [ + 174, + 133, + 826, + 258 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "E MODEL SIZE ", + "text_level": 1, + "bbox": [ + 176, + 279, + 316, + 295 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "We investigate how transformer capacity affects performance in Fig. 8. While in-context RL emerges across all model sizes investigated, we find that increasing the model depth, the model width in terms of embedding dimension, and (to a lesser extent) the number of attention heads improves performance on Dark Key-to-Door. ", + "bbox": [ + 174, + 310, + 826, + 366 + ], + "page_idx": 15 + }, + { + "type": "image", + "img_path": "images/7268b332c41895dedc19c08ef9b6099f4da74ea1f1754cc2dd5c84977a7dc825.jpg", + "image_caption": [ + "Figure 8: Model size investigations: We investigate how increasing model capacity affects AD. While in-context RL with AD emerges regardless of the model capacity, increasing the model depth and width helps improve AD until it achieves near-optimal performance. " + ], + "image_footnote": [], + "bbox": [ + 305, + 381, + 691, + 484 + ], + "page_idx": 15 + }, + { + "type": "image", + "img_path": "images/5225296573b7d46d8a1d9f73f0c47aa7acab27725bdc5f8cd9d76937d0d127b6.jpg", + "image_caption": [ + "Figure 9: Asymptotic performance of the A3C (Mnih et al., 2016) and a $\\mathbf { Q - } \\lambda$ variant of the DQN (Mnih et al., 2013) RL algorithms used to produce learning histories for the Dark and Watermaze environments. These curves show the learning histories AD is trained on. The source algorithms plotted in Fig. 4 are the same as in these plots. " + ], + "image_footnote": [], + "bbox": [ + 174, + 137, + 823, + 224 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "G LABEL SMOOTHING ABLATION ", + "text_level": 1, + "bbox": [ + 176, + 308, + 470, + 323 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "For the harder exploration task of Dark Room (Hard), we found that adding label smoothing regularization (Szegedy et al., 2016; Muller et al. ¨ , 2019) improved the in-context learning ability of AD . In Figure 10 we ablate the benefit of using label smoothing for 3 different $\\alpha$ values as well as with it turned off. Each curve in the figure denotes average performance over 5 training seeds. We can see that adding label smoothing up to a point improves the in-context learning ability of Algorithm Distillation, with performance continually increasing with the number of evaluation episodes. ", + "bbox": [ + 173, + 338, + 826, + 422 + ], + "page_idx": 16 + }, + { + "type": "image", + "img_path": "images/e07003b625836d03f8abae3f9b6de0dbf38604250cecc84a2937ab5a8c642300.jpg", + "image_caption": [ + "Figure 10: AD trained with different amounts of label smoothing on Dark Room (Hard). " + ], + "image_footnote": [], + "bbox": [ + 321, + 438, + 676, + 599 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "H $\\mathtt { R L } ^ { 2 }$ NETWORK ARCHITECTURE: TRANSFORMER VS LSTM ", + "text_level": 1, + "bbox": [ + 173, + 656, + 699, + 672 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "We compared using a transformer as the architecture for $\\mathrm { { R L ^ { 2 } } }$ instead of an LSTM. In Figure 11, we ran both transformer and LSTM $\\mathrm { { R L ^ { 2 } } }$ agents over the Dark Room environment. The curves shown are the best from a sweep over learning rate and unroll length hyperparameters. The transformer architecture is 4-layers with a model size of 256 and pre-norm layer normalization placement. While both agents reached a similar level of final performance, all $\\mathtt { R L } ^ { \\dot { 2 } }$ transformer models trained tended to be more unstable with the average return not as consistent as with an LSTM architecture. Given the poor performance of the transformer-based $\\mathrm { { R L ^ { 2 } } }$ on the simpler Dark Room setting, our other experimental settings used the LSTM. ", + "bbox": [ + 173, + 688, + 825, + 800 + ], + "page_idx": 16 + }, + { + "type": "image", + "img_path": "images/d84a5680152d9b0d5a45f824e697e85bb345213b6f295d3072548835b0a69b86.jpg", + "image_caption": [ + "LSTM v.s. Transformer ", + "Figure 11: Comparison of LSTM and Transformer architecture for $\\mathtt { R L } ^ { 2 }$ agent on Dark Room. Each curve is averaged over 5 training seeds with the shaded area representing the standard error. " + ], + "image_footnote": [], + "bbox": [ + 305, + 116, + 691, + 260 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "I NUMBER OF TRAINING TASKS IN SOURCE DATA ", + "text_level": 1, + "bbox": [ + 173, + 327, + 606, + 343 + ], + "page_idx": 17 + }, + { + "type": "image", + "img_path": "images/3f6801f52bcd71f0fcf1231b4856e8cbd97a9ff344198d24cc0710b9d3353f73.jpg", + "image_caption": [ + "Figure 12: Algorithm distillation trained on different numbers of training tasks on Dark Key-to-Door evaluated on a fixed set of test tasks for 300 episodes of evaluation. " + ], + "image_footnote": [], + "bbox": [ + 307, + 364, + 689, + 520 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "One interesting question is how many tasks AD needs to be trained on to learn an algorithm that generalizes to held out tasks. We trained AD on different numbers of Dark Key-to-Door training tasks and evaluated the resulting models on the same set of test tasks. Figure 12 shows the incontext learning plots for the resulting AD models on the set of test tasks. As a reminder, there are $8 1 ^ { 2 } = 6 5 6 1$ unique Dark Key-to-Door tasks. Models trained on 1, 9 or 18 training tasks did not show any in-context learning on test tasks. While models trained on 37, 75 and 151 tasks did not achieve good performance overall, they did exhibit some in-context learning over the course of 300 episodes. The best models were trained on 1212 and 2424 tasks which corresponds to roughly $1 8 \\%$ and $3 7 \\%$ of the total number of tasks in the Dark Key-to-Door domain. ", + "bbox": [ + 173, + 580, + 825, + 705 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "J SINGLE-STREAM ALGORITHM DISTILLATION", + "text_level": 1, + "bbox": [ + 174, + 726, + 581, + 742 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "We provide more details around the experimental setup for the single-stream result shown in Fig. 6. We showed in Fig. 4 that when AD is trained on data from a subset of the actors of a distributed source RL algorithm, the resulting model is more data efficient than the source algorithm. Here we confirm that AD can produce a faster algorithm than the one it was trained on in the single-stream setting. For this experiment we trained A3C on 2048 Dark Key-to-Door tasks for 2000 episodes each. We then trained AD on the resulting data while subsampling the learning histories by a factor of 10. More concretely, we took every 10th episode from each of the learning history, which resulted in a 200 episode compressed learning trajectory for each task. Figure 6 compares the resulting AD model evaluated on a set of test tasks to the performance of the source algorithm on these tasks. The model learned by AD learns much faster than the source algorithm confirming that Algorithm Distillation can turn a slow gradient-based algorithm into a much more data efficient in-context learning algorithm. ", + "bbox": [ + 173, + 757, + 826, + 924 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "K RANDOM MASK ", + "text_level": 1, + "bbox": [ + 174, + 102, + 341, + 117 + ], + "page_idx": 18 + }, + { + "type": "image", + "img_path": "images/1038f6e9da6d542b87e80cfa6740b77180b4083883611e385d6a893aeadde962.jpg", + "image_caption": [ + "Figure 13: Downstream performance of Algorithm Distillation with different values of random masking during training in $9 \\mathrm { { x 9 1 } }$ goal gridworld. " + ], + "image_footnote": [], + "bbox": [ + 351, + 159, + 647, + 313 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "During training, input tokens were randomly masked to avoid overfitting to training data. This plot shows the downstreams results on a $9 \\mathrm { x } 9$ Dark Key-to-Door domain with different values of this random masking. Values of $0 . 3 - 0 . 5$ perform the best with the value of 0.3 chosen for all experiments. ", + "bbox": [ + 174, + 381, + 825, + 438 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "L AD NETWORK ARCHITECTURE: TRANSFORMER VS LSTM ", + "text_level": 1, + "bbox": [ + 174, + 458, + 692, + 474 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Here we consider the importance of the Transformer architecture to the success of algorithm distillation (AD) by comparing to AD based off of an LSTM (Hochreiter & Schmidhuber, 1997). Specifically, the LSTM receives the concatenated embeddings of $\\left( o _ { i } , a _ { i } , r _ { i } \\right)$ triplets up to the most recent time step $t - 1$ . The output of the LSTM is then concatenated with the current observation $o _ { t }$ embedding and both are then fed through a multi-layer perceptron (MLP) policy torso to produce a distribution over the present action $a _ { t }$ . The LSTM hidden size (512), MLP depth (2), and MLP width (256) were swept and tuned by grid search based on downstream reward attainment. ", + "bbox": [ + 174, + 488, + 825, + 587 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Comparing Transformer AD and LSTM AD on the Dark Key-to-Door task (Figure 14), we find that both agents are capable of in-context learning, demonstrating that the success of AD is not tied to the underlying network architecture. However, we also find that the Transformer variant consistently outperforms the LSTM variant, which is why all other experiments in this paper employ the Transformer variant. This finding is consistent with the recent wider success of Transformer-based architectures over recurrent neural network (RNN)-based architectures in sequence prediction tasks. ", + "bbox": [ + 174, + 593, + 825, + 678 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "M DISTILLING ACTIONS VS PROBABILITIES ", + "text_level": 1, + "bbox": [ + 174, + 698, + 550, + 713 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "In the rest of the paper, we use (one-hot) actions taken by the source policy as the prediction target for AD. Here, we compare that choice to predicting the source policy probabilities from which that action was sampled. In other words, is it better to distill actions or probabilities? ", + "bbox": [ + 174, + 728, + 823, + 770 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Figure 15 compares all combinations of: 1) distilling actions vs probabilities, and 2) conditioning those predictions on past actions, probabilities, or both. The plot labels represent various input/output combinations, e.g. $\\mathtt { S a r - } > \\mathtt { a }$ indicates observing states, actions, and rewards while predicting actions (i.e. the main variant of AD in the rest of the paper), sar $- > \\mathrm { p }$ represents using the same observations but instead predicting probabilities, and $\\tt S a p r - > a$ represents observing states, actions, probabilities, and rewards while predicting actions, etc. ", + "bbox": [ + 173, + 776, + 825, + 861 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Two conclusions stand out from this plot. First, the original action distillation variant of AD $( s \\mathsf { a r } - \\mathsf { > a } )$ performs best, followed closely by probability distillation $( \\mathsf { s a r - > p } )$ . In the paragraphs and experiments below, we explore why action distillation outperforms probability distillation. Second, all variants that condition on past probabilities catastrophically fail. We speculate on two somewhat contradictory reasons why this might occur: 1) The AD prediction task involves a combination of inferring the current policy, as well as predicting when policy updates will occur. Since observing policy probabilities provides more information about the current policy than actions do, it is possible that this leads to AD fully focusing on inferring the current policy and ignoring the prediction of policy updates. In other words, observing only actions taken may act as a useful information bottleneck. 2) On the other hand, observing probabilities may inadvertently leak information about when policy updates occur. If the transformer context includes multiple visits to the same state, AD could learn to compare the policy probabilities to infer whether a policy update occurred in between the two visits. If AD learned during training to rely on this information, then during autoregressive evaluation, it may be “waiting“ for a policy update than never occurs. If this is the source of the issue, then a potential solution would be retraining with explicit policy update tokens and including them between evaluation episodes, however we leave this investigation to future work. ", + "bbox": [ + 174, + 867, + 825, + 924 + ], + "page_idx": 18 + }, + { + "type": "image", + "img_path": "images/454e8467c64103db58787ec5b2c5c859eefc5a040f070dda6d155b0f1e59d8e6.jpg", + "image_caption": [ + "Figure 14: Comparison between algorithm distillation with a Transformer and LSTM architecture on Dark Key-to-Door. Mean $\\pm \\nobreakspace 1 \\nobreakspace$ standard deviation over 5 training seeds and 20 evaluation seeds. 300 episodes corresponds to $1 5 \\mathrm { k }$ environment steps. " + ], + "image_footnote": [], + "bbox": [ + 356, + 102, + 642, + 275 + ], + "page_idx": 19 + }, + { + "type": "image", + "img_path": "images/a807c9a242becc876d7e1138b0bdc3096c70672568caf1845fcca6c88bec2352.jpg", + "image_caption": [ + "Figure 15: Comparison between algorithm distillation with various input/output combinations on Dark Key-to-Door. Mean $\\pm \\nobreakspace 1 \\nobreakspace$ standard deviation over 5 training seeds and 20 evaluation seeds. 1000 episodes corresponds to $5 0 \\mathrm { k }$ environment steps. " + ], + "image_footnote": [], + "bbox": [ + 356, + 371, + 640, + 544 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 652, + 825, + 819 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Now we return to the question of why action distillation $\\left( s \\mathsf { a r } - \\mathsf { > a } \\right)$ outperforms probability distillation $( s \\mathsf { a r } \\mathrm { - } \\mathsf { > p } )$ . Intuitively, distilling one-hot actions will result in a more deterministic policy than distilling probabilities when training on finite data. Indeed, the red curves in the top two plots of Figure 16 show that probability distillation (top left) converges to a higher entropy ${ \\sim } 1 . 2$ bits) policy than does action distillation (top right, ${ \\sim } 1$ bit). Noting this discrepancy, we speculated that encouraging probability distillation towards a more deterministic policy might lead to increased performance. To do so, we experimented with an entropy penalty regularizer added to the AD loss from equation 6, leading to the modified objective: ", + "bbox": [ + 173, + 825, + 825, + 924 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "", + "bbox": [ + 176, + 103, + 506, + 118 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/81eed58a4490c319f199c6610f33f8e7ff8e47c1f9af370938c7e2703e38b223.jpg", + "text": "$$\n\\tilde { \\mathcal { L } } ( \\theta , \\alpha ) : = \\sum _ { n = 1 } ^ { N } \\sum _ { t = 1 } ^ { T - 1 } - \\log P _ { \\theta } ( A = a _ { t } ^ { ( n ) } | h _ { t - 1 } ^ { ( n ) } , o _ { t } ^ { ( n ) } ) + \\alpha H \\Big [ P _ { \\theta } ( A | h _ { t - 1 } ^ { ( n ) } , o _ { t } ^ { ( n ) } ) \\Big ] ,\n$$", + "text_format": "latex", + "bbox": [ + 238, + 125, + 758, + 170 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "where $H [ \\cdot ]$ is the Shannon entropy in bits, and $\\alpha$ is a regularization weight. The top two panels of Figure 16 show the effect of increasing $\\alpha$ on decreasing policy entropy for action distillation (top right) and probability distillation (top left), throughout training. The bottom two panels show the corresponding changes in evaluation returns. Notably, entropy penalization indeed improves performance for probability distillation (bottom left), with the best performing regularization value $\\alpha = 0 . 1$ ) achieving a similar return to the original unregularized action distillation (bottom right, red). Interestingly, this is the amount of regularization that leads probability distillation to have the most similar entropy to the unregularized action distillation variant $_ { \\sim 1 }$ bit) as well. Further entropy penalization for action distillation, on the other hand, does not lead to increased performance (bottom right). Together, these results suggest that action distillation naturally leads to the “optimal“ amount of entropy regularization on its own, at least in the environments we study. ", + "bbox": [ + 173, + 174, + 825, + 328 + ], + "page_idx": 20 + }, + { + "type": "image", + "img_path": "images/1193b4fb8b446f1e1eb9a0060b68dfb52c35743a1269cf2b5dabdd292afbbd8e.jpg", + "image_caption": [ + "Figure 16: Policy entropy during training and reward during evaluation for the action distillation $( s \\mathsf { a r } - \\mathsf { > a } )$ and probability distillation $( \\mathsf { s a r - > p } )$ variants of AD on Dark Key-to-Door. Colors indicate the strength of the weight on the entropy penalty regularizer from equation 7. Mean $\\pm \\nobreakspace 1 \\nobreakspace$ standard deviation over 5 training seeds and 20 evaluation seeds. 1000 episodes corresponds to $5 0 \\mathrm { k }$ environment steps. " + ], + "image_footnote": [], + "bbox": [ + 204, + 344, + 790, + 699 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "In addition to closing gap between probability and action distillation with an entropy penalty, we also explored retuning various hyperparameters for $\\mathtt { S a r - > p }$ rather than reusing those tuned for $\\mathtt { S a r - } > \\mathtt { a }$ . We found that many hyperparameters, such as the dropout rate, attention dropout rate, and sequence mask probability, had similar optimal values in the two cases, and so retuning them did not help. However, one hyperparameter that did have an effect was the size of the transformer context window (Figure 17). While action distillation performance increased only up to a context window size of 200 steps (4 episodes) and then plateaued (Figure 17, right), probability distillation performance continued to increase up to a context window size of 300 steps (6 episodes) before plateauing (Figure 17, left). Thus, while a comparison at our default context window size of 200 steps favored action distillation, for larger context window sizes, action and probability distillation performed similarly. We leave any further explanation of why probability distillation requires larger context windows than action distillation to future work. ", + "bbox": [ + 173, + 797, + 825, + 924 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 146 + ], + "page_idx": 21 + }, + { + "type": "image", + "img_path": "images/5dfce0c0db48409f890ffe00174597da457e40c2ba659bd24f1e455836d5c726.jpg", + "image_caption": [ + "Figure 17: Context window size dependence for the action distillation $( s \\mathsf { a r } - > \\mathsf { a } )$ and probability distillation $( \\mathsf { s a r - > p } )$ variants of AD on Dark Key-to-Door. Mean $\\pm \\nobreakspace 1 \\nobreakspace$ standard deviation over 5 training seeds and 20 evaluation seeds. 1000 episodes corresponds to $5 0 \\mathrm { k }$ environment steps. " + ], + "image_footnote": [], + "bbox": [ + 205, + 159, + 787, + 329 + ], + "page_idx": 21 + }, + { + "type": "table", + "img_path": "images/f514dd11d832e9574a0edcf5736f3674ccebfff9a4e8e1f06f255808a86ab7b0.jpg", + "table_caption": [ + "N ALGORITHM DISTILLATION HYPERPARAMETERS " + ], + "table_footnote": [], + "table_body": "
HyperparameterDark RoomDark Room (Hard)Dark Key-to-DoorWatermaze
Embedding Dim.64
Number of Layers
Number of Heads4
Feedforward Dim.2048
Position EncodingsAbsolute
LayerNorm PlacementPost Norm
Dropout Rate0.1
Context Window600 tokens (200 timesteps)
Attention Dropout Rate00.50.5
Sequence Mask Prob0.5 0.30.50.30.3
Label Smoothing α00.200
", + "bbox": [ + 209, + 136, + 794, + 301 + ], + "page_idx": 22 + }, + { + "type": "table", + "img_path": "images/e8ac6cf6e32ba0ecc1c4f3c580cb0fb324bbb11f3adf0615ff00783323d68fd8.jpg", + "table_caption": [ + "Table 1: Algorithm Distillation Architecture Hyperparameters. " + ], + "table_footnote": [], + "table_body": "
HyperparameterValue
Batch Size128
OptimizerAdam
β0.9
0.99
Gradient Clip Norm Threshold1
Learning Rate ScheduleCosine Decay
Initial Value2e-6
Peak Value3e-4
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LayerHyperparameterValue
Conv Block
ConvChannel128
Kernel5
Stride2
BatchNormDecay Rate0.999
Activationeps1e-5
-ReLU
Max PoolingKernel2
Stride2
DropoutRate0.2
Network
Conv Blocks=3
FinalLinearLayerUnits256
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Hyperparameter Value
Batch Size (Num.Actors)100
入0.95
Agent Discount0.99
Entropy Bonus Weight0.01
MLPLayers3
MLPHidden Dim128
OptimizerAdam
β0.9
β20.999
E1e-6
Learning Rate1e-4
", + "bbox": [ + 385, + 162, + 614, + 327 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "O.2 DMLAB WATERMAZE ", + "text_level": 1, + "bbox": [ + 174, + 380, + 375, + 395 + ], + "page_idx": 23 + }, + { + "type": "table", + "img_path": "images/c6208479610944fd2e323eb9eb8f9e52bd2ee673a26b8d09ab2e3b658afafaa8.jpg", + "table_caption": [ + "Table 5: Source $\\mathrm { D Q N } ( \\mathrm { Q } - \\lambda )$ Algorithm Hyperparameters for Watermaze. " + ], + "table_footnote": [], + "table_body": "
HyperparameterValue
Batch Size8
Rollout Length40
Rollout Overlap31
Number of Actors16
Reply Buffer Capacity1e5
Offline Data Fraction0.7
入0.75
E0.01
Agent Discount0.9
Target Update Period50
ResNet Channels[32, 64, 64]
ResNetKernels[3,3,3]
ResNet Strides[1, 1, 1]
Pool Kernels[3,3,3]
Pool Strides[2,2,2]
OptimizerAdam
β0.9
阳0.999
E1e-6
Gradient Clip Norm Threshold10
Learning Rate1e-4
", + "bbox": [ + 354, + 409, + 645, + 710 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "P RL2 HYPERPARAMETERS ", + "text_level": 1, + "bbox": [ + 174, + 101, + 416, + 118 + ], + "page_idx": 24 + }, + { + "type": "table", + "img_path": "images/01ed28fe8446e8cdcd1095ccb00ddd17165acbb87d9edf991ea013f83836d792.jpg", + "table_caption": [ + "Table 6: RL2 Hyperparameters used in “Dark” Environments. " + ], + "table_footnote": [], + "table_body": "
HyperparameterValue
RL AlgorithmA3C
Learning Rate3e-4
Batch Size256
Unroll Length20
LSTM Hidden Dim. LSTMNumber ofLayers256
Episodes Per Trial2 10
", + "bbox": [ + 375, + 137, + 622, + 265 + ], + "page_idx": 24 + }, + { + "type": "table", + "img_path": "images/0b34856e782e56abcd8b605cce7e941a355371610c7746391fae238ffcd69ea8.jpg", + "table_caption": [], + "table_footnote": [ + "Table 7: RL2 Hyperparameters used in the Watermaze Environment. " + ], + "table_body": "
HyperparameterValue
RL AlgorithmDQN(Q-入)
Learning Rate1e-4
Batch Size96
Unroll Length40
LSTM Hidden Dim.256
LSTMNumber of Layers1
Episodes Per Trial30
", + "bbox": [ + 356, + 310, + 640, + 439 + ], + "page_idx": 24 + }, + { + "type": "image", + "img_path": "images/ea1fc762f193fd44d789c64e6914ec2d901af77612e21bb3dcbce6fbfe7da485.jpg", + "image_caption": [ + "0 1700Figure 18: Attention maps for AD from five separate seeds. White and gray colors correspond to low and high attention. Red and blue colors indicate that those transitions correspond to an episode restart and a positive reward token, respectively. The left column plots attention for an AD transformer after 200 time-steps of evaluation (when the context is initially filled). The right column plots attention after 1900 steps (38 episodes) of evaluation. Each episode has a length of 50 steps. From these patterns, it is evident that AD attends to tokens across several episodes to predict its next action. 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A striking property of pre-trained transformers is their ability to adapt to downstream", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 668, + 506, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 506, + 681 + ], + "score": 1.0, + "content": "tasks through prompt conditioning or in-context learning. 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Algorithm Distillation treats learning to", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 268, + 470, + 279 + ], + "spans": [ + { + "bbox": [ + 141, + 268, + 470, + 279 + ], + "score": 1.0, + "content": "reinforcement learn as an across-episode sequential prediction problem. A dataset", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 277, + 470, + 291 + ], + "spans": [ + { + "bbox": [ + 141, + 277, + 470, + 291 + ], + "score": 1.0, + "content": "of learning histories is generated by a source RL algorithm, and then a causal", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 290, + 469, + 302 + ], + "spans": [ + { + "bbox": [ + 141, + 290, + 469, + 302 + ], + "score": 1.0, + "content": "transformer is trained by autoregressively predicting actions given their preceding", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 300, + 470, + 313 + ], + "spans": [ + { + "bbox": [ + 141, + 300, + 470, + 313 + ], + "score": 1.0, + "content": "learning histories as context. Unlike sequential policy prediction architectures that", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 311, + 470, + 324 + ], + "spans": [ + { + "bbox": [ + 141, + 311, + 470, + 324 + ], + "score": 1.0, + "content": "distill post-learning or expert sequences, AD is able to improve its policy entirely", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 322, + 469, + 334 + ], + "spans": [ + { + "bbox": [ + 141, + 322, + 469, + 334 + ], + "score": 1.0, + "content": "in-context without updating its network parameters. We demonstrate that AD can", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 333, + 470, + 345 + ], + "spans": [ + { + "bbox": [ + 141, + 333, + 470, + 345 + ], + "score": 1.0, + "content": "reinforcement learn in-context in a variety of environments with sparse rewards,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 142, + 344, + 469, + 356 + ], + "spans": [ + { + "bbox": [ + 142, + 344, + 469, + 356 + ], + "score": 1.0, + "content": "combinatorial task structure, and pixel-based observations, and find that AD learns", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 355, + 458, + 367 + ], + "spans": [ + { + "bbox": [ + 141, + 355, + 458, + 367 + ], + "score": 1.0, + "content": "a more data-efficient RL algorithm than the one that generated the source data.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 12.5, + "bbox_fs": [ + 141, + 234, + 470, + 367 + ] + }, + { + "type": "image", + "bbox": [ + 141, + 391, + 469, + 531 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 141, + 391, + 469, + 531 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 141, + 391, + 469, + 531 + ], + "spans": [ + { + "bbox": [ + 141, + 391, + 469, + 531 + ], + "score": 0.967, + "type": "image", + "image_path": "679c16b167e631679ba20610706e8543cd9d762d7dab80268efc9673f356c188.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 141, + 391, + 469, + 437.6666666666667 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 141, + 437.6666666666667, + 469, + 484.33333333333337 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 141, + 484.33333333333337, + 469, + 531.0 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 547, + 506, + 598 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 547, + 506, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 506, + 559 + ], + "score": 1.0, + "content": "Figure 1: Algorithm Distillation (AD) has two steps – (i) a dataset of learning histories is collected from", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 558, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 106, + 558, + 505, + 569 + ], + "score": 1.0, + "content": "individual single-task RL algorithms solving different tasks; (ii) a causal transformer predicts actions from", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 567, + 506, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 506, + 580 + ], + "score": 1.0, + "content": "these histories using across-episodic contexts. 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A striking property of pre-trained transformers is their ability to adapt to downstream", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 668, + 506, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 506, + 681 + ], + "score": 1.0, + "content": "tasks through prompt conditioning or in-context learning. After pre-training on large offline datasets,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 678, + 505, + 692 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 505, + 692 + ], + "score": 1.0, + "content": "large transformers have been shown to generalize to downstream tasks in text completion (Brown", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 690, + 499, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 690, + 499, + 703 + ], + "score": 1.0, + "content": "et al., 2020), language understanding (Devlin et al., 2018), and image generation (Yu et al., 2022).", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 646, + 506, + 703 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 182 + ], + "lines": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "Recent work demonstrated that transformers can also learn policies from offline data by treating offline", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 505, + 105 + ], + "score": 1.0, + "content": "Reinforcement Learning (RL) as a sequential prediction problem. While Chen et al. (2021) showed", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 506, + 117 + ], + "score": 1.0, + "content": "that transformers can learn single-task policies from offline RL data via imitation learning, subsequent", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 505, + 127 + ], + "score": 1.0, + "content": "work showed that transformers can also extract multi-task policies in both same-domain (Lee et al.,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 125, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 125, + 505, + 140 + ], + "score": 1.0, + "content": "2022) and cross-domain settings (Reed et al., 2022). These works suggest a promising paradigm", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "score": 1.0, + "content": "for extracting generalist multi-task policies – first collect a large and diverse dataset of environment", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 148, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 505, + 161 + ], + "score": 1.0, + "content": "interactions, then extract a policy from the data via sequential modeling. We refer to the family", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 158, + 505, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 505, + 173 + ], + "score": 1.0, + "content": "of approaches that learns policies from offline RL data via imitation learning as Offline Policy", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 171, + 299, + 182 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 299, + 182 + ], + "score": 1.0, + "content": "Distillation, or simply Policy Distillation1 (PD).", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 107, + 187, + 505, + 275 + ], + "lines": [ + { + "bbox": [ + 106, + 187, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 505, + 199 + ], + "score": 1.0, + "content": "Despite its simplicity and scalability, a substantial drawback of PD is that the resulting policy does", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "score": 1.0, + "content": "not improve incrementally from additional interaction with the environment. For instance, the Multi-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 208, + 505, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 505, + 223 + ], + "score": 1.0, + "content": "Game Decision Transformer (MGDT, Lee et al., 2022) learns a return-conditioned policy that plays", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 221, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 221, + 505, + 232 + ], + "score": 1.0, + "content": "many Atari games while Gato (Reed et al., 2022) learns a policy that solves tasks across diverse", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 231, + 506, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 506, + 243 + ], + "score": 1.0, + "content": "environments by inferring tasks through context, but neither method can improve its policy in-context", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 241, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 505, + 254 + ], + "score": 1.0, + "content": "through trial and error. MGDT adapts the transformer to new tasks by finetuning the model weights", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 253, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 505, + 265 + ], + "score": 1.0, + "content": "while Gato requires prompting with an expert demonstration to adapt to a new task. In short, Policy", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 263, + 424, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 424, + 276 + ], + "score": 1.0, + "content": "Distillation methods learn policies but not Reinforcement Learning algorithms.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 12.5 + }, + { + "type": "text", + "bbox": [ + 107, + 281, + 505, + 325 + ], + "lines": [ + { + "bbox": [ + 105, + 281, + 506, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 506, + 294 + ], + "score": 1.0, + "content": "We hypothesize that the reason Policy Distillation does not improve through trial and error is that it", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 292, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 106, + 292, + 505, + 304 + ], + "score": 1.0, + "content": "trains on data that does not show learning progress. Current methods either learn policies from data", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 302, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 505, + 316 + ], + "score": 1.0, + "content": "that contains no learning (e.g. by distilling fixed expert policies) or data with learning (e.g. the replay", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 313, + 482, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 482, + 326 + ], + "score": 1.0, + "content": "buffer of an RL agent) but with a context size that is too small to capture policy improvement.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 107, + 330, + 505, + 430 + ], + "lines": [ + { + "bbox": [ + 106, + 329, + 506, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 506, + 343 + ], + "score": 1.0, + "content": "Our key observation is that the sequential nature of learning within RL algorithm training could,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 342, + 506, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 506, + 354 + ], + "score": 1.0, + "content": "in principle, make it possible to model the process of reinforcement learning itself as a causal", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 352, + 506, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 506, + 367 + ], + "score": 1.0, + "content": "sequence prediction problem. Specifically, if a transformer’s context is long enough to include policy", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 363, + 506, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 506, + 376 + ], + "score": 1.0, + "content": "improvement due to learning updates it should be able to represent not only a fixed policy but a", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 374, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 505, + 387 + ], + "score": 1.0, + "content": "policy improvement operator by attending to states, actions and rewards from previous episodes. This", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 385, + 506, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 506, + 399 + ], + "score": 1.0, + "content": "opens the possibility that any RL algorithm can be distilled into a sufficiently powerful sequence", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 396, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 505, + 410 + ], + "score": 1.0, + "content": "model such as a transformer via imitation learning, converting it into an in-context RL algorithm. By", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 408, + 504, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 408, + 504, + 419 + ], + "score": 1.0, + "content": "in-context RL we mean that the transformer should improve its policy through trial and error within", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 418, + 412, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 412, + 432 + ], + "score": 1.0, + "content": "the environment by attending to its context, without updating its parameters.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 435, + 505, + 534 + ], + "lines": [ + { + "bbox": [ + 106, + 435, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 106, + 435, + 505, + 447 + ], + "score": 1.0, + "content": "We present Algorithm Distillation (AD), a method that learns an in-context policy improvement", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 446, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 506, + 459 + ], + "score": 1.0, + "content": "operator by optimizing a causal sequence prediction loss on the learning histories of an RL algorithm.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 458, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 505, + 470 + ], + "score": 1.0, + "content": "AD has two components. First, a large multi-task dataset is generated by saving the training histories", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 468, + 505, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 505, + 481 + ], + "score": 1.0, + "content": "of an RL algorithm on many individual tasks. Next, a transformer models actions causally using the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 478, + 506, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 506, + 493 + ], + "score": 1.0, + "content": "preceding learning history as its context. Since the policy improves throughout the course of training", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 489, + 505, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 505, + 503 + ], + "score": 1.0, + "content": "of the source RL algorithm, AD is forced to learn the improvement operator in order to accurately", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 500, + 506, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 506, + 515 + ], + "score": 1.0, + "content": "model the actions at any given point in the training history. Crucially, the transformer context size", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 511, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 525 + ], + "score": 1.0, + "content": "must be sufficiently large (i.e. across-episodic) to capture improvement in the training data. The full", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 523, + 215, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 215, + 536 + ], + "score": 1.0, + "content": "method is shown in Fig. 1.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 540, + 505, + 628 + ], + "lines": [ + { + "bbox": [ + 105, + 539, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 505, + 552 + ], + "score": 1.0, + "content": "We show that by imitating gradient-based RL algorithms using a causal transformer with sufficiently", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 551, + 506, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 506, + 563 + ], + "score": 1.0, + "content": "large contexts, AD can reinforcement learn new tasks entirely in-context. We evaluate AD across", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 561, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 505, + 574 + ], + "score": 1.0, + "content": "a number of partially observed environments that require exploration, including the pixel-based", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 573, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 106, + 573, + 505, + 584 + ], + "score": 1.0, + "content": "Watermaze (Morris, 1981) from DMLab (Beattie et al., 2016). We show that AD is capable of", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 584, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 505, + 596 + ], + "score": 1.0, + "content": "in-context exploration, temporal credit assignment, and generalization. We also show that AD learns", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 104, + 594, + 506, + 608 + ], + "spans": [ + { + "bbox": [ + 104, + 594, + 506, + 608 + ], + "score": 1.0, + "content": "a more data-efficient algorithm than the one that generated the source data for transformer training.", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 604, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 619 + ], + "score": 1.0, + "content": "To the best of our knowledge, AD is the first method to demonstrate in-context reinforcement learning", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 617, + 353, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 353, + 628 + ], + "score": 1.0, + "content": "via sequential modeling of offline data with an imitation loss.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 42.5 + }, + { + "type": "title", + "bbox": [ + 108, + 644, + 200, + 657 + ], + "lines": [ + { + "bbox": [ + 104, + 642, + 201, + 659 + ], + "spans": [ + { + "bbox": [ + 104, + 642, + 201, + 659 + ], + "score": 1.0, + "content": "2 BACKGROUND", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 47 + }, + { + "type": "text", + "bbox": [ + 107, + 669, + 505, + 703 + ], + "lines": [ + { + "bbox": [ + 105, + 669, + 506, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 669, + 506, + 681 + ], + "score": 1.0, + "content": "Partially Observable Markov Decision Processes: A Markov Decision Process (MDP) consists of", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 680, + 505, + 693 + ], + "spans": [ + { + "bbox": [ + 105, + 680, + 131, + 693 + ], + "score": 1.0, + "content": "states", + "type": "text" + }, + { + "bbox": [ + 131, + 681, + 155, + 690 + ], + "score": 0.88, + "content": "s \\in S", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 680, + 189, + 693 + ], + "score": 1.0, + "content": ", actions", + "type": "text" + }, + { + "bbox": [ + 190, + 681, + 216, + 691 + ], + "score": 0.9, + "content": "a \\in { \\mathcal { A } }", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 680, + 253, + 693 + ], + "score": 1.0, + "content": ", rewards", + "type": "text" + }, + { + "bbox": [ + 253, + 681, + 279, + 691 + ], + "score": 0.89, + "content": "r \\in \\mathcal { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 680, + 351, + 693 + ], + "score": 1.0, + "content": ", a discount factor", + "type": "text" + }, + { + "bbox": [ + 351, + 682, + 358, + 692 + ], + "score": 0.8, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 680, + 505, + 693 + ], + "score": 1.0, + "content": ", and a transition probability function", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 107, + 691, + 505, + 704 + ], + "spans": [ + { + "bbox": [ + 107, + 691, + 162, + 703 + ], + "score": 0.92, + "content": "p ( s _ { t + 1 } | s _ { t } , a _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 692, + 193, + 704 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 193, + 693, + 198, + 701 + ], + "score": 0.78, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 692, + 355, + 704 + ], + "score": 1.0, + "content": "is an integer denoting the timestep and", + "type": "text" + }, + { + "bbox": [ + 356, + 692, + 383, + 703 + ], + "score": 0.92, + "content": "( \\cal { S } , \\cal { A } )", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 692, + 505, + 704 + ], + "score": 1.0, + "content": "are state and action spaces. In", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 49 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 712, + 505, + 731 + ], + "lines": [ + { + "bbox": [ + 118, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 118, + 709, + 505, + 723 + ], + "score": 1.0, + "content": "1What we refer to as Policy Distillation is similar to Rusu et al. (2016) but the policy is distilled from offline", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 721, + 208, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 208, + 732 + ], + "score": 1.0, + "content": "data, not a teacher network.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2023", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 182 + ], + "lines": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "Recent work demonstrated that transformers can also learn policies from offline data by treating offline", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 505, + 105 + ], + "score": 1.0, + "content": "Reinforcement Learning (RL) as a sequential prediction problem. While Chen et al. (2021) showed", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 506, + 117 + ], + "score": 1.0, + "content": "that transformers can learn single-task policies from offline RL data via imitation learning, subsequent", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 505, + 127 + ], + "score": 1.0, + "content": "work showed that transformers can also extract multi-task policies in both same-domain (Lee et al.,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 125, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 125, + 505, + 140 + ], + "score": 1.0, + "content": "2022) and cross-domain settings (Reed et al., 2022). These works suggest a promising paradigm", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "score": 1.0, + "content": "for extracting generalist multi-task policies – first collect a large and diverse dataset of environment", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 148, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 505, + 161 + ], + "score": 1.0, + "content": "interactions, then extract a policy from the data via sequential modeling. We refer to the family", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 158, + 505, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 505, + 173 + ], + "score": 1.0, + "content": "of approaches that learns policies from offline RL data via imitation learning as Offline Policy", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 171, + 299, + 182 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 299, + 182 + ], + "score": 1.0, + "content": "Distillation, or simply Policy Distillation1 (PD).", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 4, + "bbox_fs": [ + 105, + 83, + 506, + 182 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 187, + 505, + 275 + ], + "lines": [ + { + "bbox": [ + 106, + 187, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 505, + 199 + ], + "score": 1.0, + "content": "Despite its simplicity and scalability, a substantial drawback of PD is that the resulting policy does", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "score": 1.0, + "content": "not improve incrementally from additional interaction with the environment. For instance, the Multi-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 208, + 505, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 505, + 223 + ], + "score": 1.0, + "content": "Game Decision Transformer (MGDT, Lee et al., 2022) learns a return-conditioned policy that plays", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 221, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 221, + 505, + 232 + ], + "score": 1.0, + "content": "many Atari games while Gato (Reed et al., 2022) learns a policy that solves tasks across diverse", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 231, + 506, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 506, + 243 + ], + "score": 1.0, + "content": "environments by inferring tasks through context, but neither method can improve its policy in-context", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 241, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 505, + 254 + ], + "score": 1.0, + "content": "through trial and error. MGDT adapts the transformer to new tasks by finetuning the model weights", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 253, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 505, + 265 + ], + "score": 1.0, + "content": "while Gato requires prompting with an expert demonstration to adapt to a new task. In short, Policy", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 263, + 424, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 424, + 276 + ], + "score": 1.0, + "content": "Distillation methods learn policies but not Reinforcement Learning algorithms.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 187, + 506, + 276 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 281, + 505, + 325 + ], + "lines": [ + { + "bbox": [ + 105, + 281, + 506, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 506, + 294 + ], + "score": 1.0, + "content": "We hypothesize that the reason Policy Distillation does not improve through trial and error is that it", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 292, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 106, + 292, + 505, + 304 + ], + "score": 1.0, + "content": "trains on data that does not show learning progress. Current methods either learn policies from data", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 302, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 505, + 316 + ], + "score": 1.0, + "content": "that contains no learning (e.g. by distilling fixed expert policies) or data with learning (e.g. the replay", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 313, + 482, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 482, + 326 + ], + "score": 1.0, + "content": "buffer of an RL agent) but with a context size that is too small to capture policy improvement.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 281, + 506, + 326 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 330, + 505, + 430 + ], + "lines": [ + { + "bbox": [ + 106, + 329, + 506, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 506, + 343 + ], + "score": 1.0, + "content": "Our key observation is that the sequential nature of learning within RL algorithm training could,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 342, + 506, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 506, + 354 + ], + "score": 1.0, + "content": "in principle, make it possible to model the process of reinforcement learning itself as a causal", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 352, + 506, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 506, + 367 + ], + "score": 1.0, + "content": "sequence prediction problem. Specifically, if a transformer’s context is long enough to include policy", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 363, + 506, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 506, + 376 + ], + "score": 1.0, + "content": "improvement due to learning updates it should be able to represent not only a fixed policy but a", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 374, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 505, + 387 + ], + "score": 1.0, + "content": "policy improvement operator by attending to states, actions and rewards from previous episodes. This", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 385, + 506, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 506, + 399 + ], + "score": 1.0, + "content": "opens the possibility that any RL algorithm can be distilled into a sufficiently powerful sequence", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 396, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 505, + 410 + ], + "score": 1.0, + "content": "model such as a transformer via imitation learning, converting it into an in-context RL algorithm. By", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 408, + 504, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 408, + 504, + 419 + ], + "score": 1.0, + "content": "in-context RL we mean that the transformer should improve its policy through trial and error within", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 418, + 412, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 412, + 432 + ], + "score": 1.0, + "content": "the environment by attending to its context, without updating its parameters.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 329, + 506, + 432 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 435, + 505, + 534 + ], + "lines": [ + { + "bbox": [ + 106, + 435, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 106, + 435, + 505, + 447 + ], + "score": 1.0, + "content": "We present Algorithm Distillation (AD), a method that learns an in-context policy improvement", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 446, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 506, + 459 + ], + "score": 1.0, + "content": "operator by optimizing a causal sequence prediction loss on the learning histories of an RL algorithm.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 458, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 505, + 470 + ], + "score": 1.0, + "content": "AD has two components. 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For", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 520, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 505, + 532 + ], + "score": 1.0, + "content": "example, large language models like GPT-3 (Brown et al., 2020) or Gopher (Rae et al., 2021) can", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 530, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 505, + 543 + ], + "score": 1.0, + "content": "be directed at solving tasks such as text completion, code generation, and text summarization by", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 542, + 506, + 554 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 506, + 554 + ], + "score": 1.0, + "content": "specifying the task through language as a prompt. 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Eqn. (3) suggests", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "that, similar to a policy, an algorithm can be unrolled in an environment to generate sequences of ob-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 505, + 116 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 389, + 116 + ], + "score": 1.0, + "content": "servations, rewards, and actions. 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Eqn. (3) suggests", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "that, similar to a policy, an algorithm can be unrolled in an environment to generate sequences of ob-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 505, + 116 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 389, + 116 + ], + "score": 1.0, + "content": "servations, rewards, and actions. 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We denote the dataset of", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 176, + 235, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 188, + 189 + ], + "score": 1.0, + "content": "learning histories as", + "type": "text" + }, + { + "bbox": [ + 189, + 177, + 198, + 186 + ], + "score": 0.8, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 176, + 235, + 189 + ], + "score": 1.0, + "content": "in Eq. 5.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 107, + 193, + 506, + 282 + ], + "lines": [ + { + "bbox": [ + 105, + 192, + 507, + 206 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 398, + 206 + ], + "score": 1.0, + "content": "Training the Sequence Model: Once a dataset of learning histories", + "type": "text" + }, + { + "bbox": [ + 399, + 194, + 408, + 203 + ], + "score": 0.78, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 192, + 507, + 206 + ], + "score": 1.0, + "content": "is collected, a sequen-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 204, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 106, + 204, + 505, + 217 + ], + "score": 1.0, + "content": "tial prediction model is trained to predict actions given the preceding histories. We utilize the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 215, + 506, + 227 + ], + "spans": [ + { + "bbox": [ + 106, + 215, + 506, + 227 + ], + "score": 1.0, + "content": "GPT (Radford et al., 2018) causal transformer model for sequential action prediction, but AD is", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 225, + 506, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 506, + 239 + ], + "score": 1.0, + "content": "compatible with any sequence model including RNNs (Williams & Zipser, 1989). For instance,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 237, + 506, + 249 + ], + "spans": [ + { + "bbox": [ + 106, + 237, + 196, + 249 + ], + "score": 1.0, + "content": "we show in Appendix", + "type": "text" + }, + { + "bbox": [ + 196, + 237, + 205, + 247 + ], + "score": 0.45, + "content": "\\mathrm { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 237, + 506, + 249 + ], + "score": 1.0, + "content": "that AD can also be achieved with an LSTM (Hochreiter & Schmidhuber,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 247, + 506, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 506, + 260 + ], + "score": 1.0, + "content": "1997), though less effectively than AD with causal transformers. Since causal transformer train-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 260, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 106, + 260, + 505, + 271 + ], + "score": 1.0, + "content": "ing and inference are quadratic in the sequence length, we sample across-episodic subsequences", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 268, + 493, + 284 + ], + "spans": [ + { + "bbox": [ + 106, + 270, + 259, + 282 + ], + "score": 0.9, + "content": "\\bar { h _ { j } } ^ { - } = \\left( o _ { j } , r _ { j } , a _ { j } \\ldots , o _ { j + c } , r _ { j + c } , a _ { j + c } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 268, + 298, + 284 + ], + "score": 1.0, + "content": "of length", + "type": "text" + }, + { + "bbox": [ + 299, + 270, + 325, + 280 + ], + "score": 0.9, + "content": "c < T", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 268, + 348, + 284 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 348, + 270, + 357, + 280 + ], + "score": 0.77, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 268, + 493, + 284 + ], + "score": 1.0, + "content": "rather than training full histories.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 12.5 + }, + { + "type": "title", + "bbox": [ + 108, + 298, + 244, + 311 + ], + "lines": [ + { + "bbox": [ + 105, + 297, + 245, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 245, + 313 + ], + "score": 1.0, + "content": "4 EXPERIMENTAL SETUP", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "title", + "bbox": [ + 108, + 325, + 201, + 336 + ], + "lines": [ + { + "bbox": [ + 105, + 324, + 202, + 337 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 202, + 337 + ], + "score": 1.0, + "content": "4.1 ENVIRONMENTS", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 346, + 364, + 434 + ], + "lines": [ + { + "bbox": [ + 106, + 346, + 366, + 358 + ], + "spans": [ + { + "bbox": [ + 106, + 346, + 366, + 358 + ], + "score": 1.0, + "content": "To investigate the in-context RL capabilities of AD and the base-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 357, + 364, + 368 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 364, + 368 + ], + "score": 1.0, + "content": "lines (see next section), we focus on environments that cannot be", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 368, + 365, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 368, + 365, + 380 + ], + "score": 1.0, + "content": "solved through zero-shot generalization after pre-training. Specif-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 379, + 366, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 366, + 391 + ], + "score": 1.0, + "content": "ically, we require that each environment supports many tasks,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 390, + 365, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 365, + 401 + ], + "score": 1.0, + "content": "that the tasks cannot be inferred easily from the observation, and", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 400, + 365, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 365, + 414 + ], + "score": 1.0, + "content": "that episodes are short enough to feasibly train across-episodic", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 412, + 364, + 424 + ], + "spans": [ + { + "bbox": [ + 106, + 412, + 364, + 424 + ], + "score": 1.0, + "content": "causal transformers - for more details regarding environments see", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 423, + 334, + 435 + ], + "spans": [ + { + "bbox": [ + 106, + 423, + 334, + 435 + ], + "score": 1.0, + "content": "Appendix B. We list the evaluation environments below:", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 107, + 439, + 365, + 505 + ], + "lines": [ + { + "bbox": [ + 106, + 439, + 364, + 451 + ], + "spans": [ + { + "bbox": [ + 106, + 439, + 364, + 451 + ], + "score": 1.0, + "content": "Adversarial Bandit: a multi-armed bandit with 10 arms and 100", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 449, + 365, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 293, + 462 + ], + "score": 1.0, + "content": "trials similar to the environment considered in", + "type": "text" + }, + { + "bbox": [ + 294, + 450, + 312, + 461 + ], + "score": 0.6, + "content": "\\mathrm { { R L ^ { 2 } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 449, + 365, + 462 + ], + "score": 1.0, + "content": "(Duan et al.,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 461, + 366, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 366, + 473 + ], + "score": 1.0, + "content": "2016). However, during evaluation the reward is out of distribu-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 472, + 365, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 472, + 332, + 484 + ], + "score": 1.0, + "content": "tion. Reward is more likely distributed under odd arms", + "type": "text" + }, + { + "bbox": [ + 333, + 472, + 352, + 483 + ], + "score": 0.88, + "content": "9 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 472, + 365, + 484 + ], + "score": 1.0, + "content": "of", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 483, + 365, + 495 + ], + "spans": [ + { + "bbox": [ + 106, + 483, + 365, + 495 + ], + "score": 1.0, + "content": "the time during training. At evaluation, the opposite happens -", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 494, + 352, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 283, + 506 + ], + "score": 1.0, + "content": "reward appears more often under even arms", + "type": "text" + }, + { + "bbox": [ + 283, + 495, + 303, + 505 + ], + "score": 0.89, + "content": "9 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 494, + 352, + 506 + ], + "score": 1.0, + "content": "of the time.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 29.5 + }, + { + "type": "image", + "bbox": [ + 384, + 360, + 492, + 441 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 384, + 360, + 492, + 441 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 384, + 360, + 492, + 441 + ], + "spans": [ + { + "bbox": [ + 384, + 360, + 492, + 441 + ], + "score": 0.969, + "type": "image", + "image_path": "a3da0f2d56a2dea7e0ac773cbc4072974da4d3b19ed7b19d036e66b3977c8b9e.jpg" + } + ] + } + ], + "index": 33.5, + "virtual_lines": [ + { + "bbox": [ + 384, + 360, + 492, + 400.5 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 384, + 400.5, + 492, + 441.0 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 371, + 450, + 505, + 490 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 371, + 450, + 506, + 461 + ], + "spans": [ + { + "bbox": [ + 371, + 450, + 506, + 461 + ], + "score": 1.0, + "content": "Figure 2: Agent view from the DM-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 371, + 460, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 371, + 460, + 505, + 471 + ], + "score": 1.0, + "content": "Lab Watermaze environment. The", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 371, + 470, + 505, + 481 + ], + "spans": [ + { + "bbox": [ + 371, + 470, + 505, + 481 + ], + "score": 1.0, + "content": "task is to find a hidden platform that", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 371, + 480, + 447, + 491 + ], + "spans": [ + { + "bbox": [ + 371, + 480, + 447, + 491 + ], + "score": 1.0, + "content": "elevates once found.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36.5 + } + ], + "index": 35.0 + }, + { + "type": "text", + "bbox": [ + 107, + 511, + 505, + 588 + ], + "lines": [ + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "score": 1.0, + "content": "Dark Room: a 2D discrete POMDP where an agent spawns in a room and must find a goal location.", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 522, + 506, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 228, + 536 + ], + "score": 1.0, + "content": "The agent only knows its own", + "type": "text" + }, + { + "bbox": [ + 228, + 523, + 252, + 534 + ], + "score": 0.91, + "content": "( x , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 522, + 506, + 536 + ], + "score": 1.0, + "content": "coordinates but does not know the goal location and must infer", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 532, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 250, + 546 + ], + "score": 1.0, + "content": "it from the reward. The room size is", + "type": "text" + }, + { + "bbox": [ + 250, + 534, + 274, + 544 + ], + "score": 0.9, + "content": "9 \\times 9", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 532, + 506, + 546 + ], + "score": 1.0, + "content": ", the possible actions are one step left, right, up, down, and", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 104, + 544, + 506, + 557 + ], + "spans": [ + { + "bbox": [ + 104, + 544, + 506, + 557 + ], + "score": 1.0, + "content": "no-op, the episode length is 20, and the agent resets at the center of the map. We test two environment", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 555, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 294, + 568 + ], + "score": 1.0, + "content": "variants – Dark Room where the agent receives", + "type": "text" + }, + { + "bbox": [ + 294, + 556, + 319, + 565 + ], + "score": 0.89, + "content": "r = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 555, + 505, + 568 + ], + "score": 1.0, + "content": "every time the goal is reached and Dark Room", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 566, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 271, + 578 + ], + "score": 1.0, + "content": "Hard, a hard exploration variant with a", + "type": "text" + }, + { + "bbox": [ + 271, + 566, + 306, + 577 + ], + "score": 0.9, + "content": "1 7 \\times 1 7", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 566, + 407, + 578 + ], + "score": 1.0, + "content": "size and sparse reward", + "type": "text" + }, + { + "bbox": [ + 407, + 567, + 434, + 577 + ], + "score": 0.88, + "content": "\\mathbf { \\boldsymbol { r } } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 566, + 505, + 578 + ], + "score": 1.0, + "content": "exactly once for", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 577, + 302, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 225, + 590 + ], + "score": 1.0, + "content": "reaching the goal). When not", + "type": "text" + }, + { + "bbox": [ + 226, + 578, + 249, + 587 + ], + "score": 0.89, + "content": "r = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 577, + 273, + 590 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 273, + 578, + 297, + 587 + ], + "score": 0.89, + "content": "r = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 577, + 302, + 590 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 106, + 594, + 505, + 660 + ], + "lines": [ + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "Dark Key-to-Door: similar to Dark Room but this environment requires an agent to first find an", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 604, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 304, + 618 + ], + "score": 1.0, + "content": "invisible key upon which it receives a reward of", + "type": "text" + }, + { + "bbox": [ + 304, + 605, + 329, + 615 + ], + "score": 0.9, + "content": "r = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 604, + 505, + 618 + ], + "score": 1.0, + "content": "once and then open an invisible door upon", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 616, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 219, + 628 + ], + "score": 1.0, + "content": "which it receives a reward of", + "type": "text" + }, + { + "bbox": [ + 219, + 617, + 244, + 626 + ], + "score": 0.9, + "content": "r = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 616, + 388, + 628 + ], + "score": 1.0, + "content": "once again. Otherwise, the reward is", + "type": "text" + }, + { + "bbox": [ + 388, + 617, + 412, + 626 + ], + "score": 0.9, + "content": "r = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 616, + 483, + 628 + ], + "score": 1.0, + "content": ". The room size is", + "type": "text" + }, + { + "bbox": [ + 483, + 616, + 505, + 627 + ], + "score": 0.89, + "content": "9 \\times 9", + "type": "inline_equation" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 626, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 273, + 639 + ], + "score": 1.0, + "content": "making the task space combinatorial with", + "type": "text" + }, + { + "bbox": [ + 273, + 626, + 322, + 637 + ], + "score": 0.91, + "content": "8 1 ^ { 2 } = 6 5 6 1", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 626, + 506, + 639 + ], + "score": 1.0, + "content": "possible tasks. This environment is similar to", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 638, + 507, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 507, + 650 + ], + "score": 1.0, + "content": "the one considered in Chen et al. (2021) except the key and door are invisible and the reward is semi-", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 648, + 507, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 136, + 662 + ], + "score": 1.0, + "content": "sparse", + "type": "text" + }, + { + "bbox": [ + 136, + 649, + 161, + 659 + ], + "score": 0.87, + "content": "\\boldsymbol { r } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 648, + 507, + 662 + ], + "score": 1.0, + "content": "for both key and the door). The agent is randomly reset. The episode length is 50 steps.", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 48.5 + }, + { + "type": "text", + "bbox": [ + 107, + 666, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 666, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 505, + 677 + ], + "score": 1.0, + "content": "DMLab Watermaze: a partially observable 3D visual DMLab environment based on the classic", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 106, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "Morris Watermaze (Morris, 1981). The task is to navigate the water maze to find a randomly spawned", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 106, + 688, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 506, + 700 + ], + "score": 1.0, + "content": "trap-door. The maze walls have color patterns that can be used to remember the goal location.", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 260, + 712 + ], + "score": 1.0, + "content": "Observations are pixel images of size", + "type": "text" + }, + { + "bbox": [ + 261, + 699, + 311, + 709 + ], + "score": 0.91, + "content": "7 2 \\times 9 6 \\times 3", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 698, + 505, + 712 + ], + "score": 1.0, + "content": ". There are 8 possible actions in total, including", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "score": 1.0, + "content": "going forward, backward, left, or right, rotating left or right, and rotating left or right while going", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 106, + 721, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 732 + ], + "score": 1.0, + "content": "forward. The episode length is 50, and the agent resets at the center of the map. Similar to Dark", + "type": "text" + } + ], + "index": 57 + } + ], + "index": 54.5 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2023", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "score": 1.0, + "content": "5", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 504, + 105 + ], + "lines": [], + "index": 0.5, + "bbox_fs": [ + 105, + 82, + 505, + 105 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 110, + 505, + 188 + ], + "lines": [ + { + "bbox": [ + 105, + 110, + 506, + 123 + ], + "spans": [ + { + "bbox": [ + 105, + 110, + 406, + 123 + ], + "score": 1.0, + "content": "Dataset Generation: A dataset of learning histories is collected by training", + "type": "text" + }, + { + "bbox": [ + 407, + 111, + 417, + 121 + ], + "score": 0.8, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 110, + 506, + 123 + ], + "score": 1.0, + "content": "individual single-task", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 122, + 505, + 134 + ], + "spans": [ + { + "bbox": [ + 106, + 122, + 505, + 134 + ], + "score": 1.0, + "content": "gradient-based RL algorithms. To prevent overfitting to any specific task during sequence model", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 133, + 505, + 144 + ], + "spans": [ + { + "bbox": [ + 106, + 133, + 167, + 144 + ], + "score": 1.0, + "content": "training, a task", + "type": "text" + }, + { + "bbox": [ + 167, + 133, + 180, + 142 + ], + "score": 0.75, + "content": "\\mathcal { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 133, + 505, + 144 + ], + "score": 1.0, + "content": "is sampled randomly from a task distribution for each RL run. The data generation", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 144, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 505, + 156 + ], + "score": 1.0, + "content": "step is RL algorithm agnostic - any RL algorithm can be distilled. We show results distilling UCB", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "score": 1.0, + "content": "exploration (Lai & Robbins, 1985), an on-policy actor-critic (Mnih et al., 2016), and an off-policy", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 506, + 177 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 506, + 177 + ], + "score": 1.0, + "content": "DQN (Mnih et al., 2013), in both distributed and single-stream settings. We denote the dataset of", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 176, + 235, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 188, + 189 + ], + "score": 1.0, + "content": "learning histories as", + "type": "text" + }, + { + "bbox": [ + 189, + 177, + 198, + 186 + ], + "score": 0.8, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 176, + 235, + 189 + ], + "score": 1.0, + "content": "in Eq. 5.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5, + "bbox_fs": [ + 105, + 110, + 506, + 189 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 193, + 506, + 282 + ], + "lines": [ + { + "bbox": [ + 105, + 192, + 507, + 206 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 398, + 206 + ], + "score": 1.0, + "content": "Training the Sequence Model: Once a dataset of learning histories", + "type": "text" + }, + { + "bbox": [ + 399, + 194, + 408, + 203 + ], + "score": 0.78, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 192, + 507, + 206 + ], + "score": 1.0, + "content": "is collected, a sequen-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 204, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 106, + 204, + 505, + 217 + ], + "score": 1.0, + "content": "tial prediction model is trained to predict actions given the preceding histories. We utilize the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 215, + 506, + 227 + ], + "spans": [ + { + "bbox": [ + 106, + 215, + 506, + 227 + ], + "score": 1.0, + "content": "GPT (Radford et al., 2018) causal transformer model for sequential action prediction, but AD is", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 225, + 506, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 506, + 239 + ], + "score": 1.0, + "content": "compatible with any sequence model including RNNs (Williams & Zipser, 1989). For instance,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 237, + 506, + 249 + ], + "spans": [ + { + "bbox": [ + 106, + 237, + 196, + 249 + ], + "score": 1.0, + "content": "we show in Appendix", + "type": "text" + }, + { + "bbox": [ + 196, + 237, + 205, + 247 + ], + "score": 0.45, + "content": "\\mathrm { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 237, + 506, + 249 + ], + "score": 1.0, + "content": "that AD can also be achieved with an LSTM (Hochreiter & Schmidhuber,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 247, + 506, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 506, + 260 + ], + "score": 1.0, + "content": "1997), though less effectively than AD with causal transformers. Since causal transformer train-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 260, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 106, + 260, + 505, + 271 + ], + "score": 1.0, + "content": "ing and inference are quadratic in the sequence length, we sample across-episodic subsequences", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 268, + 493, + 284 + ], + "spans": [ + { + "bbox": [ + 106, + 270, + 259, + 282 + ], + "score": 0.9, + "content": "\\bar { h _ { j } } ^ { - } = \\left( o _ { j } , r _ { j } , a _ { j } \\ldots , o _ { j + c } , r _ { j + c } , a _ { j + c } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 268, + 298, + 284 + ], + "score": 1.0, + "content": "of length", + "type": "text" + }, + { + "bbox": [ + 299, + 270, + 325, + 280 + ], + "score": 0.9, + "content": "c < T", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 268, + 348, + 284 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 348, + 270, + 357, + 280 + ], + "score": 0.77, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 268, + 493, + 284 + ], + "score": 1.0, + "content": "rather than training full histories.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 192, + 507, + 284 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 298, + 244, + 311 + ], + "lines": [ + { + "bbox": [ + 105, + 297, + 245, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 245, + 313 + ], + "score": 1.0, + "content": "4 EXPERIMENTAL SETUP", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "title", + "bbox": [ + 108, + 325, + 201, + 336 + ], + "lines": [ + { + "bbox": [ + 105, + 324, + 202, + 337 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 202, + 337 + ], + "score": 1.0, + "content": "4.1 ENVIRONMENTS", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 346, + 364, + 434 + ], + "lines": [ + { + "bbox": [ + 106, + 346, + 366, + 358 + ], + "spans": [ + { + "bbox": [ + 106, + 346, + 366, + 358 + ], + "score": 1.0, + "content": "To investigate the in-context RL capabilities of AD and the base-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 357, + 364, + 368 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 364, + 368 + ], + "score": 1.0, + "content": "lines (see next section), we focus on environments that cannot be", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 368, + 365, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 368, + 365, + 380 + ], + "score": 1.0, + "content": "solved through zero-shot generalization after pre-training. Specif-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 379, + 366, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 366, + 391 + ], + "score": 1.0, + "content": "ically, we require that each environment supports many tasks,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 390, + 365, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 365, + 401 + ], + "score": 1.0, + "content": "that the tasks cannot be inferred easily from the observation, and", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 400, + 365, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 365, + 414 + ], + "score": 1.0, + "content": "that episodes are short enough to feasibly train across-episodic", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 412, + 364, + 424 + ], + "spans": [ + { + "bbox": [ + 106, + 412, + 364, + 424 + ], + "score": 1.0, + "content": "causal transformers - for more details regarding environments see", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 423, + 334, + 435 + ], + "spans": [ + { + "bbox": [ + 106, + 423, + 334, + 435 + ], + "score": 1.0, + "content": "Appendix B. We list the evaluation environments below:", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 346, + 366, + 435 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 439, + 365, + 505 + ], + "lines": [ + { + "bbox": [ + 106, + 439, + 364, + 451 + ], + "spans": [ + { + "bbox": [ + 106, + 439, + 364, + 451 + ], + "score": 1.0, + "content": "Adversarial Bandit: a multi-armed bandit with 10 arms and 100", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 449, + 365, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 293, + 462 + ], + "score": 1.0, + "content": "trials similar to the environment considered in", + "type": "text" + }, + { + "bbox": [ + 294, + 450, + 312, + 461 + ], + "score": 0.6, + "content": "\\mathrm { { R L ^ { 2 } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 449, + 365, + 462 + ], + "score": 1.0, + "content": "(Duan et al.,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 461, + 366, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 366, + 473 + ], + "score": 1.0, + "content": "2016). However, during evaluation the reward is out of distribu-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 472, + 365, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 472, + 332, + 484 + ], + "score": 1.0, + "content": "tion. Reward is more likely distributed under odd arms", + "type": "text" + }, + { + "bbox": [ + 333, + 472, + 352, + 483 + ], + "score": 0.88, + "content": "9 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 472, + 365, + 484 + ], + "score": 1.0, + "content": "of", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 483, + 365, + 495 + ], + "spans": [ + { + "bbox": [ + 106, + 483, + 365, + 495 + ], + "score": 1.0, + "content": "the time during training. At evaluation, the opposite happens -", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 494, + 352, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 283, + 506 + ], + "score": 1.0, + "content": "reward appears more often under even arms", + "type": "text" + }, + { + "bbox": [ + 283, + 495, + 303, + 505 + ], + "score": 0.89, + "content": "9 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 494, + 352, + 506 + ], + "score": 1.0, + "content": "of the time.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 439, + 366, + 506 + ] + }, + { + "type": "image", + "bbox": [ + 384, + 360, + 492, + 441 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 384, + 360, + 492, + 441 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 384, + 360, + 492, + 441 + ], + "spans": [ + { + "bbox": [ + 384, + 360, + 492, + 441 + ], + "score": 0.969, + "type": "image", + "image_path": "a3da0f2d56a2dea7e0ac773cbc4072974da4d3b19ed7b19d036e66b3977c8b9e.jpg" + } + ] + } + ], + "index": 33.5, + "virtual_lines": [ + { + "bbox": [ + 384, + 360, + 492, + 400.5 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 384, + 400.5, + 492, + 441.0 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 371, + 450, + 505, + 490 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 371, + 450, + 506, + 461 + ], + "spans": [ + { + "bbox": [ + 371, + 450, + 506, + 461 + ], + "score": 1.0, + "content": "Figure 2: Agent view from the DM-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 371, + 460, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 371, + 460, + 505, + 471 + ], + "score": 1.0, + "content": "Lab Watermaze environment. The", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 371, + 470, + 505, + 481 + ], + "spans": [ + { + "bbox": [ + 371, + 470, + 505, + 481 + ], + "score": 1.0, + "content": "task is to find a hidden platform that", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 371, + 480, + 447, + 491 + ], + "spans": [ + { + "bbox": [ + 371, + 480, + 447, + 491 + ], + "score": 1.0, + "content": "elevates once found.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36.5 + } + ], + "index": 35.0 + }, + { + "type": "text", + "bbox": [ + 107, + 511, + 505, + 588 + ], + "lines": [ + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "score": 1.0, + "content": "Dark Room: a 2D discrete POMDP where an agent spawns in a room and must find a goal location.", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 522, + 506, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 228, + 536 + ], + "score": 1.0, + "content": "The agent only knows its own", + "type": "text" + }, + { + "bbox": [ + 228, + 523, + 252, + 534 + ], + "score": 0.91, + "content": "( x , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 522, + 506, + 536 + ], + "score": 1.0, + "content": "coordinates but does not know the goal location and must infer", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 532, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 250, + 546 + ], + "score": 1.0, + "content": "it from the reward. The room size is", + "type": "text" + }, + { + "bbox": [ + 250, + 534, + 274, + 544 + ], + "score": 0.9, + "content": "9 \\times 9", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 532, + 506, + 546 + ], + "score": 1.0, + "content": ", the possible actions are one step left, right, up, down, and", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 104, + 544, + 506, + 557 + ], + "spans": [ + { + "bbox": [ + 104, + 544, + 506, + 557 + ], + "score": 1.0, + "content": "no-op, the episode length is 20, and the agent resets at the center of the map. We test two environment", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 555, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 294, + 568 + ], + "score": 1.0, + "content": "variants – Dark Room where the agent receives", + "type": "text" + }, + { + "bbox": [ + 294, + 556, + 319, + 565 + ], + "score": 0.89, + "content": "r = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 555, + 505, + 568 + ], + "score": 1.0, + "content": "every time the goal is reached and Dark Room", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 566, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 271, + 578 + ], + "score": 1.0, + "content": "Hard, a hard exploration variant with a", + "type": "text" + }, + { + "bbox": [ + 271, + 566, + 306, + 577 + ], + "score": 0.9, + "content": "1 7 \\times 1 7", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 566, + 407, + 578 + ], + "score": 1.0, + "content": "size and sparse reward", + "type": "text" + }, + { + "bbox": [ + 407, + 567, + 434, + 577 + ], + "score": 0.88, + "content": "\\mathbf { \\boldsymbol { r } } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 566, + 505, + 578 + ], + "score": 1.0, + "content": "exactly once for", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 577, + 302, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 225, + 590 + ], + "score": 1.0, + "content": "reaching the goal). When not", + "type": "text" + }, + { + "bbox": [ + 226, + 578, + 249, + 587 + ], + "score": 0.89, + "content": "r = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 577, + 273, + 590 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 273, + 578, + 297, + 587 + ], + "score": 0.89, + "content": "r = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 577, + 302, + 590 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 42, + "bbox_fs": [ + 104, + 511, + 506, + 590 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 594, + 505, + 660 + ], + "lines": [ + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "Dark Key-to-Door: similar to Dark Room but this environment requires an agent to first find an", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 604, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 304, + 618 + ], + "score": 1.0, + "content": "invisible key upon which it receives a reward of", + "type": "text" + }, + { + "bbox": [ + 304, + 605, + 329, + 615 + ], + "score": 0.9, + "content": "r = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 604, + 505, + 618 + ], + "score": 1.0, + "content": "once and then open an invisible door upon", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 616, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 219, + 628 + ], + "score": 1.0, + "content": "which it receives a reward of", + "type": "text" + }, + { + "bbox": [ + 219, + 617, + 244, + 626 + ], + "score": 0.9, + "content": "r = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 616, + 388, + 628 + ], + "score": 1.0, + "content": "once again. Otherwise, the reward is", + "type": "text" + }, + { + "bbox": [ + 388, + 617, + 412, + 626 + ], + "score": 0.9, + "content": "r = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 616, + 483, + 628 + ], + "score": 1.0, + "content": ". The room size is", + "type": "text" + }, + { + "bbox": [ + 483, + 616, + 505, + 627 + ], + "score": 0.89, + "content": "9 \\times 9", + "type": "inline_equation" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 626, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 273, + 639 + ], + "score": 1.0, + "content": "making the task space combinatorial with", + "type": "text" + }, + { + "bbox": [ + 273, + 626, + 322, + 637 + ], + "score": 0.91, + "content": "8 1 ^ { 2 } = 6 5 6 1", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 626, + 506, + 639 + ], + "score": 1.0, + "content": "possible tasks. This environment is similar to", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 638, + 507, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 507, + 650 + ], + "score": 1.0, + "content": "the one considered in Chen et al. (2021) except the key and door are invisible and the reward is semi-", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 648, + 507, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 136, + 662 + ], + "score": 1.0, + "content": "sparse", + "type": "text" + }, + { + "bbox": [ + 136, + 649, + 161, + 659 + ], + "score": 0.87, + "content": "\\boldsymbol { r } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 648, + 507, + 662 + ], + "score": 1.0, + "content": "for both key and the door). The agent is randomly reset. The episode length is 50 steps.", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 48.5, + "bbox_fs": [ + 105, + 594, + 507, + 662 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 666, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 666, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 505, + 677 + ], + "score": 1.0, + "content": "DMLab Watermaze: a partially observable 3D visual DMLab environment based on the classic", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 106, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "Morris Watermaze (Morris, 1981). The task is to navigate the water maze to find a randomly spawned", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 106, + 688, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 506, + 700 + ], + "score": 1.0, + "content": "trap-door. The maze walls have color patterns that can be used to remember the goal location.", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 260, + 712 + ], + "score": 1.0, + "content": "Observations are pixel images of size", + "type": "text" + }, + { + "bbox": [ + 261, + 699, + 311, + 709 + ], + "score": 0.91, + "content": "7 2 \\times 9 6 \\times 3", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 698, + 505, + 712 + ], + "score": 1.0, + "content": ". There are 8 possible actions in total, including", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "score": 1.0, + "content": "going forward, backward, left, or right, rotating left or right, and rotating left or right while going", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 106, + 721, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 732 + ], + "score": 1.0, + "content": "forward. The episode length is 50, and the agent resets at the center of the map. 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The", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 178, + 506, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 178, + 506, + 190 + ], + "score": 1.0, + "content": "source data for AD comes from learning histories from UCB (Lai & Robbins, 1985). During training, the reward", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 189, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 189, + 211, + 200 + ], + "score": 1.0, + "content": "is distributed under odd arms", + "type": "text" + }, + { + "bbox": [ + 211, + 189, + 229, + 199 + ], + "score": 0.88, + "content": "9 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 189, + 345, + 200 + ], + "score": 1.0, + "content": "of the time and under even arms", + "type": "text" + }, + { + "bbox": [ + 345, + 189, + 363, + 199 + ], + "score": 0.88, + "content": "9 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 189, + 506, + 200 + ], + "score": 1.0, + "content": "of the time during evaluation. Both AD", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 197, + 504, + 210 + ], + "spans": [ + { + "bbox": [ + 104, + 197, + 121, + 210 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 122, + 198, + 138, + 208 + ], + "score": 0.69, + "content": "\\mathrm { { R L } ^ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 197, + 487, + 210 + ], + "score": 1.0, + "content": "can in-context learn in-distribution tasks, but AD generalizes better out of distribution. Running", + "type": "text" + }, + { + "bbox": [ + 487, + 198, + 504, + 209 + ], + "score": 0.42, + "content": "\\mathrm { { R L } ^ { 2 } }", + "type": "inline_equation" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 208, + 505, + 220 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 505, + 220 + ], + "score": 1.0, + "content": "with a transformer generally doesn’t offer an advantage over the original LSTM variant. ED performs poorly", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 218, + 448, + 229 + ], + "spans": [ + { + "bbox": [ + 106, + 218, + 284, + 229 + ], + "score": 1.0, + "content": "both in and out of distribution relative to AD and", + "type": "text" + }, + { + "bbox": [ + 284, + 218, + 301, + 228 + ], + "score": 0.72, + "content": "\\mathrm { { R L } ^ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 218, + 448, + 229 + ], + "score": 1.0, + "content": ". Scores are normalized relative to UCB.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5.5 + } + ], + "index": 3.25 + }, + { + "type": "text", + "bbox": [ + 107, + 250, + 504, + 284 + ], + "lines": [ + { + "bbox": [ + 106, + 251, + 505, + 263 + ], + "spans": [ + { + "bbox": [ + 106, + 251, + 505, + 263 + ], + "score": 1.0, + "content": "Room, the agent cannot see the location of the goal from the observations and must infer it through", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 262, + 506, + 274 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 164, + 274 + ], + "score": 1.0, + "content": "the reward of", + "type": "text" + }, + { + "bbox": [ + 165, + 263, + 191, + 272 + ], + "score": 0.9, + "content": "r = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 262, + 254, + 274 + ], + "score": 1.0, + "content": "if reached and", + "type": "text" + }, + { + "bbox": [ + 255, + 263, + 281, + 272 + ], + "score": 0.9, + "content": "r = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 262, + 506, + 274 + ], + "score": 1.0, + "content": "otherwise; however, the goal space is continuous and", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 272, + 294, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 294, + 285 + ], + "score": 1.0, + "content": "therefore there are an infinite number of goals.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10 + }, + { + "type": "title", + "bbox": [ + 107, + 298, + 180, + 309 + ], + "lines": [ + { + "bbox": [ + 105, + 297, + 181, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 181, + 311 + ], + "score": 1.0, + "content": "4.2 BASELINES", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 318, + 505, + 385 + ], + "lines": [ + { + "bbox": [ + 106, + 319, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 106, + 319, + 505, + 331 + ], + "score": 1.0, + "content": "The main aim of this work is to investigate to what extent AD reinforcement learns in-context relative", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 330, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 330, + 505, + 342 + ], + "score": 1.0, + "content": "to prior related work. AD is mostly closely related to Policy Distillation, where a policy is learned", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 341, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 106, + 341, + 505, + 353 + ], + "score": 1.0, + "content": "with a sequential model from offline interaction data. In-context online meta-RL is also related", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 352, + 505, + 364 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 505, + 364 + ], + "score": 1.0, + "content": "though not directly comparable to AD, since AD is an in-context offline meta-RL method. Still, we", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 363, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 505, + 374 + ], + "score": 1.0, + "content": "consider both types of baselines to better contextualize our work. For a more detailed discussion of", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 373, + 430, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 430, + 386 + ], + "score": 1.0, + "content": "these baseline choices we refer the reader to Appendix C. Our baselines include:", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 108, + 390, + 503, + 424 + ], + "lines": [ + { + "bbox": [ + 106, + 390, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 181, + 403 + ], + "score": 1.0, + "content": "Expert Distillation", + "type": "text" + }, + { + "bbox": [ + 182, + 391, + 202, + 402 + ], + "score": 0.53, + "content": "( E D )", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 390, + 505, + 403 + ], + "score": 1.0, + "content": ": this baseline is exactly the same as AD but the source data consists of expert", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 401, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 505, + 414 + ], + "score": 1.0, + "content": "trajectories only, rather than learning histories. ED is most similar to Gato (Reed et al., 2022) except", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 413, + 483, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 413, + 483, + 425 + ], + "score": 1.0, + "content": "ED models state-action-reward sequences like AD, while Gato models state-action sequences.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 108, + 429, + 504, + 463 + ], + "lines": [ + { + "bbox": [ + 106, + 429, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 429, + 505, + 443 + ], + "score": 1.0, + "content": "Source Algorithm: we compare AD to the gradient-based source RL algorithm that generates the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 440, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 106, + 440, + 506, + 453 + ], + "score": 1.0, + "content": "training data for distillation. We include running the source algorithm from scratch as a baseline to", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 451, + 430, + 465 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 430, + 465 + ], + "score": 1.0, + "content": "compare the data-efficiency of in-context RL to the in-weights source algorithm.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 468, + 505, + 523 + ], + "lines": [ + { + "bbox": [ + 106, + 466, + 507, + 481 + ], + "spans": [ + { + "bbox": [ + 106, + 467, + 123, + 479 + ], + "score": 0.86, + "content": "R L ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 124, + 466, + 507, + 481 + ], + "score": 1.0, + "content": "(Duan et al., 2016): an online meta-RL algorithm where exploration and fast in-context adapta-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 478, + 506, + 492 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 382, + 492 + ], + "score": 1.0, + "content": "tion are learned jointly by maximizing a multi-episodic value function.", + "type": "text" + }, + { + "bbox": [ + 382, + 478, + 401, + 490 + ], + "score": 0.59, + "content": "\\mathrm { { R L } ^ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 478, + 506, + 492 + ], + "score": 1.0, + "content": "is not directly comparable", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 489, + 505, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 505, + 503 + ], + "score": 1.0, + "content": "to AD for similar reasons to why online and offline RL algorithms are not directly comparable – RL2", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 500, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 438, + 515 + ], + "score": 1.0, + "content": "gets to interact with the environment during training while AD does not. We use", + "type": "text" + }, + { + "bbox": [ + 438, + 500, + 456, + 512 + ], + "score": 0.64, + "content": "\\mathrm { { R L } ^ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 500, + 505, + 515 + ], + "score": 1.0, + "content": "asymptotic", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 512, + 320, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 320, + 524 + ], + "score": 1.0, + "content": "performance as an approximate upper bound for AD.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27 + }, + { + "type": "title", + "bbox": [ + 107, + 537, + 187, + 549 + ], + "lines": [ + { + "bbox": [ + 105, + 536, + 189, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 189, + 551 + ], + "score": 1.0, + "content": "4.3 EVALUATION", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 558, + 505, + 668 + ], + "lines": [ + { + "bbox": [ + 105, + 557, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 262, + 570 + ], + "score": 1.0, + "content": "After pre-training, the AD transformer", + "type": "text" + }, + { + "bbox": [ + 263, + 559, + 275, + 569 + ], + "score": 0.88, + "content": "P _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 557, + 505, + 570 + ], + "score": 1.0, + "content": "can reinforcement learn in-context. 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ED performs poorly", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 218, + 448, + 229 + ], + "spans": [ + { + "bbox": [ + 106, + 218, + 284, + 229 + ], + "score": 1.0, + "content": "both in and out of distribution relative to AD and", + "type": "text" + }, + { + "bbox": [ + 284, + 218, + 301, + 228 + ], + "score": 0.72, + "content": "\\mathrm { { R L } ^ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 218, + 448, + 229 + ], + "score": 1.0, + "content": ". 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AD is mostly closely related to Policy Distillation, where a policy is learned", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 341, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 106, + 341, + 505, + 353 + ], + "score": 1.0, + "content": "with a sequential model from offline interaction data. In-context online meta-RL is also related", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 352, + 505, + 364 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 505, + 364 + ], + "score": 1.0, + "content": "though not directly comparable to AD, since AD is an in-context offline meta-RL method. Still, we", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 363, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 505, + 374 + ], + "score": 1.0, + "content": "consider both types of baselines to better contextualize our work. For a more detailed discussion of", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 373, + 430, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 430, + 386 + ], + "score": 1.0, + "content": "these baseline choices we refer the reader to Appendix C. 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ED is most similar to Gato (Reed et al., 2022) except", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 413, + 483, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 413, + 483, + 425 + ], + "score": 1.0, + "content": "ED models state-action-reward sequences like AD, while Gato models state-action sequences.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 390, + 505, + 425 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 429, + 504, + 463 + ], + "lines": [ + { + "bbox": [ + 106, + 429, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 429, + 505, + 443 + ], + "score": 1.0, + "content": "Source Algorithm: we compare AD to the gradient-based source RL algorithm that generates the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 440, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 106, + 440, + 506, + 453 + ], + "score": 1.0, + "content": "training data for distillation. We include running the source algorithm from scratch as a baseline to", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 451, + 430, + 465 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 430, + 465 + ], + "score": 1.0, + "content": "compare the data-efficiency of in-context RL to the in-weights source algorithm.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23, + "bbox_fs": [ + 106, + 429, + 506, + 465 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 468, + 505, + 523 + ], + "lines": [ + { + "bbox": [ + 106, + 466, + 507, + 481 + ], + "spans": [ + { + "bbox": [ + 106, + 467, + 123, + 479 + ], + "score": 0.86, + "content": "R L ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 124, + 466, + 507, + 481 + ], + "score": 1.0, + "content": "(Duan et al., 2016): an online meta-RL algorithm where exploration and fast in-context adapta-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 478, + 506, + 492 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 382, + 492 + ], + "score": 1.0, + "content": "tion are learned jointly by maximizing a multi-episodic value function.", + "type": "text" + }, + { + "bbox": [ + 382, + 478, + 401, + 490 + ], + "score": 0.59, + "content": "\\mathrm { { R L } ^ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 478, + 506, + 492 + ], + "score": 1.0, + "content": "is not directly comparable", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 489, + 505, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 505, + 503 + ], + "score": 1.0, + "content": "to AD for similar reasons to why online and offline RL algorithms are not directly comparable – RL2", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 500, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 438, + 515 + ], + "score": 1.0, + "content": "gets to interact with the environment during training while AD does not. We use", + "type": "text" + }, + { + "bbox": [ + 438, + 500, + 456, + 512 + ], + "score": 0.64, + "content": "\\mathrm { { R L } ^ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 500, + 505, + 515 + ], + "score": 1.0, + "content": "asymptotic", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 512, + 320, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 320, + 524 + ], + "score": 1.0, + "content": "performance as an approximate upper bound for AD.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 466, + 507, + 524 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 537, + 187, + 549 + ], + "lines": [ + { + "bbox": [ + 105, + 536, + 189, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 189, + 551 + ], + "score": 1.0, + "content": "4.3 EVALUATION", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 558, + 505, + 668 + ], + "lines": [ + { + "bbox": [ + 105, + 557, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 262, + 570 + ], + "score": 1.0, + "content": "After pre-training, the AD transformer", + "type": "text" + }, + { + "bbox": [ + 263, + 559, + 275, + 569 + ], + "score": 0.88, + "content": "P _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 557, + 505, + 570 + ], + "score": 1.0, + "content": "can reinforcement learn in-context. 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The transformer’s performance is then evaluated in terms", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 613, + 505, + 625 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 505, + 625 + ], + "score": 1.0, + "content": "of its ability to maximize return. For all evaluation runs, we average results across 5 training seeds", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 623, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 106, + 623, + 351, + 636 + ], + "score": 1.0, + "content": "with 20 evaluation seeds each for a total of 100 seeds. A task", + "type": "text" + }, + { + "bbox": [ + 351, + 624, + 364, + 634 + ], + "score": 0.72, + "content": "\\mathcal { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 623, + 505, + 636 + ], + "score": 1.0, + "content": "is sampled uniformly from the test", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 634, + 506, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 634, + 506, + 648 + ], + "score": 1.0, + "content": "task distribution and fixed for each evaluation seed. The aggregate statistics reported reflect multi-task", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 645, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 506, + 658 + ], + "score": 1.0, + "content": "performance. We evaluate for 1000 and 160 episodes for the Dark and Watermaze environments", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 657, + 453, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 453, + 669 + ], + "score": 1.0, + "content": "respectively and plot performance as a function of total environment steps at test-time.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 35.5, + "bbox_fs": [ + 105, + 557, + 506, + 669 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 684, + 200, + 697 + ], + "lines": [ + { + "bbox": [ + 105, + 683, + 201, + 699 + ], + "spans": [ + { + "bbox": [ + 105, + 683, + 201, + 699 + ], + "score": 1.0, + "content": "5 EXPERIMENTS", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "The main research question of this work is whether an in-weights RL algorithm can be amortized into", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 721, + 506, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 506, + 732 + ], + "score": 1.0, + "content": "an in-context one via Algorithm Distillation. The in-context RL algorithm should behave in a similar", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 227, + 506, + 240 + ], + "spans": [ + { + "bbox": [ + 105, + 227, + 506, + 240 + ], + "score": 1.0, + "content": "way as the in-weights one and exhibit exploration, credit assignment, and generalization capabilities.", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 238, + 505, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 238, + 505, + 252 + ], + "score": 1.0, + "content": "We begin our analysis in a clean and simple experimental setting where all three properties are", + "type": "text", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 250, + 388, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 388, + 261 + ], + "score": 1.0, + "content": "required to solve the task - the Adversarial Bandit described in Sec. 4.", + "type": "text", + "cross_page": true + } + ], + "index": 11 + } + ], + "index": 42.5, + "bbox_fs": [ + 105, + 709, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 107, + 79, + 504, + 146 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 107, + 79, + 504, + 146 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 79, + 504, + 146 + ], + "spans": [ + { + "bbox": [ + 107, + 79, + 504, + 146 + ], + "score": 0.955, + "type": "image", + "image_path": "9ee4c7c970fbe6d5f893cf896ce23072080991bcd6e0b43221766e63d5068738.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 107, + 79, + 504, + 101.33333333333333 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 107, + 101.33333333333333, + 504, + 123.66666666666666 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 107, + 123.66666666666666, + 504, + 146.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 155, + 505, + 216 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 154, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 264, + 167 + ], + "score": 1.0, + "content": "Figure 4: Main results: we evaluate AD,", + "type": "text" + }, + { + "bbox": [ + 264, + 155, + 281, + 165 + ], + "score": 0.65, + "content": "\\mathrm { { R L } ^ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 154, + 506, + 167 + ], + "score": 1.0, + "content": ", ED, and the source algorithm on environments that require", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 166, + 505, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 166, + 505, + 177 + ], + "score": 1.0, + "content": "memory and exploration. In these environments, an agent must reach a goal location that can only be inferred", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 176, + 506, + 187 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 506, + 187 + ], + "score": 1.0, + "content": "through a binary reward. AD is consistently able to in-context reinforcement learn across all environments and is", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 186, + 505, + 196 + ], + "spans": [ + { + "bbox": [ + 106, + 186, + 505, + 196 + ], + "score": 1.0, + "content": "more data-efficient than the A3C (“Dark” environments) (Mnih et al., 2016) or DQN (Watermaze) (Mnih et al.,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 195, + 506, + 207 + ], + "spans": [ + { + "bbox": [ + 105, + 195, + 327, + 207 + ], + "score": 1.0, + "content": "2013) source algorithm it distilled. We report the mean return", + "type": "text" + }, + { + "bbox": [ + 327, + 196, + 343, + 205 + ], + "score": 0.81, + "content": "\\pm \\nobreakspace 1 \\nobreakspace", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 195, + 506, + 207 + ], + "score": 1.0, + "content": "standard deviation over 5 training seeds with", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 205, + 176, + 216 + ], + "spans": [ + { + "bbox": [ + 106, + 205, + 176, + 216 + ], + "score": 1.0, + "content": "20 test seeds each.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5.5 + } + ], + "index": 3.25 + }, + { + "type": "text", + "bbox": [ + 107, + 227, + 505, + 260 + ], + "lines": [ + { + "bbox": [ + 105, + 227, + 506, + 240 + ], + "spans": [ + { + "bbox": [ + 105, + 227, + 506, + 240 + ], + "score": 1.0, + "content": "way as the in-weights one and exhibit exploration, credit assignment, and generalization capabilities.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 238, + 505, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 238, + 505, + 252 + ], + "score": 1.0, + "content": "We begin our analysis in a clean and simple experimental setting where all three properties are", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 250, + 388, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 388, + 261 + ], + "score": 1.0, + "content": "required to solve the task - the Adversarial Bandit described in Sec. 4.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 266, + 505, + 388 + ], + "lines": [ + { + "bbox": [ + 102, + 261, + 509, + 290 + ], + "spans": [ + { + "bbox": [ + 102, + 261, + 353, + 290 + ], + "score": 1.0, + "content": "To generate the source data, we sample a set of training tasks Bound algorithm (Lai & Robbins, 1985), and save its learning", + "type": "text" + }, + { + "bbox": [ + 354, + 265, + 395, + 280 + ], + "score": 0.93, + "content": "\\{ \\mathcal { M } _ { j } \\} _ { j = 1 } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 395, + 261, + 509, + 290 + ], + "score": 1.0, + "content": ", run the Upper Confidence We then train a transformer", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 287, + 506, + 301 + ], + "spans": [ + { + "bbox": [ + 105, + 287, + 377, + 301 + ], + "score": 1.0, + "content": "to predict actions as described in Alg. 1. We evaluate AD, ED, and", + "type": "text" + }, + { + "bbox": [ + 377, + 288, + 396, + 299 + ], + "score": 0.81, + "content": "\\mathrm { { R L } ^ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 287, + 506, + 301 + ], + "score": 1.0, + "content": "and normalize their scores", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 298, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 104, + 298, + 267, + 313 + ], + "score": 1.0, + "content": "relative to UCB and a random policy", + "type": "text" + }, + { + "bbox": [ + 267, + 299, + 389, + 312 + ], + "score": 0.9, + "content": "( r - r _ { r a n d . } ) / ( r _ { U C B } - r _ { r a n d . } )", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 298, + 506, + 313 + ], + "score": 1.0, + "content": ". The results are shown in", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 309, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 241, + 324 + ], + "score": 1.0, + "content": "Fig. 3. We find that both AD and", + "type": "text" + }, + { + "bbox": [ + 241, + 310, + 260, + 321 + ], + "score": 0.49, + "content": "\\mathtt { R L } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 309, + 505, + 324 + ], + "score": 1.0, + "content": "can reliably in-context learn tasks sampled from the training", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 321, + 505, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 505, + 334 + ], + "score": 1.0, + "content": "distribution while ED cannot, though ED does do better than random guessing when evaluated", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 332, + 505, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 332, + 505, + 345 + ], + "score": 1.0, + "content": "in-distribution. However, AD can also in-context learn to solve out of distribution tasks whereas", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 343, + 505, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 505, + 356 + ], + "score": 1.0, + "content": "the other methods cannot. This experiment shows that AD can explore the bandit arms, can assign", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 355, + 505, + 366 + ], + "spans": [ + { + "bbox": [ + 106, + 355, + 505, + 366 + ], + "score": 1.0, + "content": "credit by exploiting an arm once reached, and can generalize well out of distribution nearly as well as", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 365, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 505, + 378 + ], + "score": 1.0, + "content": "UCB. We now move beyond the bandit setting and investigate similar research questions in more", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 376, + 497, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 497, + 389 + ], + "score": 1.0, + "content": "challenging RL environments and present our results as answers to a series of research questions.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 16.5 + }, + { + "type": "text", + "bbox": [ + 107, + 395, + 505, + 516 + ], + "lines": [ + { + "bbox": [ + 105, + 394, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 506, + 408 + ], + "score": 1.0, + "content": "Does Algorithm Distillation exhibit in-context reinforcement learning? To answer this question,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 407, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 505, + 419 + ], + "score": 1.0, + "content": "we first generate source data for Algorithm Distillation. In the Dark Room and Dark Key-to-Door", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 418, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 106, + 418, + 505, + 429 + ], + "score": 1.0, + "content": "environments we collect 2000 learning histories with an Asynchronous Advantage Actor-Critic", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 427, + 506, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 506, + 442 + ], + "score": 1.0, + "content": "(A3C) (Mnih et al., 2016) with 100 actors, while in DMLab Watermaze we collect 4000 learning", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 438, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 104, + 438, + 506, + 453 + ], + "score": 1.0, + "content": "histories with a distributed DQN with 16 parallel actors (see Appendix F for asymptotic learning", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 451, + 506, + 463 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 506, + 463 + ], + "score": 1.0, + "content": "curves of the source algorithm and Appendix O for hyperparameters). Shown in Fig. 4, AD in-context", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 460, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 506, + 474 + ], + "score": 1.0, + "content": "reinforcement learns in all of the environments. In contrast, ED fails to explore and learn in-context", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 471, + 507, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 214, + 486 + ], + "score": 1.0, + "content": "in most settings. We use", + "type": "text" + }, + { + "bbox": [ + 215, + 471, + 234, + 483 + ], + "score": 0.71, + "content": "\\mathrm { { R L } ^ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 471, + 507, + 486 + ], + "score": 1.0, + "content": "trained for 1 billion environment steps as a proxy for the upper", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 482, + 506, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 293, + 496 + ], + "score": 1.0, + "content": "bound of performance for a meta-RL method.", + "type": "text" + }, + { + "bbox": [ + 294, + 483, + 313, + 494 + ], + "score": 0.37, + "content": "\\mathrm { { R L ^ { 2 } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 482, + 506, + 496 + ], + "score": 1.0, + "content": "achieves a near-optimal asymptotic score in all", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 494, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 494, + 505, + 505 + ], + "score": 1.0, + "content": "the environments except for Dark Room (Hard). Despite learning from offline data, AD matches", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 505, + 469, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 505, + 152, + 518 + ], + "score": 1.0, + "content": "asymptotic", + "type": "text" + }, + { + "bbox": [ + 153, + 505, + 171, + 516 + ], + "score": 0.65, + "content": "\\mathtt { R L } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 505, + 381, + 518 + ], + "score": 1.0, + "content": "on the Dark environments and approaches it (within", + "type": "text" + }, + { + "bbox": [ + 382, + 505, + 401, + 516 + ], + "score": 0.86, + "content": "1 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 505, + 469, + 518 + ], + "score": 1.0, + "content": ") on Watermaze.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 522, + 505, + 555 + ], + "lines": [ + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 316, + 535 + ], + "score": 1.0, + "content": "Credit-assignment: In Dark Room, the agent receives", + "type": "text" + }, + { + "bbox": [ + 316, + 523, + 341, + 532 + ], + "score": 0.91, + "content": "r = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 522, + 505, + 535 + ], + "score": 1.0, + "content": "each time it visits the goal location. Even", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 533, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 533, + 505, + 545 + ], + "score": 1.0, + "content": "though AD is trained to condition only on single timestep reward and not episodic return tokens, it is", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 543, + 484, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 484, + 556 + ], + "score": 1.0, + "content": "still able to maximize the reward, which suggests that AD has learned to do credit assignment.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 561, + 505, + 594 + ], + "lines": [ + { + "bbox": [ + 105, + 560, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 506, + 574 + ], + "score": 1.0, + "content": "Exploration: Dark Room (Hard) tests the agents exploration capability. Since the reward is sparse", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 110, + 572, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 110, + 572, + 135, + 582 + ], + "score": 0.87, + "content": "r = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 572, + 399, + 584 + ], + "score": 1.0, + "content": "exactly once), most of the learning history has reward values of", + "type": "text" + }, + { + "bbox": [ + 400, + 572, + 425, + 582 + ], + "score": 0.89, + "content": "r = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 572, + 505, + 584 + ], + "score": 1.0, + "content": ". Nevertheless, AD", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 583, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 506, + 595 + ], + "score": 1.0, + "content": "infers the goal from previous episodes in its context which means it has learned to explore and exploit.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 599, + 505, + 633 + ], + "lines": [ + { + "bbox": [ + 105, + 598, + 507, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 507, + 613 + ], + "score": 1.0, + "content": "Generalization: Dark Key-to-Door tests in-distribution generalization with a combinatorial task space.", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 609, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 256, + 624 + ], + "score": 1.0, + "content": "While the environment has a total of", + "type": "text" + }, + { + "bbox": [ + 257, + 611, + 287, + 621 + ], + "score": 0.87, + "content": "\\sim 6 . 5 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 609, + 505, + 624 + ], + "score": 1.0, + "content": "tasks, less than 2k were seen during training. During", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 622, + 493, + 634 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 493, + 634 + ], + "score": 1.0, + "content": "evaluation, AD both generalizes and achieves near-optimal performance on mostly unseen tasks.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 107, + 641, + 505, + 696 + ], + "lines": [ + { + "bbox": [ + 107, + 641, + 506, + 653 + ], + "spans": [ + { + "bbox": [ + 107, + 641, + 506, + 653 + ], + "score": 1.0, + "content": "Can Algorithm Distillation learn from pixel-based observations? DMLab Watermaze is a pixel-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 652, + 505, + 664 + ], + "spans": [ + { + "bbox": [ + 106, + 652, + 505, + 664 + ], + "score": 1.0, + "content": "based environment that is larger than the Dark environments with tasks sampled from a continuous", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 662, + 505, + 676 + ], + "spans": [ + { + "bbox": [ + 105, + 662, + 505, + 676 + ], + "score": 1.0, + "content": "uniform distribution. The environment is partially observable in two ways - the goal is invisible until", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 673, + 507, + 687 + ], + "spans": [ + { + "bbox": [ + 105, + 673, + 507, + 687 + ], + "score": 1.0, + "content": "the agent has reached it and the first-person view limits the agent’s field of vision. Shown in Fig. 4,", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 684, + 425, + 698 + ], + "spans": [ + { + "bbox": [ + 105, + 684, + 425, + 698 + ], + "score": 1.0, + "content": "AD maximizes the episodic return with in-context RL while ED does not learn.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 44 + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "Can AD learn a more data-efficient RL algorithm than the one that produced the source", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "data? In Fig. 4, AD is significantly more data-efficient than the source algorithm. This gain is a", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "score": 1.0, + "content": "byproduct of distilling a multi-stream algorithm into a single-stream one. The source algorithms", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 48 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2023", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 107, + 79, + 504, + 146 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 107, + 79, + 504, + 146 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 79, + 504, + 146 + ], + "spans": [ + { + "bbox": [ + 107, + 79, + 504, + 146 + ], + "score": 0.955, + "type": "image", + "image_path": "9ee4c7c970fbe6d5f893cf896ce23072080991bcd6e0b43221766e63d5068738.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 107, + 79, + 504, + 101.33333333333333 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 107, + 101.33333333333333, + 504, + 123.66666666666666 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 107, + 123.66666666666666, + 504, + 146.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 155, + 505, + 216 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 154, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 264, + 167 + ], + "score": 1.0, + "content": "Figure 4: Main results: we evaluate AD,", + "type": "text" + }, + { + "bbox": [ + 264, + 155, + 281, + 165 + ], + "score": 0.65, + "content": "\\mathrm { { R L } ^ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 154, + 506, + 167 + ], + "score": 1.0, + "content": ", ED, and the source algorithm on environments that require", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 166, + 505, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 166, + 505, + 177 + ], + "score": 1.0, + "content": "memory and exploration. In these environments, an agent must reach a goal location that can only be inferred", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 176, + 506, + 187 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 506, + 187 + ], + "score": 1.0, + "content": "through a binary reward. AD is consistently able to in-context reinforcement learn across all environments and is", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 186, + 505, + 196 + ], + "spans": [ + { + "bbox": [ + 106, + 186, + 505, + 196 + ], + "score": 1.0, + "content": "more data-efficient than the A3C (“Dark” environments) (Mnih et al., 2016) or DQN (Watermaze) (Mnih et al.,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 195, + 506, + 207 + ], + "spans": [ + { + "bbox": [ + 105, + 195, + 327, + 207 + ], + "score": 1.0, + "content": "2013) source algorithm it distilled. We report the mean return", + "type": "text" + }, + { + "bbox": [ + 327, + 196, + 343, + 205 + ], + "score": 0.81, + "content": "\\pm \\nobreakspace 1 \\nobreakspace", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 195, + 506, + 207 + ], + "score": 1.0, + "content": "standard deviation over 5 training seeds with", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 205, + 176, + 216 + ], + "spans": [ + { + "bbox": [ + 106, + 205, + 176, + 216 + ], + "score": 1.0, + "content": "20 test seeds each.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5.5 + } + ], + "index": 3.25 + }, + { + "type": "text", + "bbox": [ + 107, + 227, + 505, + 260 + ], + "lines": [], + "index": 10, + "bbox_fs": [ + 105, + 227, + 506, + 261 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 266, + 505, + 388 + ], + "lines": [ + { + "bbox": [ + 102, + 261, + 509, + 290 + ], + "spans": [ + { + "bbox": [ + 102, + 261, + 353, + 290 + ], + "score": 1.0, + "content": "To generate the source data, we sample a set of training tasks Bound algorithm (Lai & Robbins, 1985), and save its learning", + "type": "text" + }, + { + "bbox": [ + 354, + 265, + 395, + 280 + ], + "score": 0.93, + "content": "\\{ \\mathcal { M } _ { j } \\} _ { j = 1 } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 395, + 261, + 509, + 290 + ], + "score": 1.0, + "content": ", run the Upper Confidence We then train a transformer", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 287, + 506, + 301 + ], + "spans": [ + { + "bbox": [ + 105, + 287, + 377, + 301 + ], + "score": 1.0, + "content": "to predict actions as described in Alg. 1. We evaluate AD, ED, and", + "type": "text" + }, + { + "bbox": [ + 377, + 288, + 396, + 299 + ], + "score": 0.81, + "content": "\\mathrm { { R L } ^ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 287, + 506, + 301 + ], + "score": 1.0, + "content": "and normalize their scores", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 298, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 104, + 298, + 267, + 313 + ], + "score": 1.0, + "content": "relative to UCB and a random policy", + "type": "text" + }, + { + "bbox": [ + 267, + 299, + 389, + 312 + ], + "score": 0.9, + "content": "( r - r _ { r a n d . } ) / ( r _ { U C B } - r _ { r a n d . } )", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 298, + 506, + 313 + ], + "score": 1.0, + "content": ". The results are shown in", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 309, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 241, + 324 + ], + "score": 1.0, + "content": "Fig. 3. We find that both AD and", + "type": "text" + }, + { + "bbox": [ + 241, + 310, + 260, + 321 + ], + "score": 0.49, + "content": "\\mathtt { R L } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 309, + 505, + 324 + ], + "score": 1.0, + "content": "can reliably in-context learn tasks sampled from the training", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 321, + 505, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 505, + 334 + ], + "score": 1.0, + "content": "distribution while ED cannot, though ED does do better than random guessing when evaluated", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 332, + 505, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 332, + 505, + 345 + ], + "score": 1.0, + "content": "in-distribution. However, AD can also in-context learn to solve out of distribution tasks whereas", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 343, + 505, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 505, + 356 + ], + "score": 1.0, + "content": "the other methods cannot. This experiment shows that AD can explore the bandit arms, can assign", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 355, + 505, + 366 + ], + "spans": [ + { + "bbox": [ + 106, + 355, + 505, + 366 + ], + "score": 1.0, + "content": "credit by exploiting an arm once reached, and can generalize well out of distribution nearly as well as", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 365, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 505, + 378 + ], + "score": 1.0, + "content": "UCB. We now move beyond the bandit setting and investigate similar research questions in more", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 376, + 497, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 497, + 389 + ], + "score": 1.0, + "content": "challenging RL environments and present our results as answers to a series of research questions.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 16.5, + "bbox_fs": [ + 102, + 261, + 509, + 389 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 395, + 505, + 516 + ], + "lines": [ + { + "bbox": [ + 105, + 394, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 506, + 408 + ], + "score": 1.0, + "content": "Does Algorithm Distillation exhibit in-context reinforcement learning? To answer this question,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 407, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 505, + 419 + ], + "score": 1.0, + "content": "we first generate source data for Algorithm Distillation. In the Dark Room and Dark Key-to-Door", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 418, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 106, + 418, + 505, + 429 + ], + "score": 1.0, + "content": "environments we collect 2000 learning histories with an Asynchronous Advantage Actor-Critic", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 427, + 506, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 506, + 442 + ], + "score": 1.0, + "content": "(A3C) (Mnih et al., 2016) with 100 actors, while in DMLab Watermaze we collect 4000 learning", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 438, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 104, + 438, + 506, + 453 + ], + "score": 1.0, + "content": "histories with a distributed DQN with 16 parallel actors (see Appendix F for asymptotic learning", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 451, + 506, + 463 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 506, + 463 + ], + "score": 1.0, + "content": "curves of the source algorithm and Appendix O for hyperparameters). Shown in Fig. 4, AD in-context", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 460, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 506, + 474 + ], + "score": 1.0, + "content": "reinforcement learns in all of the environments. In contrast, ED fails to explore and learn in-context", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 471, + 507, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 214, + 486 + ], + "score": 1.0, + "content": "in most settings. We use", + "type": "text" + }, + { + "bbox": [ + 215, + 471, + 234, + 483 + ], + "score": 0.71, + "content": "\\mathrm { { R L } ^ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 471, + 507, + 486 + ], + "score": 1.0, + "content": "trained for 1 billion environment steps as a proxy for the upper", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 482, + 506, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 293, + 496 + ], + "score": 1.0, + "content": "bound of performance for a meta-RL method.", + "type": "text" + }, + { + "bbox": [ + 294, + 483, + 313, + 494 + ], + "score": 0.37, + "content": "\\mathrm { { R L ^ { 2 } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 482, + 506, + 496 + ], + "score": 1.0, + "content": "achieves a near-optimal asymptotic score in all", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 494, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 494, + 505, + 505 + ], + "score": 1.0, + "content": "the environments except for Dark Room (Hard). Despite learning from offline data, AD matches", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 505, + 469, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 505, + 152, + 518 + ], + "score": 1.0, + "content": "asymptotic", + "type": "text" + }, + { + "bbox": [ + 153, + 505, + 171, + 516 + ], + "score": 0.65, + "content": "\\mathtt { R L } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 505, + 381, + 518 + ], + "score": 1.0, + "content": "on the Dark environments and approaches it (within", + "type": "text" + }, + { + "bbox": [ + 382, + 505, + 401, + 516 + ], + "score": 0.86, + "content": "1 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 505, + 469, + 518 + ], + "score": 1.0, + "content": ") on Watermaze.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 27, + "bbox_fs": [ + 104, + 394, + 507, + 518 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 522, + 505, + 555 + ], + "lines": [ + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 316, + 535 + ], + "score": 1.0, + "content": "Credit-assignment: In Dark Room, the agent receives", + "type": "text" + }, + { + "bbox": [ + 316, + 523, + 341, + 532 + ], + "score": 0.91, + "content": "r = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 522, + 505, + 535 + ], + "score": 1.0, + "content": "each time it visits the goal location. Even", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 533, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 533, + 505, + 545 + ], + "score": 1.0, + "content": "though AD is trained to condition only on single timestep reward and not episodic return tokens, it is", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 543, + 484, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 484, + 556 + ], + "score": 1.0, + "content": "still able to maximize the reward, which suggests that AD has learned to do credit assignment.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34, + "bbox_fs": [ + 105, + 522, + 505, + 556 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 561, + 505, + 594 + ], + "lines": [ + { + "bbox": [ + 105, + 560, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 506, + 574 + ], + "score": 1.0, + "content": "Exploration: Dark Room (Hard) tests the agents exploration capability. Since the reward is sparse", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 110, + 572, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 110, + 572, + 135, + 582 + ], + "score": 0.87, + "content": "r = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 572, + 399, + 584 + ], + "score": 1.0, + "content": "exactly once), most of the learning history has reward values of", + "type": "text" + }, + { + "bbox": [ + 400, + 572, + 425, + 582 + ], + "score": 0.89, + "content": "r = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 572, + 505, + 584 + ], + "score": 1.0, + "content": ". Nevertheless, AD", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 583, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 506, + 595 + ], + "score": 1.0, + "content": "infers the goal from previous episodes in its context which means it has learned to explore and exploit.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 560, + 506, + 595 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 599, + 505, + 633 + ], + "lines": [ + { + "bbox": [ + 105, + 598, + 507, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 507, + 613 + ], + "score": 1.0, + "content": "Generalization: Dark Key-to-Door tests in-distribution generalization with a combinatorial task space.", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 609, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 256, + 624 + ], + "score": 1.0, + "content": "While the environment has a total of", + "type": "text" + }, + { + "bbox": [ + 257, + 611, + 287, + 621 + ], + "score": 0.87, + "content": "\\sim 6 . 5 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 609, + 505, + 624 + ], + "score": 1.0, + "content": "tasks, less than 2k were seen during training. During", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 622, + 493, + 634 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 493, + 634 + ], + "score": 1.0, + "content": "evaluation, AD both generalizes and achieves near-optimal performance on mostly unseen tasks.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 40, + "bbox_fs": [ + 105, + 598, + 507, + 634 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 641, + 505, + 696 + ], + "lines": [ + { + "bbox": [ + 107, + 641, + 506, + 653 + ], + "spans": [ + { + "bbox": [ + 107, + 641, + 506, + 653 + ], + "score": 1.0, + "content": "Can Algorithm Distillation learn from pixel-based observations? DMLab Watermaze is a pixel-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 652, + 505, + 664 + ], + "spans": [ + { + "bbox": [ + 106, + 652, + 505, + 664 + ], + "score": 1.0, + "content": "based environment that is larger than the Dark environments with tasks sampled from a continuous", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 662, + 505, + 676 + ], + "spans": [ + { + "bbox": [ + 105, + 662, + 505, + 676 + ], + "score": 1.0, + "content": "uniform distribution. The environment is partially observable in two ways - the goal is invisible until", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 673, + 507, + 687 + ], + "spans": [ + { + "bbox": [ + 105, + 673, + 507, + 687 + ], + "score": 1.0, + "content": "the agent has reached it and the first-person view limits the agent’s field of vision. Shown in Fig. 4,", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 684, + 425, + 698 + ], + "spans": [ + { + "bbox": [ + 105, + 684, + 425, + 698 + ], + "score": 1.0, + "content": "AD maximizes the episodic return with in-context RL while ED does not learn.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 44, + "bbox_fs": [ + 105, + 641, + 507, + 698 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "Can AD learn a more data-efficient RL algorithm than the one that produced the source", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "data? In Fig. 4, AD is significantly more data-efficient than the source algorithm. This gain is a", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "score": 1.0, + "content": "byproduct of distilling a multi-stream algorithm into a single-stream one. The source algorithms", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 223, + 506, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 506, + 236 + ], + "score": 1.0, + "content": "(A3C and DQN) are distributed, which means they run many actors in parallel to achieve good", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 233, + 506, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 233, + 506, + 248 + ], + "score": 1.0, + "content": "performance.3 A distributed RL algorithm may not be very data-efficient in aggregate but each", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 244, + 506, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 244, + 506, + 259 + ], + "score": 1.0, + "content": "individual actor can be data-efficient. 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Therefore, AD can be more data-efficient", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 360, + 337, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 337, + 374 + ], + "score": 1.0, + "content": "than both a multi and single-stream source RL algorithm.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17 + }, + { + "type": "image", + "bbox": [ + 365, + 267, + 483, + 360 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 365, + 267, + 483, + 360 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 365, + 267, + 483, + 360 + ], + "spans": [ + { + "bbox": [ + 365, + 267, + 483, + 360 + ], + "score": 0.945, + "type": "image", + "image_path": "c652f1346777c272cd0157cbac3dddfaa6bac8bab0398dde13e052045c590f1e.jpg" + } + ] + } + ], + "index": 17.0, + "virtual_lines": [ + { + "bbox": [ + 365, + 267, + 483, + 313.5 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 365, + 313.5, + 483, + 360.0 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 344, + 375, + 505, + 435 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 344, + 375, + 506, + 385 + ], + "spans": [ + { + "bbox": [ + 344, + 375, + 506, + 385 + ], + "score": 1.0, + "content": "Figure 6: Single-stream Algorithm Distil-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 343, + 384, + 506, + 397 + ], + "spans": [ + { + "bbox": [ + 343, + 384, + 506, + 397 + ], + "score": 1.0, + "content": "lation: AD trained on the learning history", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 343, + 394, + 506, + 406 + ], + "spans": [ + { + "bbox": [ + 343, + 394, + 506, + 406 + ], + "score": 1.0, + "content": "from an A3C agent with only one actor (i.e.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 343, + 405, + 506, + 416 + ], + "spans": [ + { + "bbox": [ + 343, + 405, + 506, + 416 + ], + "score": 1.0, + "content": "single-stream). 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To answer the research", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 507, + 337, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 507, + 337, + 518 + ], + "score": 1.0, + "content": "question, we sample policies from the hold-out test-set", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 518, + 337, + 530 + ], + "spans": [ + { + "bbox": [ + 106, + 518, + 337, + 530 + ], + "score": 1.0, + "content": "data along different points of the source algorithm history", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 104, + 527, + 338, + 542 + ], + "spans": [ + { + "bbox": [ + 104, + 527, + 338, + 542 + ], + "score": 1.0, + "content": "- from a near-random policy to a near-expert policy. We", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 539, + 337, + 550 + ], + "spans": [ + { + "bbox": [ + 106, + 539, + 337, + 550 + ], + "score": 1.0, + "content": "then pre-fill the context for both AD and ED with this", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 550, + 337, + 563 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 337, + 563 + ], + "score": 1.0, + "content": "policy data, and run both methods in the environment in", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 561, + 337, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 337, + 573 + ], + "score": 1.0, + "content": "Dark Room (Fig. 5). While ED maintains the performance", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 572, + 337, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 337, + 585 + ], + "score": 1.0, + "content": "of the input policy, AD improves every policy in-context", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 106, + 583, + 337, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 337, + 596 + ], + "score": 1.0, + "content": "until it is near-optimal. 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In-context", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 343, + 574, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 343, + 574, + 506, + 586 + ], + "score": 1.0, + "content": "RL only emerges once the context size is", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 343, + 585, + 468, + 596 + ], + "spans": [ + { + "bbox": [ + 343, + 585, + 468, + 596 + ], + "score": 1.0, + "content": "large enough and across-episodic.", + "type": "text" + } + ], + "index": 58 + } + ], + "index": 56.5 + } + ], + "index": 51.25 + }, + { + "type": "text", + "bbox": [ + 106, + 613, + 505, + 701 + ], + "lines": [ + { + "bbox": [ + 106, + 613, + 505, + 625 + ], + "spans": [ + { + "bbox": [ + 106, + 613, + 505, + 625 + ], + "score": 1.0, + "content": "What context size is required for in-context RL to emerge? We’ve hypothesized that AD", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 105, + 624, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 505, + 636 + ], + "score": 1.0, + "content": "requires sufficiently long (i.e. across-episodic) contexts to in-context reinforcement learn. We test this", + "type": "text" + } + ], + "index": 60 + }, + { + "bbox": [ + 106, + 635, + 506, + 647 + ], + "spans": [ + { + "bbox": [ + 106, + 635, + 506, + 647 + ], + "score": 1.0, + "content": "hypothesis by training several AD variants with different context sizes on the Dark Room environment.", + "type": "text" + } + ], + "index": 61 + }, + { + "bbox": [ + 106, + 646, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 106, + 646, + 505, + 658 + ], + "score": 1.0, + "content": "We plot the learning curves of these different variants in Fig. 7 and find that multi-episodic contexts", + "type": "text" + } + ], + "index": 62 + }, + { + "bbox": [ + 106, + 657, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 106, + 657, + 506, + 669 + ], + "score": 1.0, + "content": "of 2-4 episodes are necessary to learn a near-optimal in-context RL algorithm. 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The reason", + "type": "text" + } + ], + "index": 64 + }, + { + "bbox": [ + 106, + 679, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 106, + 679, + 505, + 691 + ], + "score": 1.0, + "content": "for this is likely that the context is large enough to retrain across-episodic information – e.g., at the", + "type": "text" + } + ], + "index": 65 + }, + { + "bbox": [ + 105, + 689, + 503, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 503, + 702 + ], + "score": 1.0, + "content": "start of a new episode, the context will be filled with transitions from most of the previous episode.", + "type": "text" + } + ], + "index": 66 + } + ], + "index": 62.5 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 711, + 504, + 731 + ], + "lines": [ + { + "bbox": [ + 117, + 709, + 506, + 724 + ], + "spans": [ + { + "bbox": [ + 117, + 709, + 506, + 724 + ], + "score": 1.0, + "content": "3Indeed, current state-of-the-art 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In-context", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 343, + 574, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 343, + 574, + 506, + 586 + ], + "score": 1.0, + "content": "RL only emerges once the context size is", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 343, + 585, + 468, + 596 + ], + "spans": [ + { + "bbox": [ + 343, + 585, + 468, + 596 + ], + "score": 1.0, + "content": "large enough and across-episodic.", + "type": "text" + } + ], + "index": 58 + } + ], + "index": 56.5 + } + ], + "index": 51.25 + }, + { + "type": "text", + "bbox": [ + 106, + 613, + 505, + 701 + ], + "lines": [ + { + "bbox": [ + 106, + 613, + 505, + 625 + ], + "spans": [ + { + "bbox": [ + 106, + 613, + 505, + 625 + ], + "score": 1.0, + "content": "What context size is required for in-context RL to emerge? We’ve hypothesized that AD", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 105, + 624, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 505, + 636 + ], + "score": 1.0, + "content": "requires sufficiently long (i.e. across-episodic) contexts to in-context reinforcement learn. We test this", + "type": "text" + } + ], + "index": 60 + }, + { + "bbox": [ + 106, + 635, + 506, + 647 + ], + "spans": [ + { + "bbox": [ + 106, + 635, + 506, + 647 + ], + "score": 1.0, + "content": "hypothesis by training several AD variants with different context sizes on the Dark Room environment.", + "type": "text" + } + ], + "index": 61 + }, + { + "bbox": [ + 106, + 646, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 106, + 646, + 505, + 658 + ], + "score": 1.0, + "content": "We plot the learning curves of these different variants in Fig. 7 and find that multi-episodic contexts", + "type": "text" + } + ], + "index": 62 + }, + { + "bbox": [ + 106, + 657, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 106, + 657, + 506, + 669 + ], + "score": 1.0, + "content": "of 2-4 episodes are necessary to learn a near-optimal in-context RL algorithm. Initial signs of", + "type": "text" + } + ], + "index": 63 + }, + { + "bbox": [ + 104, + 667, + 506, + 681 + ], + "spans": [ + { + "bbox": [ + 104, + 667, + 506, + 681 + ], + "score": 1.0, + "content": "in-context RL begin to emerge when the context size is roughly the length of an episode. The reason", + "type": "text" + } + ], + "index": 64 + }, + { + "bbox": [ + 106, + 679, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 106, + 679, + 505, + 691 + ], + "score": 1.0, + "content": "for this is likely that the context is large enough to retrain across-episodic information – e.g., at the", + "type": "text" + } + ], + "index": 65 + }, + { + "bbox": [ + 105, + 689, + 503, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 503, + 702 + ], + "score": 1.0, + "content": "start of a new episode, the context will be filled with transitions from most of the previous episode.", + "type": "text" + } + ], + "index": 66 + } + ], + "index": 62.5, + "bbox_fs": [ + 104, + 613, + 506, + 702 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 81, + 211, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 213, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 213, + 96 + ], + "score": 1.0, + "content": "6 RELATED WORK", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 98, + 505, + 296 + ], + "lines": [ + { + "bbox": [ + 105, + 97, + 505, + 111 + ], + "spans": [ + { + "bbox": [ + 105, + 97, + 505, + 111 + ], + "score": 1.0, + "content": "Offline Policy Distillation: Most closely related to our work are the recent advances in learning", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 108, + 505, + 122 + ], + "spans": [ + { + "bbox": [ + 105, + 108, + 505, + 122 + ], + "score": 1.0, + "content": "policies from offline environment interaction data with transformers, which we have been referring to", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 119, + 505, + 132 + ], + "spans": [ + { + "bbox": [ + 105, + 119, + 505, + 132 + ], + "score": 1.0, + "content": "as Policy Distillation (PD). Initial PD architectures such as Decision Transformer (DT) (Chen et al.,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 131, + 506, + 143 + ], + "spans": [ + { + "bbox": [ + 106, + 131, + 506, + 143 + ], + "score": 1.0, + "content": "2021) and Trajectory Transformer (Janner et al., 2021) showed that transformers can learn single-task", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 142, + 506, + 154 + ], + "spans": [ + { + "bbox": [ + 106, + 142, + 506, + 154 + ], + "score": 1.0, + "content": "policies from offline data. Subsequently the Multi-Game Decision Transformer (MGDT) (Lee et al.,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 153, + 506, + 165 + ], + "spans": [ + { + "bbox": [ + 106, + 153, + 506, + 165 + ], + "score": 1.0, + "content": "2022) and Gato (Reed et al., 2022) showed that PD architectures can also learn multi-task same", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 164, + 506, + 176 + ], + "spans": [ + { + "bbox": [ + 106, + 164, + 506, + 176 + ], + "score": 1.0, + "content": "domain and cross-domain policies, respectively. Importantly, these prior methods use contexts", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 175, + 506, + 187 + ], + "spans": [ + { + "bbox": [ + 106, + 175, + 506, + 187 + ], + "score": 1.0, + "content": "substantially smaller than an episode length, which is likely the reason in-context RL was not", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 185, + 506, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 185, + 506, + 199 + ], + "score": 1.0, + "content": "observed in these works. Instead, they rely on alternate ways to adapt to new tasks - MGDT", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 196, + 506, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 196, + 506, + 209 + ], + "score": 1.0, + "content": "finetunes the model parameters while Gato gets prompted with expert demonstrations to adapt to", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 208, + 506, + 219 + ], + "spans": [ + { + "bbox": [ + 106, + 208, + 506, + 219 + ], + "score": 1.0, + "content": "downstream tasks. AD adapts in-context without finetuning and does not rely on demonstrations.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 218, + 506, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 506, + 231 + ], + "score": 1.0, + "content": "A number of recent works have explored more generalized PD architectures (Furuta et al., 2021),", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 230, + 506, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 506, + 241 + ], + "score": 1.0, + "content": "prompt conditioning (Xu et al., 2022), and online gradient-based RL (Zheng et al., 2022). Some PD", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 240, + 506, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 506, + 253 + ], + "score": 1.0, + "content": "architectures such as DT and MGDT are instantiations of Upside Down RL (UDRL) Schmidhuber", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 252, + 505, + 263 + ], + "spans": [ + { + "bbox": [ + 106, + 252, + 505, + 263 + ], + "score": 1.0, + "content": "(2019); Srivastava et al. (2019) where rather than learning a value function, a policy is conditioned", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 262, + 506, + 275 + ], + "spans": [ + { + "bbox": [ + 106, + 262, + 506, + 275 + ], + "score": 1.0, + "content": "directly on the desired return. However, AD is not explicitly doing UDRL since it is not conditioned", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 274, + 506, + 285 + ], + "spans": [ + { + "bbox": [ + 106, + 274, + 506, + 285 + ], + "score": 1.0, + "content": "on returns. In fact, return maximization is not specified anywhere in the AD objective but rather", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 284, + 404, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 404, + 297 + ], + "score": 1.0, + "content": "emerges implicitly by modeling the learning histories of an RL algorithm.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 107, + 302, + 505, + 476 + ], + "lines": [ + { + "bbox": [ + 106, + 301, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 301, + 505, + 313 + ], + "score": 1.0, + "content": "Meta Reinforcement Learning: AD falls into the category of methods that learn to reinforcement", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 312, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 312, + 505, + 324 + ], + "score": 1.0, + "content": "learn, also known as meta-RL. Specifically, AD is an in-context offline meta-RL method. This general", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 322, + 506, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 506, + 336 + ], + "score": 1.0, + "content": "idea of learning the policy improvement process has a long history in reinforcement learning, but has", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 332, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 332, + 505, + 348 + ], + "score": 1.0, + "content": "been limited to meta-learning hyper-parameters4 until recently (Ishii et al., 2002). In-context deep", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 345, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 505, + 357 + ], + "score": 1.0, + "content": "meta-RL methods introduced by Wang et al. (2016) and Duan et al. (2016) are usually trained in", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 355, + 506, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 355, + 506, + 369 + ], + "score": 1.0, + "content": "the online setting by maximizing multi-episodic value functions with memory-based architectures", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 367, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 106, + 367, + 505, + 379 + ], + "score": 1.0, + "content": "through environment interactions (Ni et al., 2022). Meta-RL through multi-episodic value functions", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 377, + 506, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 506, + 390 + ], + "score": 1.0, + "content": "has been done in both on-policy (Duan et al., 2016) and off-policy (Rakelly et al., 2019; Fakoor", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 389, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 505, + 402 + ], + "score": 1.0, + "content": "et al., 2020) settings. Another common approach to online meta-RL includes optimization-based", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 400, + 506, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 506, + 412 + ], + "score": 1.0, + "content": "methods that find good network parameter initializations for meta-RL (Hochreiter et al., 2001;", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 410, + 506, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 506, + 423 + ], + "score": 1.0, + "content": "Finn et al., 2017; Nichol et al., 2018) and adapt by taking additional gradient steps. Like other", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 423, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 423, + 505, + 434 + ], + "score": 1.0, + "content": "in-context meta-RL approaches, AD is gradient-free - it adapts to downstream tasks without updating", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 432, + 505, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 505, + 445 + ], + "score": 1.0, + "content": "its network parameters. Recent works have proposed learning to reinforcement learn from offline", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 444, + 504, + 456 + ], + "spans": [ + { + "bbox": [ + 106, + 444, + 504, + 456 + ], + "score": 1.0, + "content": "datasets, or offline meta-RL, using Bayesian RL (Dorfman et al., 2021) and optimization-based", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 454, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 505, + 468 + ], + "score": 1.0, + "content": "meta-RL (Mitchell et al., 2021). Given the difficulty of offline meta-RL, Pong et al. (2022) proposed", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 465, + 286, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 286, + 477 + ], + "score": 1.0, + "content": "a hybrid offline-online strategy for meta-RL.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 26.5 + }, + { + "type": "text", + "bbox": [ + 107, + 483, + 505, + 636 + ], + "lines": [ + { + "bbox": [ + 105, + 482, + 506, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 506, + 495 + ], + "score": 1.0, + "content": "In-Context Learning with Transformers: In this work, we make the distinction between in-context", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 494, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 494, + 505, + 505 + ], + "score": 1.0, + "content": "learning and incremental or in-context learning. In-context learning involves learning from a provided", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 504, + 506, + 517 + ], + "spans": [ + { + "bbox": [ + 104, + 504, + 506, + 517 + ], + "score": 1.0, + "content": "prompt or demonstration while incremental in-context learning involves learning from one’s own", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 516, + 504, + 527 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 504, + 527 + ], + "score": 1.0, + "content": "behavior through trial and error. While many recent works have demonstrated the former, it is much", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 527, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 106, + 527, + 505, + 538 + ], + "score": 1.0, + "content": "less common to see methods that exhibit the latter. Arguably, the most impressive demonstrations of", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 537, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 505, + 550 + ], + "score": 1.0, + "content": "in-context learning to date have been shown in the text completion setting (Radford et al., 2018; Chen", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 548, + 504, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 548, + 504, + 560 + ], + "score": 1.0, + "content": "et al., 2020; Brown et al., 2020) through prompt conditioning. Similar methodology was recently", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 559, + 506, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 506, + 572 + ], + "score": 1.0, + "content": "extended to show powerful composability properties in text-conditioned image generation (Yu et al.,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 570, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 106, + 570, + 505, + 582 + ], + "score": 1.0, + "content": "2022). Recent work showed that transformers can also learn simple algorithm classes, such as linear", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 581, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 506, + 595 + ], + "score": 1.0, + "content": "regression, in-context in a small-scale setting (Garg et al., 2022). Like prior in-context learning", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 592, + 505, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 505, + 604 + ], + "score": 1.0, + "content": "methods, Garg et al. (2022) required initializing the transformer prompt with expert examples. While", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 602, + 506, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 506, + 615 + ], + "score": 1.0, + "content": "the aforementioned approaches were examples of in-context learning, a recent work (Chen et al.,", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 613, + 505, + 627 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 505, + 627 + ], + "score": 1.0, + "content": "2022) demonstrated incremental in-context learning for hyperparameter optimization by treating", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 626, + 452, + 637 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 452, + 637 + ], + "score": 1.0, + "content": "hyperparameter optimization as a sequential prediction problem with a score function.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 41.5 + }, + { + "type": "title", + "bbox": [ + 108, + 648, + 195, + 660 + ], + "lines": [ + { + "bbox": [ + 104, + 645, + 197, + 664 + ], + "spans": [ + { + "bbox": [ + 104, + 645, + 197, + 664 + ], + "score": 1.0, + "content": "7 CONCLUSION", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 49 + }, + { + "type": "text", + "bbox": [ + 108, + 668, + 505, + 712 + ], + "lines": [ + { + "bbox": [ + 106, + 668, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 106, + 668, + 505, + 679 + ], + "score": 1.0, + "content": "We have demonstrated that can distill RL algorithms by modeling their learning histories causally", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 678, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 505, + 690 + ], + "score": 1.0, + "content": "with imitation learning and that AD can learn more data-efficient algorithms than those that generated", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 690, + 505, + 703 + ], + "spans": [ + { + "bbox": [ + 106, + 690, + 505, + 703 + ], + "score": 1.0, + "content": "the source data. We hope that AD inspires further investigation into in-context reinforcement learning", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 700, + 228, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 228, + 713 + ], + "score": 1.0, + "content": "from the research community.", + "type": "text" + } + ], + "index": 53 + } + ], + "index": 51.5 + } + ], + "page_idx": 8, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 119, + 721, + 404, + 732 + ], + "lines": [ + { + "bbox": [ + 119, + 719, + 405, + 734 + ], + "spans": [ + { + "bbox": [ + 119, + 719, + 405, + 734 + ], + "score": 1.0, + "content": "4For a recent example of meta-optimization in RL see Flennerhag et al. (2022).", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2023", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "9", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 81, + 211, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 213, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 213, + 96 + ], + "score": 1.0, + "content": "6 RELATED WORK", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 98, + 505, + 296 + ], + "lines": [ + { + "bbox": [ + 105, + 97, + 505, + 111 + ], + "spans": [ + { + "bbox": [ + 105, + 97, + 505, + 111 + ], + "score": 1.0, + "content": "Offline Policy Distillation: Most closely related to our work are the recent advances in learning", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 108, + 505, + 122 + ], + "spans": [ + { + "bbox": [ + 105, + 108, + 505, + 122 + ], + "score": 1.0, + "content": "policies from offline environment interaction data with transformers, which we have been referring to", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 119, + 505, + 132 + ], + "spans": [ + { + "bbox": [ + 105, + 119, + 505, + 132 + ], + "score": 1.0, + "content": "as Policy Distillation (PD). Initial PD architectures such as Decision Transformer (DT) (Chen et al.,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 131, + 506, + 143 + ], + "spans": [ + { + "bbox": [ + 106, + 131, + 506, + 143 + ], + "score": 1.0, + "content": "2021) and Trajectory Transformer (Janner et al., 2021) showed that transformers can learn single-task", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 142, + 506, + 154 + ], + "spans": [ + { + "bbox": [ + 106, + 142, + 506, + 154 + ], + "score": 1.0, + "content": "policies from offline data. Subsequently the Multi-Game Decision Transformer (MGDT) (Lee et al.,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 153, + 506, + 165 + ], + "spans": [ + { + "bbox": [ + 106, + 153, + 506, + 165 + ], + "score": 1.0, + "content": "2022) and Gato (Reed et al., 2022) showed that PD architectures can also learn multi-task same", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 164, + 506, + 176 + ], + "spans": [ + { + "bbox": [ + 106, + 164, + 506, + 176 + ], + "score": 1.0, + "content": "domain and cross-domain policies, respectively. Importantly, these prior methods use contexts", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 175, + 506, + 187 + ], + "spans": [ + { + "bbox": [ + 106, + 175, + 506, + 187 + ], + "score": 1.0, + "content": "substantially smaller than an episode length, which is likely the reason in-context RL was not", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 185, + 506, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 185, + 506, + 199 + ], + "score": 1.0, + "content": "observed in these works. Instead, they rely on alternate ways to adapt to new tasks - MGDT", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 196, + 506, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 196, + 506, + 209 + ], + "score": 1.0, + "content": "finetunes the model parameters while Gato gets prompted with expert demonstrations to adapt to", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 208, + 506, + 219 + ], + "spans": [ + { + "bbox": [ + 106, + 208, + 506, + 219 + ], + "score": 1.0, + "content": "downstream tasks. AD adapts in-context without finetuning and does not rely on demonstrations.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 218, + 506, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 506, + 231 + ], + "score": 1.0, + "content": "A number of recent works have explored more generalized PD architectures (Furuta et al., 2021),", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 230, + 506, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 506, + 241 + ], + "score": 1.0, + "content": "prompt conditioning (Xu et al., 2022), and online gradient-based RL (Zheng et al., 2022). Some PD", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 240, + 506, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 506, + 253 + ], + "score": 1.0, + "content": "architectures such as DT and MGDT are instantiations of Upside Down RL (UDRL) Schmidhuber", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 252, + 505, + 263 + ], + "spans": [ + { + "bbox": [ + 106, + 252, + 505, + 263 + ], + "score": 1.0, + "content": "(2019); Srivastava et al. (2019) where rather than learning a value function, a policy is conditioned", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 262, + 506, + 275 + ], + "spans": [ + { + "bbox": [ + 106, + 262, + 506, + 275 + ], + "score": 1.0, + "content": "directly on the desired return. However, AD is not explicitly doing UDRL since it is not conditioned", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 274, + 506, + 285 + ], + "spans": [ + { + "bbox": [ + 106, + 274, + 506, + 285 + ], + "score": 1.0, + "content": "on returns. In fact, return maximization is not specified anywhere in the AD objective but rather", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 284, + 404, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 404, + 297 + ], + "score": 1.0, + "content": "emerges implicitly by modeling the learning histories of an RL algorithm.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 9.5, + "bbox_fs": [ + 105, + 97, + 506, + 297 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 302, + 505, + 476 + ], + "lines": [ + { + "bbox": [ + 106, + 301, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 301, + 505, + 313 + ], + "score": 1.0, + "content": "Meta Reinforcement Learning: AD falls into the category of methods that learn to reinforcement", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 312, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 312, + 505, + 324 + ], + "score": 1.0, + "content": "learn, also known as meta-RL. Specifically, AD is an in-context offline meta-RL method. This general", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 322, + 506, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 506, + 336 + ], + "score": 1.0, + "content": "idea of learning the policy improvement process has a long history in reinforcement learning, but has", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 332, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 332, + 505, + 348 + ], + "score": 1.0, + "content": "been limited to meta-learning hyper-parameters4 until recently (Ishii et al., 2002). In-context deep", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 345, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 505, + 357 + ], + "score": 1.0, + "content": "meta-RL methods introduced by Wang et al. (2016) and Duan et al. (2016) are usually trained in", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 355, + 506, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 355, + 506, + 369 + ], + "score": 1.0, + "content": "the online setting by maximizing multi-episodic value functions with memory-based architectures", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 367, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 106, + 367, + 505, + 379 + ], + "score": 1.0, + "content": "through environment interactions (Ni et al., 2022). Meta-RL through multi-episodic value functions", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 377, + 506, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 506, + 390 + ], + "score": 1.0, + "content": "has been done in both on-policy (Duan et al., 2016) and off-policy (Rakelly et al., 2019; Fakoor", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 389, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 505, + 402 + ], + "score": 1.0, + "content": "et al., 2020) settings. Another common approach to online meta-RL includes optimization-based", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 400, + 506, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 506, + 412 + ], + "score": 1.0, + "content": "methods that find good network parameter initializations for meta-RL (Hochreiter et al., 2001;", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 410, + 506, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 506, + 423 + ], + "score": 1.0, + "content": "Finn et al., 2017; Nichol et al., 2018) and adapt by taking additional gradient steps. Like other", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 423, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 423, + 505, + 434 + ], + "score": 1.0, + "content": "in-context meta-RL approaches, AD is gradient-free - it adapts to downstream tasks without updating", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 432, + 505, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 505, + 445 + ], + "score": 1.0, + "content": "its network parameters. Recent works have proposed learning to reinforcement learn from offline", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 444, + 504, + 456 + ], + "spans": [ + { + "bbox": [ + 106, + 444, + 504, + 456 + ], + "score": 1.0, + "content": "datasets, or offline meta-RL, using Bayesian RL (Dorfman et al., 2021) and optimization-based", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 454, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 505, + 468 + ], + "score": 1.0, + "content": "meta-RL (Mitchell et al., 2021). Given the difficulty of offline meta-RL, Pong et al. (2022) proposed", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 465, + 286, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 286, + 477 + ], + "score": 1.0, + "content": "a hybrid offline-online strategy for meta-RL.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 301, + 506, + 477 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 483, + 505, + 636 + ], + "lines": [ + { + "bbox": [ + 105, + 482, + 506, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 506, + 495 + ], + "score": 1.0, + "content": "In-Context Learning with Transformers: In this work, we make the distinction between in-context", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 494, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 494, + 505, + 505 + ], + "score": 1.0, + "content": "learning and incremental or in-context learning. In-context learning involves learning from a provided", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 504, + 506, + 517 + ], + "spans": [ + { + "bbox": [ + 104, + 504, + 506, + 517 + ], + "score": 1.0, + "content": "prompt or demonstration while incremental in-context learning involves learning from one’s own", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 516, + 504, + 527 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 504, + 527 + ], + "score": 1.0, + "content": "behavior through trial and error. While many recent works have demonstrated the former, it is much", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 527, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 106, + 527, + 505, + 538 + ], + "score": 1.0, + "content": "less common to see methods that exhibit the latter. Arguably, the most impressive demonstrations of", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 537, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 505, + 550 + ], + "score": 1.0, + "content": "in-context learning to date have been shown in the text completion setting (Radford et al., 2018; Chen", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 548, + 504, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 548, + 504, + 560 + ], + "score": 1.0, + "content": "et al., 2020; Brown et al., 2020) through prompt conditioning. Similar methodology was recently", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 559, + 506, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 506, + 572 + ], + "score": 1.0, + "content": "extended to show powerful composability properties in text-conditioned image generation (Yu et al.,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 570, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 106, + 570, + 505, + 582 + ], + "score": 1.0, + "content": "2022). Recent work showed that transformers can also learn simple algorithm classes, such as linear", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 581, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 506, + 595 + ], + "score": 1.0, + "content": "regression, in-context in a small-scale setting (Garg et al., 2022). Like prior in-context learning", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 592, + 505, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 505, + 604 + ], + "score": 1.0, + "content": "methods, Garg et al. (2022) required initializing the transformer prompt with expert examples. While", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 602, + 506, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 506, + 615 + ], + "score": 1.0, + "content": "the aforementioned approaches were examples of in-context learning, a recent work (Chen et al.,", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 613, + 505, + 627 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 505, + 627 + ], + "score": 1.0, + "content": "2022) demonstrated incremental in-context learning for hyperparameter optimization by treating", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 626, + 452, + 637 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 452, + 637 + ], + "score": 1.0, + "content": "hyperparameter optimization as a sequential prediction problem with a score function.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 41.5, + "bbox_fs": [ + 104, + 482, + 506, + 637 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 648, + 195, + 660 + ], + "lines": [ + { + "bbox": [ + 104, + 645, + 197, + 664 + ], + "spans": [ + { + "bbox": [ + 104, + 645, + 197, + 664 + ], + "score": 1.0, + "content": "7 CONCLUSION", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 49 + }, + { + "type": "text", + "bbox": [ + 108, + 668, + 505, + 712 + ], + "lines": [ + { + "bbox": [ + 106, + 668, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 106, + 668, + 505, + 679 + ], + "score": 1.0, + "content": "We have demonstrated that can distill RL algorithms by modeling their learning histories causally", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 678, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 505, + 690 + ], + "score": 1.0, + "content": "with imitation learning and that AD can learn more data-efficient algorithms than those that generated", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 690, + 505, + 703 + ], + "spans": [ + { + "bbox": [ + 106, + 690, + 505, + 703 + ], + "score": 1.0, + "content": "the source data. We hope that AD inspires further investigation into in-context reinforcement learning", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 700, + 228, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 228, + 713 + ], + "score": 1.0, + "content": "from the research community.", + "type": "text" + } + ], + "index": 53 + } + ], + "index": 51.5, + "bbox_fs": [ + 105, + 668, + 505, + 713 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 82, + 175, + 93 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 176, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 176, + 95 + ], + "score": 1.0, + "content": "REFERENCES", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 100, + 506, + 145 + ], + "lines": [ + { + "bbox": [ + 106, + 99, + 507, + 113 + ], + "spans": [ + { + "bbox": [ + 106, + 99, + 507, + 113 + ], + "score": 1.0, + "content": "Charles Beattie, Joel Z. 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In International Conference on", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 115, + 104, + 327, + 117 + ], + "spans": [ + { + "bbox": [ + 115, + 104, + 327, + 117 + ], + "score": 1.0, + "content": "Machine Learning, pp. 24631–24645. PMLR, 2022.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1, + "bbox_fs": [ + 106, + 81, + 506, + 117 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 123, + 506, + 168 + ], + "lines": [ + { + "bbox": [ + 105, + 122, + 507, + 136 + ], + "spans": [ + { + "bbox": [ + 105, + 122, + 507, + 136 + ], + "score": 1.0, + "content": "Jiahui Yu, Yuanzhong Xu, Jing Yu Koh, Thang Luong, Gunjan Baid, Zirui Wang, Vijay Vasudevan,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 116, + 135, + 506, + 146 + ], + "spans": [ + { + "bbox": [ + 116, + 135, + 506, + 146 + ], + "score": 1.0, + "content": "Alexander Ku, Yinfei Yang, Burcu Karagol Ayan, Ben Hutchinson, Wei Han, Zarana Parekh,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 115, + 144, + 506, + 158 + ], + "spans": [ + { + "bbox": [ + 115, + 144, + 506, + 158 + ], + "score": 1.0, + "content": "Xin Li, Han Zhang, Jason Baldridge, and Yonghui Wu. Scaling Autoregressive Models for", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 116, + 156, + 306, + 168 + ], + "spans": [ + { + "bbox": [ + 116, + 156, + 306, + 168 + ], + "score": 1.0, + "content": "Content-Rich Text-to-Image Generation, 2022.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5, + "bbox_fs": [ + 105, + 122, + 507, + 168 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 175, + 504, + 197 + ], + "lines": [ + { + "bbox": [ + 106, + 174, + 506, + 188 + ], + "spans": [ + { + "bbox": [ + 106, + 174, + 506, + 188 + ], + "score": 1.0, + "content": "Qinqing Zheng, Amy Zhang, and Aditya Grover. Online decision transformer. arXiv preprint", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 115, + 186, + 219, + 198 + ], + "spans": [ + { + "bbox": [ + 115, + 186, + 219, + 198 + ], + "score": 1.0, + "content": "arXiv:2202.05607, 2022.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5, + "bbox_fs": [ + 106, + 174, + 506, + 198 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 109, + 81, + 288, + 93 + ], + "lines": [ + { + "bbox": [ + 106, + 80, + 290, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 80, + 290, + 96 + ], + "score": 1.0, + "content": "A DISCUSSION AND LIMITATIONS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 107, + 505, + 217 + ], + "lines": [ + { + "bbox": [ + 105, + 106, + 506, + 120 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 506, + 120 + ], + "score": 1.0, + "content": "Discussion: While AD is more data-efficient than the source algorithm, we found that the source", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 119, + 506, + 130 + ], + "spans": [ + { + "bbox": [ + 106, + 119, + 506, + 130 + ], + "score": 1.0, + "content": "algorithm achieves a slightly higher asymptotic score in harder environments. This presents a trade-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 129, + 505, + 142 + ], + "spans": [ + { + "bbox": [ + 106, + 129, + 505, + 142 + ], + "score": 1.0, + "content": "off between data-efficiency and asymptotic optimality when using AD that should be taken into", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 141, + 505, + 152 + ], + "spans": [ + { + "bbox": [ + 106, + 141, + 505, + 152 + ], + "score": 1.0, + "content": "consideration when applying AD to a specific problem. In terms of long-term consequences, AD", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 151, + 506, + 164 + ], + "spans": [ + { + "bbox": [ + 106, + 151, + 506, + 164 + ], + "score": 1.0, + "content": "presents a path for converting narrow single-task RL agents into multi-task generalist ones. To date,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 163, + 506, + 174 + ], + "spans": [ + { + "bbox": [ + 106, + 163, + 506, + 174 + ], + "score": 1.0, + "content": "deep RL research has mostly focused on powerful single-task agents Mnih et al. (2015); Schrittwieser", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 173, + 505, + 185 + ], + "spans": [ + { + "bbox": [ + 106, + 173, + 505, + 185 + ], + "score": 1.0, + "content": "et al. (2019); Hessel et al. (2021). These algorithms have produced powerful but data-inefficient", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 183, + 506, + 198 + ], + "spans": [ + { + "bbox": [ + 105, + 183, + 506, + 198 + ], + "score": 1.0, + "content": "agents, which has limited their applicability beyond narrow domains. AD offers a path toward training", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 195, + 506, + 208 + ], + "spans": [ + { + "bbox": [ + 106, + 195, + 506, + 208 + ], + "score": 1.0, + "content": "substantially more data-efficient, though perhaps less optimal, generalist agents by distilling narrow", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 206, + 327, + 218 + ], + "spans": [ + { + "bbox": [ + 106, + 206, + 327, + 218 + ], + "score": 1.0, + "content": "RL algorithms into sequence models like transformers.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 5.5 + }, + { + "type": "text", + "bbox": [ + 107, + 223, + 505, + 311 + ], + "lines": [ + { + "bbox": [ + 105, + 222, + 505, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 505, + 236 + ], + "score": 1.0, + "content": "Limitations: While AD is a general method, in this work we’ve shown that AD generalizes to", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 234, + 505, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 234, + 505, + 246 + ], + "score": 1.0, + "content": "new tasks within the same domain. Showing the ability to distill cross-domain RL algorithms and", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 245, + 505, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 505, + 258 + ], + "score": 1.0, + "content": "generalize to new domains would be an interesting line if inquiry for future work. AD also requires", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 256, + 505, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 256, + 505, + 268 + ], + "score": 1.0, + "content": "storing many learning histories which could take up significant memory, though this may not be too", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 267, + 505, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 505, + 279 + ], + "score": 1.0, + "content": "much of an issue since the learning histories can be stored on disk rather than RAM. Perhaps the main", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 277, + 507, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 507, + 290 + ], + "score": 1.0, + "content": "limitation of AD is that most RL environments of interest have long episodes and modeling multi-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 289, + 505, + 301 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 505, + 301 + ], + "score": 1.0, + "content": "episodic context requires more powerful long-horizon sequential models than the ones considered in", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 300, + 387, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 387, + 312 + ], + "score": 1.0, + "content": "this work. We believe this is a promising direction for future research.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 14.5 + }, + { + "type": "title", + "bbox": [ + 108, + 329, + 300, + 342 + ], + "lines": [ + { + "bbox": [ + 105, + 327, + 302, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 302, + 344 + ], + "score": 1.0, + "content": "B ENVIRONMENT CONSIDERATIONS", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 355, + 505, + 410 + ], + "lines": [ + { + "bbox": [ + 106, + 355, + 505, + 367 + ], + "spans": [ + { + "bbox": [ + 106, + 355, + 505, + 367 + ], + "score": 1.0, + "content": "In this work, we consider environments where zero-shot generalization is difficult, so the agent", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 365, + 506, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 506, + 379 + ], + "score": 1.0, + "content": "must learn through trial and error. We also want environments where overfitting to any particular", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 377, + 506, + 389 + ], + "spans": [ + { + "bbox": [ + 106, + 377, + 506, + 389 + ], + "score": 1.0, + "content": "task is difficult to ensure our method is general. A final practical consideration is that we consider", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 389, + 507, + 401 + ], + "spans": [ + { + "bbox": [ + 106, + 389, + 507, + 401 + ], + "score": 1.0, + "content": "environments wher across-episodic histories can be feasibly modeled with a causal transformer.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 399, + 450, + 411 + ], + "spans": [ + { + "bbox": [ + 106, + 399, + 450, + 411 + ], + "score": 1.0, + "content": "Given these considerations, our evaluation environments need to satisfy three criteria:", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 129, + 422, + 505, + 565 + ], + "lines": [ + { + "bbox": [ + 131, + 423, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 131, + 423, + 505, + 434 + ], + "score": 1.0, + "content": "1. Supports many tasks: The environments must be multi-task to ensure that our agent and the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 142, + 434, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 142, + 434, + 505, + 444 + ], + "score": 1.0, + "content": "baselines do not overfit to any single task and instead is able to in-context reinforcement", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 141, + 444, + 333, + 457 + ], + "spans": [ + { + "bbox": [ + 141, + 444, + 333, + 457 + ], + "score": 1.0, + "content": "learn across many tasks within a given domain.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 128, + 460, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 128, + 460, + 505, + 473 + ], + "score": 1.0, + "content": "2. Task must be hard to infer: To ensure that the downstream tasks are hard to generalize to in", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 142, + 473, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 142, + 473, + 505, + 483 + ], + "score": 1.0, + "content": "zero-shot, we use environments that require exploration. Namely, we require environments", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 142, + 483, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 142, + 483, + 505, + 495 + ], + "score": 1.0, + "content": "where either the task can only be inferred from the reward and not the observation, or tasks", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 142, + 494, + 258, + 506 + ], + "spans": [ + { + "bbox": [ + 142, + 494, + 258, + 506 + ], + "score": 1.0, + "content": "that are partially observable.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 128, + 510, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 128, + 510, + 505, + 523 + ], + "score": 1.0, + "content": "3. Supports multi-episodic contexts: Lastly, we impose a practical constraint - the environment", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 141, + 522, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 141, + 522, + 505, + 534 + ], + "score": 1.0, + "content": "episodes must be short enough such that a normal GPT-like transformer can fit multiple", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 141, + 532, + 506, + 545 + ], + "spans": [ + { + "bbox": [ + 141, + 532, + 506, + 545 + ], + "score": 1.0, + "content": "episodes in its context. Since this work introduces AD as a method, we wish to investigate it", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 141, + 543, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 141, + 543, + 505, + 556 + ], + "score": 1.0, + "content": "in the cleanest possible setting using a canonical architecture. We leave investigating AD", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 142, + 554, + 462, + 566 + ], + "spans": [ + { + "bbox": [ + 142, + 554, + 462, + 566 + ], + "score": 1.0, + "content": "with more complex architectures that scale to longer sequences for future work.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 30.5 + }, + { + "type": "text", + "bbox": [ + 107, + 577, + 505, + 665 + ], + "lines": [ + { + "bbox": [ + 105, + 576, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 505, + 589 + ], + "score": 1.0, + "content": "Prior related works (Chen et al., 2021; Lee et al., 2022; Janner et al., 2021; Reed et al., 2022)", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 588, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 505, + 600 + ], + "score": 1.0, + "content": "evaluated on Atari, OpenAI gym, and as well as other environments. However, Atari and OpenAI", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "score": 1.0, + "content": "gym don’t satisfy at least one of the above criteria. Atari and OpenAI gym episodes are often long", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "score": 1.0, + "content": "and can contain thousands or more transitions per episode, so it’s technically challenging to populate", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 621, + 505, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 505, + 633 + ], + "score": 1.0, + "content": "a causal transformer’s context with across-episode histories. Indeed, the prior related works only", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 632, + 505, + 643 + ], + "spans": [ + { + "bbox": [ + 106, + 632, + 505, + 643 + ], + "score": 1.0, + "content": "considered within-episode context lengths. Additionally, it is often easy to infer the task from either", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 643, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 505, + 655 + ], + "score": 1.0, + "content": "the observation or the dense reward alone in both Atari and OpenAI gym, which reduces the need for", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 654, + 492, + 666 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 492, + 666 + ], + "score": 1.0, + "content": "exploration. For these reasons, we evaluate in environments that satisfy all three criteria instead.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 40.5 + }, + { + "type": "title", + "bbox": [ + 107, + 683, + 318, + 696 + ], + "lines": [ + { + "bbox": [ + 105, + 682, + 320, + 698 + ], + "spans": [ + { + "bbox": [ + 105, + 682, + 320, + 698 + ], + "score": 1.0, + "content": "C CLOSELY RELATED PRIOR METHODS", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 45 + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "In our main results we use Expert Distillation (ED) as a baseline. Here, we discuss how the most", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 721, + 486, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 486, + 733 + ], + "score": 1.0, + "content": "closely related methods differ from AD and why ED is sufficient to support the paper’s claims.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 46.5 + } + ], + "page_idx": 13, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2023", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 752, + 310, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 13 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 109, + 81, + 288, + 93 + ], + "lines": [ + { + "bbox": [ + 106, + 80, + 290, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 80, + 290, + 96 + ], + "score": 1.0, + "content": "A DISCUSSION AND LIMITATIONS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 107, + 505, + 217 + ], + "lines": [ + { + "bbox": [ + 105, + 106, + 506, + 120 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 506, + 120 + ], + "score": 1.0, + "content": "Discussion: While AD is more data-efficient than the source algorithm, we found that the source", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 119, + 506, + 130 + ], + "spans": [ + { + "bbox": [ + 106, + 119, + 506, + 130 + ], + "score": 1.0, + "content": "algorithm achieves a slightly higher asymptotic score in harder environments. This presents a trade-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 129, + 505, + 142 + ], + "spans": [ + { + "bbox": [ + 106, + 129, + 505, + 142 + ], + "score": 1.0, + "content": "off between data-efficiency and asymptotic optimality when using AD that should be taken into", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 141, + 505, + 152 + ], + "spans": [ + { + "bbox": [ + 106, + 141, + 505, + 152 + ], + "score": 1.0, + "content": "consideration when applying AD to a specific problem. In terms of long-term consequences, AD", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 151, + 506, + 164 + ], + "spans": [ + { + "bbox": [ + 106, + 151, + 506, + 164 + ], + "score": 1.0, + "content": "presents a path for converting narrow single-task RL agents into multi-task generalist ones. To date,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 163, + 506, + 174 + ], + "spans": [ + { + "bbox": [ + 106, + 163, + 506, + 174 + ], + "score": 1.0, + "content": "deep RL research has mostly focused on powerful single-task agents Mnih et al. (2015); Schrittwieser", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 173, + 505, + 185 + ], + "spans": [ + { + "bbox": [ + 106, + 173, + 505, + 185 + ], + "score": 1.0, + "content": "et al. (2019); Hessel et al. (2021). These algorithms have produced powerful but data-inefficient", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 183, + 506, + 198 + ], + "spans": [ + { + "bbox": [ + 105, + 183, + 506, + 198 + ], + "score": 1.0, + "content": "agents, which has limited their applicability beyond narrow domains. AD offers a path toward training", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 195, + 506, + 208 + ], + "spans": [ + { + "bbox": [ + 106, + 195, + 506, + 208 + ], + "score": 1.0, + "content": "substantially more data-efficient, though perhaps less optimal, generalist agents by distilling narrow", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 206, + 327, + 218 + ], + "spans": [ + { + "bbox": [ + 106, + 206, + 327, + 218 + ], + "score": 1.0, + "content": "RL algorithms into sequence models like transformers.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 5.5, + "bbox_fs": [ + 105, + 106, + 506, + 218 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 223, + 505, + 311 + ], + "lines": [ + { + "bbox": [ + 105, + 222, + 505, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 505, + 236 + ], + "score": 1.0, + "content": "Limitations: While AD is a general method, in this work we’ve shown that AD generalizes to", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 234, + 505, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 234, + 505, + 246 + ], + "score": 1.0, + "content": "new tasks within the same domain. Showing the ability to distill cross-domain RL algorithms and", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 245, + 505, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 505, + 258 + ], + "score": 1.0, + "content": "generalize to new domains would be an interesting line if inquiry for future work. AD also requires", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 256, + 505, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 256, + 505, + 268 + ], + "score": 1.0, + "content": "storing many learning histories which could take up significant memory, though this may not be too", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 267, + 505, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 505, + 279 + ], + "score": 1.0, + "content": "much of an issue since the learning histories can be stored on disk rather than RAM. Perhaps the main", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 277, + 507, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 507, + 290 + ], + "score": 1.0, + "content": "limitation of AD is that most RL environments of interest have long episodes and modeling multi-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 289, + 505, + 301 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 505, + 301 + ], + "score": 1.0, + "content": "episodic context requires more powerful long-horizon sequential models than the ones considered in", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 300, + 387, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 387, + 312 + ], + "score": 1.0, + "content": "this work. We believe this is a promising direction for future research.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 14.5, + "bbox_fs": [ + 105, + 222, + 507, + 312 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 329, + 300, + 342 + ], + "lines": [ + { + "bbox": [ + 105, + 327, + 302, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 302, + 344 + ], + "score": 1.0, + "content": "B ENVIRONMENT CONSIDERATIONS", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 355, + 505, + 410 + ], + "lines": [ + { + "bbox": [ + 106, + 355, + 505, + 367 + ], + "spans": [ + { + "bbox": [ + 106, + 355, + 505, + 367 + ], + "score": 1.0, + "content": "In this work, we consider environments where zero-shot generalization is difficult, so the agent", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 365, + 506, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 506, + 379 + ], + "score": 1.0, + "content": "must learn through trial and error. We also want environments where overfitting to any particular", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 377, + 506, + 389 + ], + "spans": [ + { + "bbox": [ + 106, + 377, + 506, + 389 + ], + "score": 1.0, + "content": "task is difficult to ensure our method is general. A final practical consideration is that we consider", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 389, + 507, + 401 + ], + "spans": [ + { + "bbox": [ + 106, + 389, + 507, + 401 + ], + "score": 1.0, + "content": "environments wher across-episodic histories can be feasibly modeled with a causal transformer.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 399, + 450, + 411 + ], + "spans": [ + { + "bbox": [ + 106, + 399, + 450, + 411 + ], + "score": 1.0, + "content": "Given these considerations, our evaluation environments need to satisfy three criteria:", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 355, + 507, + 411 + ] + }, + { + "type": "list", + "bbox": [ + 129, + 422, + 505, + 565 + ], + "lines": [ + { + "bbox": [ + 131, + 423, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 131, + 423, + 505, + 434 + ], + "score": 1.0, + "content": "1. Supports many tasks: The environments must be multi-task to ensure that our agent and the", + "type": "text" + } + ], + "index": 25, + "is_list_start_line": true + }, + { + "bbox": [ + 142, + 434, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 142, + 434, + 505, + 444 + ], + "score": 1.0, + "content": "baselines do not overfit to any single task and instead is able to in-context reinforcement", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 141, + 444, + 333, + 457 + ], + "spans": [ + { + "bbox": [ + 141, + 444, + 333, + 457 + ], + "score": 1.0, + "content": "learn across many tasks within a given domain.", + "type": "text" + } + ], + "index": 27, + "is_list_end_line": true + }, + { + "bbox": [ + 128, + 460, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 128, + 460, + 505, + 473 + ], + "score": 1.0, + "content": "2. Task must be hard to infer: To ensure that the downstream tasks are hard to generalize to in", + "type": "text" + } + ], + "index": 28, + "is_list_start_line": true + }, + { + "bbox": [ + 142, + 473, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 142, + 473, + 505, + 483 + ], + "score": 1.0, + "content": "zero-shot, we use environments that require exploration. Namely, we require environments", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 142, + 483, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 142, + 483, + 505, + 495 + ], + "score": 1.0, + "content": "where either the task can only be inferred from the reward and not the observation, or tasks", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 142, + 494, + 258, + 506 + ], + "spans": [ + { + "bbox": [ + 142, + 494, + 258, + 506 + ], + "score": 1.0, + "content": "that are partially observable.", + "type": "text" + } + ], + "index": 31, + "is_list_end_line": true + }, + { + "bbox": [ + 128, + 510, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 128, + 510, + 505, + 523 + ], + "score": 1.0, + "content": "3. Supports multi-episodic contexts: Lastly, we impose a practical constraint - the environment", + "type": "text" + } + ], + "index": 32, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 522, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 141, + 522, + 505, + 534 + ], + "score": 1.0, + "content": "episodes must be short enough such that a normal GPT-like transformer can fit multiple", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 141, + 532, + 506, + 545 + ], + "spans": [ + { + "bbox": [ + 141, + 532, + 506, + 545 + ], + "score": 1.0, + "content": "episodes in its context. Since this work introduces AD as a method, we wish to investigate it", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 141, + 543, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 141, + 543, + 505, + 556 + ], + "score": 1.0, + "content": "in the cleanest possible setting using a canonical architecture. We leave investigating AD", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 142, + 554, + 462, + 566 + ], + "spans": [ + { + "bbox": [ + 142, + 554, + 462, + 566 + ], + "score": 1.0, + "content": "with more complex architectures that scale to longer sequences for future work.", + "type": "text" + } + ], + "index": 36, + "is_list_end_line": true + } + ], + "index": 30.5, + "bbox_fs": [ + 128, + 423, + 506, + 566 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 577, + 505, + 665 + ], + "lines": [ + { + "bbox": [ + 105, + 576, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 505, + 589 + ], + "score": 1.0, + "content": "Prior related works (Chen et al., 2021; Lee et al., 2022; Janner et al., 2021; Reed et al., 2022)", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 588, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 505, + 600 + ], + "score": 1.0, + "content": "evaluated on Atari, OpenAI gym, and as well as other environments. However, Atari and OpenAI", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "score": 1.0, + "content": "gym don’t satisfy at least one of the above criteria. Atari and OpenAI gym episodes are often long", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "score": 1.0, + "content": "and can contain thousands or more transitions per episode, so it’s technically challenging to populate", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 621, + 505, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 505, + 633 + ], + "score": 1.0, + "content": "a causal transformer’s context with across-episode histories. Indeed, the prior related works only", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 632, + 505, + 643 + ], + "spans": [ + { + "bbox": [ + 106, + 632, + 505, + 643 + ], + "score": 1.0, + "content": "considered within-episode context lengths. Additionally, it is often easy to infer the task from either", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 643, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 505, + 655 + ], + "score": 1.0, + "content": "the observation or the dense reward alone in both Atari and OpenAI gym, which reduces the need for", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 654, + 492, + 666 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 492, + 666 + ], + "score": 1.0, + "content": "exploration. For these reasons, we evaluate in environments that satisfy all three criteria instead.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 40.5, + "bbox_fs": [ + 105, + 576, + 505, + 666 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 683, + 318, + 696 + ], + "lines": [ + { + "bbox": [ + 105, + 682, + 320, + 698 + ], + "spans": [ + { + "bbox": [ + 105, + 682, + 320, + 698 + ], + "score": 1.0, + "content": "C CLOSELY RELATED PRIOR METHODS", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 45 + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "In our main results we use Expert Distillation (ED) as a baseline. Here, we discuss how the most", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 721, + 486, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 486, + 733 + ], + "score": 1.0, + "content": "closely related methods differ from AD and why ED is sufficient to support the paper’s claims.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 46.5, + "bbox_fs": [ + 105, + 709, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 182 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 94 + ], + "score": 1.0, + "content": "Expert Distillation (ED): ED is most similar to Gato (Reed et al., 2022), which models expert", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "sequences from a converged RL policy using a causal transformer. ED also trains a causal transformer", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 506, + 117 + ], + "score": 1.0, + "content": "to predict actions using expert policy data. There are two key differences between ED and Gato. First,", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "unlike Gato which utilizes small (relative to an episode length) within-episode contexts, ED is trained", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 507, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 507, + 139 + ], + "score": 1.0, + "content": "on the same across-episode contexts as AD, so the architectures used by ED and AD are the same.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 135, + 505, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 135, + 505, + 151 + ], + "score": 1.0, + "content": "The benefits of AD cannot therefore be attributed to across-episode contexts alone but also learning", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 148, + 505, + 160 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 505, + 160 + ], + "score": 1.0, + "content": "progress in the offline data used to train AD. Second, ED models state-action-reward sequences while", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 159, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 106, + 159, + 506, + 172 + ], + "score": 1.0, + "content": "Gato models only state-action sequences. The main difference between ED and AD is that AD is", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 169, + 401, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 401, + 183 + ], + "score": 1.0, + "content": "trained on full multi-task learning histories rather than expert policy data.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 107, + 187, + 505, + 276 + ], + "lines": [ + { + "bbox": [ + 105, + 187, + 506, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 506, + 199 + ], + "score": 1.0, + "content": "Decision Transformer (DT) (Chen et al., 2021) and Multi-Game Decision Transformer (Lee et al.,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "score": 1.0, + "content": "2022): DTs learn return-conditioned policies from single-task offline data collected by an RL agent.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "score": 1.0, + "content": "While the training data itself (an RL agent’s replay buffer) contains learning, the context sizes", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 220, + 505, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 505, + 233 + ], + "score": 1.0, + "content": "used in DT are too small to capture any learning progress or identify the task using across-episode", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 231, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 394, + 243 + ], + "score": 1.0, + "content": "information. For instance, the Atari experiments use a context of length", + "type": "text" + }, + { + "bbox": [ + 394, + 231, + 428, + 242 + ], + "score": 0.64, + "content": "3 0 - 5 0", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 231, + 471, + 243 + ], + "score": 1.0, + "content": "tokens, or", + "type": "text" + }, + { + "bbox": [ + 471, + 231, + 505, + 242 + ], + "score": 0.57, + "content": "1 0 - 1 7", + "type": "inline_equation" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 242, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 505, + 254 + ], + "score": 1.0, + "content": "transitions. Atari games can have hundreds or thousands of transitions in a single episode, which", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 252, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 505, + 266 + ], + "score": 1.0, + "content": "means these contexts capture mostly within-episode information. Additionally, very little learning", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 264, + 438, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 438, + 276 + ], + "score": 1.0, + "content": "progress happens in the underlying replay buffer data within that many transitions.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 12.5 + }, + { + "type": "text", + "bbox": [ + 107, + 280, + 505, + 357 + ], + "lines": [ + { + "bbox": [ + 105, + 280, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 505, + 293 + ], + "score": 1.0, + "content": "Another difference between DT and AD / ED is that DT learns a return-conditioned model whereas", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 292, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 505, + 304 + ], + "score": 1.0, + "content": "AD / ED are both reward-conditioned. In our setting return-conditioning alone cannot yield an", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 303, + 506, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 506, + 315 + ], + "score": 1.0, + "content": "optimal policy since the agent does not know the task until after it explores the environment and can", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 314, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 505, + 326 + ], + "score": 1.0, + "content": "identify it using across-episode contexts. Since (i) DT uses small within-episodic contexts and (ii)", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 325, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 505, + 338 + ], + "score": 1.0, + "content": "return-conditioning would not help in the environments considered, this baseline is similar to ED", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 335, + 506, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 506, + 348 + ], + "score": 1.0, + "content": "with a small within-episode context which is strictly weaker than the long across-episode context", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 347, + 214, + 358 + ], + "spans": [ + { + "bbox": [ + 106, + 347, + 214, + 358 + ], + "score": 1.0, + "content": "variant of ED we consider.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 363, + 505, + 462 + ], + "lines": [ + { + "bbox": [ + 106, + 363, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 505, + 375 + ], + "score": 1.0, + "content": "Trajectory Transformer (TTO) (Janner et al., 2021): Like AD, TTO also models state-action-reward", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 374, + 506, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 506, + 387 + ], + "score": 1.0, + "content": "tokens but in addition to predicting actions it also learns a world model by predicting states and", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 385, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 506, + 398 + ], + "score": 1.0, + "content": "rewards. To maximize return, TTO then uses beam search to select high-reward actions. However,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 396, + 506, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 506, + 409 + ], + "score": 1.0, + "content": "in our setting, TTO will run into the same problem as DT. To model rewards accurately it will", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 408, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 506, + 420 + ], + "score": 1.0, + "content": "need longer across-episodic contexts since one environment supports many tasks. Similar to DT,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 417, + 506, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 506, + 431 + ], + "score": 1.0, + "content": "MGDT, and Gato, TTO uses smaller within-episode contexts. For this reason, TTO will fare no better", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 430, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 430, + 505, + 441 + ], + "score": 1.0, + "content": "than DT, MGDT, or ED in the settings we consider. We also note that in contrast to TTO, AD is", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 440, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 506, + 453 + ], + "score": 1.0, + "content": "model-free. In AD, actions are sampled from the transformer history-conditioned predictions and", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 451, + 455, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 455, + 464 + ], + "score": 1.0, + "content": "return maximization emerges from modeling the learning histories of an RL algorithm.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 468, + 505, + 534 + ], + "lines": [ + { + "bbox": [ + 105, + 467, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 505, + 480 + ], + "score": 1.0, + "content": "To summarize, AD differs from prior methods mainly because its context is across-episodic and hence", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 478, + 505, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 505, + 492 + ], + "score": 1.0, + "content": "large enough to capture learning progress and task information. AD could further be augmented by", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 490, + 505, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 490, + 505, + 503 + ], + "score": 1.0, + "content": "learning world models like TTO or conditioning on returns like DT, but these investigations would be", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 501, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 505, + 513 + ], + "score": 1.0, + "content": "well suited for future work since they are tangential to the main research question addressed in this", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 512, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 505, + 524 + ], + "score": 1.0, + "content": "work – whether in-context RL can emerge by imitating the learning histories of an RL algorithm with", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 523, + 229, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 229, + 536 + ], + "score": 1.0, + "content": "long across-episodic contexts.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 35.5 + }, + { + "type": "text", + "bbox": [ + 107, + 540, + 504, + 595 + ], + "lines": [ + { + "bbox": [ + 105, + 539, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 506, + 552 + ], + "score": 1.0, + "content": "AD is also closely related to prior work in in-context meta-RL. While both AD and in-context", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 551, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 505, + 563 + ], + "score": 1.0, + "content": "meta-RL model across-episodic histories with memory-based architectures, prior in-context meta-RL", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 561, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 505, + 574 + ], + "score": 1.0, + "content": "algorithms, such as RL2 (Duan et al., 2016) are trained online and rely on learning multi-episodic", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 570, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 506, + 587 + ], + "score": 1.0, + "content": "value functions with TD learning while AD is trained offline and uses a supervised imitation learning", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 583, + 148, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 148, + 597 + ], + "score": 1.0, + "content": "objective.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 41 + } + ], + "page_idx": 14, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2023", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 310, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 13 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 182 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 94 + ], + "score": 1.0, + "content": "Expert Distillation (ED): ED is most similar to Gato (Reed et al., 2022), which models expert", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "sequences from a converged RL policy using a causal transformer. ED also trains a causal transformer", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 506, + 117 + ], + "score": 1.0, + "content": "to predict actions using expert policy data. There are two key differences between ED and Gato. First,", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "unlike Gato which utilizes small (relative to an episode length) within-episode contexts, ED is trained", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 507, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 507, + 139 + ], + "score": 1.0, + "content": "on the same across-episode contexts as AD, so the architectures used by ED and AD are the same.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 135, + 505, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 135, + 505, + 151 + ], + "score": 1.0, + "content": "The benefits of AD cannot therefore be attributed to across-episode contexts alone but also learning", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 148, + 505, + 160 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 505, + 160 + ], + "score": 1.0, + "content": "progress in the offline data used to train AD. Second, ED models state-action-reward sequences while", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 159, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 106, + 159, + 506, + 172 + ], + "score": 1.0, + "content": "Gato models only state-action sequences. The main difference between ED and AD is that AD is", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 169, + 401, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 401, + 183 + ], + "score": 1.0, + "content": "trained on full multi-task learning histories rather than expert policy data.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 4, + "bbox_fs": [ + 105, + 82, + 507, + 183 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 187, + 505, + 276 + ], + "lines": [ + { + "bbox": [ + 105, + 187, + 506, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 506, + 199 + ], + "score": 1.0, + "content": "Decision Transformer (DT) (Chen et al., 2021) and Multi-Game Decision Transformer (Lee et al.,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "score": 1.0, + "content": "2022): DTs learn return-conditioned policies from single-task offline data collected by an RL agent.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "score": 1.0, + "content": "While the training data itself (an RL agent’s replay buffer) contains learning, the context sizes", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 220, + 505, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 505, + 233 + ], + "score": 1.0, + "content": "used in DT are too small to capture any learning progress or identify the task using across-episode", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 231, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 394, + 243 + ], + "score": 1.0, + "content": "information. For instance, the Atari experiments use a context of length", + "type": "text" + }, + { + "bbox": [ + 394, + 231, + 428, + 242 + ], + "score": 0.64, + "content": "3 0 - 5 0", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 231, + 471, + 243 + ], + "score": 1.0, + "content": "tokens, or", + "type": "text" + }, + { + "bbox": [ + 471, + 231, + 505, + 242 + ], + "score": 0.57, + "content": "1 0 - 1 7", + "type": "inline_equation" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 242, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 505, + 254 + ], + "score": 1.0, + "content": "transitions. Atari games can have hundreds or thousands of transitions in a single episode, which", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 252, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 505, + 266 + ], + "score": 1.0, + "content": "means these contexts capture mostly within-episode information. Additionally, very little learning", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 264, + 438, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 438, + 276 + ], + "score": 1.0, + "content": "progress happens in the underlying replay buffer data within that many transitions.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 187, + 506, + 276 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 280, + 505, + 357 + ], + "lines": [ + { + "bbox": [ + 105, + 280, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 505, + 293 + ], + "score": 1.0, + "content": "Another difference between DT and AD / ED is that DT learns a return-conditioned model whereas", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 292, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 505, + 304 + ], + "score": 1.0, + "content": "AD / ED are both reward-conditioned. In our setting return-conditioning alone cannot yield an", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 303, + 506, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 506, + 315 + ], + "score": 1.0, + "content": "optimal policy since the agent does not know the task until after it explores the environment and can", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 314, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 505, + 326 + ], + "score": 1.0, + "content": "identify it using across-episode contexts. 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To maximize return, TTO then uses beam search to select high-reward actions. However,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 396, + 506, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 506, + 409 + ], + "score": 1.0, + "content": "in our setting, TTO will run into the same problem as DT. To model rewards accurately it will", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 408, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 506, + 420 + ], + "score": 1.0, + "content": "need longer across-episodic contexts since one environment supports many tasks. Similar to DT,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 417, + 506, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 506, + 431 + ], + "score": 1.0, + "content": "MGDT, and Gato, TTO uses smaller within-episode contexts. For this reason, TTO will fare no better", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 430, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 430, + 505, + 441 + ], + "score": 1.0, + "content": "than DT, MGDT, or ED in the settings we consider. We also note that in contrast to TTO, AD is", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 440, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 506, + 453 + ], + "score": 1.0, + "content": "model-free. In AD, actions are sampled from the transformer history-conditioned predictions and", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 451, + 455, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 455, + 464 + ], + "score": 1.0, + "content": "return maximization emerges from modeling the learning histories of an RL algorithm.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 363, + 506, + 464 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 468, + 505, + 534 + ], + "lines": [ + { + "bbox": [ + 105, + 467, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 505, + 480 + ], + "score": 1.0, + "content": "To summarize, AD differs from prior methods mainly because its context is across-episodic and hence", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 478, + 505, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 505, + 492 + ], + "score": 1.0, + "content": "large enough to capture learning progress and task information. 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Here we", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "confirm that AD can produce a faster algorithm than the one it was trained on in the single-stream", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 644, + 506, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 506, + 656 + ], + "score": 1.0, + "content": "setting. 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We trained AD on different numbers of Dark Key-to-Door training", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 483, + 506, + 494 + ], + "spans": [ + { + "bbox": [ + 106, + 483, + 506, + 494 + ], + "score": 1.0, + "content": "tasks and evaluated the resulting models on the same set of test tasks. Figure 12 shows the in-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 493, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 506, + 506 + ], + "score": 1.0, + "content": "context learning plots for the resulting AD models on the set of test tasks. As a reminder, there are", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 502, + 506, + 517 + ], + "spans": [ + { + "bbox": [ + 106, + 503, + 155, + 514 + ], + "score": 0.9, + "content": "8 1 ^ { 2 } = 6 5 6 1", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 502, + 506, + 517 + ], + "score": 1.0, + "content": "unique Dark Key-to-Door tasks. 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While models trained on 37, 75 and 151 tasks did not achieve", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 525, + 507, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 507, + 538 + ], + "score": 1.0, + "content": "good performance overall, they did exhibit some in-context learning over the course of 300 episodes.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 536, + 506, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 437, + 550 + ], + "score": 1.0, + "content": "The best models were trained on 1212 and 2424 tasks which corresponds to roughly", + "type": "text" + }, + { + "bbox": [ + 437, + 537, + 456, + 548 + ], + "score": 0.88, + "content": "1 8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 536, + 473, + 550 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 474, + 537, + 493, + 547 + ], + "score": 0.88, + "content": "3 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 536, + 506, + 550 + ], + "score": 1.0, + "content": "of", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 546, + 343, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 343, + 560 + ], + "score": 1.0, + "content": "the total number of tasks in the Dark Key-to-Door domain.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 460, + 507, + 560 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 575, + 356, + 588 + ], + "lines": [ + { + "bbox": [ + 104, + 574, + 358, + 590 + ], + "spans": [ + { + "bbox": [ + 104, + 574, + 358, + 590 + ], + "score": 1.0, + "content": "J SINGLE-STREAM ALGORITHM DISTILLATION", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 106, + 600, + 506, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 600, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 506, + 613 + ], + "score": 1.0, + "content": "We provide more details around the experimental setup for the single-stream result shown in Fig. 6.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 610, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 506, + 623 + ], + "score": 1.0, + "content": "We showed in Fig. 4 that when AD is trained on data from a subset of the actors of a distributed", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 621, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 506, + 635 + ], + "score": 1.0, + "content": "source RL algorithm, the resulting model is more data efficient than the source algorithm. Here we", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "confirm that AD can produce a faster algorithm than the one it was trained on in the single-stream", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 644, + 506, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 506, + 656 + ], + "score": 1.0, + "content": "setting. 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However, we also find that the Transformer variant", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 503, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 505, + 516 + ], + "score": 1.0, + "content": "consistently outperforms the LSTM variant, which is why all other experiments in this paper employ", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 515, + 505, + 526 + ], + "spans": [ + { + "bbox": [ + 106, + 515, + 505, + 526 + ], + "score": 1.0, + "content": "the Transformer variant. This finding is consistent with the recent wider success of Transformer-based", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 525, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 506, + 538 + ], + "score": 1.0, + "content": "architectures over recurrent neural network (RNN)-based architectures in sequence prediction tasks.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26.5 + }, + { + "type": "title", + "bbox": [ + 107, + 553, + 337, + 565 + ], + "lines": [ + { + "bbox": [ + 105, + 552, + 338, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 338, + 567 + ], + "score": 1.0, + "content": "M DISTILLING ACTIONS VS PROBABILITIES", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 577, + 504, + 610 + ], + "lines": [ + { + "bbox": [ + 105, + 576, + 506, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 506, + 591 + ], + "score": 1.0, + "content": "In the rest of the paper, we use (one-hot) actions taken by the source policy as the prediction target", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 588, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 505, + 600 + ], + "score": 1.0, + "content": "for AD. Here, we compare that choice to predicting the source policy probabilities from which that", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 599, + 429, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 599, + 429, + 612 + ], + "score": 1.0, + "content": "action was sampled. In other words, is it better to distill actions or probabilities?", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 106, + 615, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 105, + 615, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 505, + 629 + ], + "score": 1.0, + "content": "Figure 15 compares all combinations of: 1) distilling actions vs probabilities, and 2) conditioning", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 627, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 505, + 640 + ], + "score": 1.0, + "content": "those predictions on past actions, probabilities, or both. The plot labels represent various input/output", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 639, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 181, + 650 + ], + "score": 1.0, + "content": "combinations, e.g.", + "type": "text" + }, + { + "bbox": [ + 182, + 639, + 219, + 649 + ], + "score": 0.33, + "content": "\\mathtt { S a r - } > \\mathtt { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 639, + 505, + 650 + ], + "score": 1.0, + "content": "indicates observing states, actions, and rewards while predicting actions", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 649, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 331, + 662 + ], + "score": 1.0, + "content": "(i.e. the main variant of AD in the rest of the paper), sar", + "type": "text" + }, + { + "bbox": [ + 331, + 650, + 350, + 660 + ], + "score": 0.6, + "content": "- > \\mathrm { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 649, + 505, + 662 + ], + "score": 1.0, + "content": "represents using the same observations", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 659, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 265, + 673 + ], + "score": 1.0, + "content": "but instead predicting probabilities, and", + "type": "text" + }, + { + "bbox": [ + 265, + 662, + 308, + 671 + ], + "score": 0.84, + "content": "\\tt S a p r - > a", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 659, + 506, + 673 + ], + "score": 1.0, + "content": "represents observing states, actions, probabilities,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 670, + 274, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 274, + 684 + ], + "score": 1.0, + "content": "and rewards while predicting actions, etc.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 36.5 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "Two conclusions stand out from this plot. First, the original action distillation variant of AD", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 107, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 107, + 700, + 149, + 710 + ], + "score": 0.59, + "content": "( s \\mathsf { a r } - \\mathsf { > a } )", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 699, + 385, + 711 + ], + "score": 1.0, + "content": "performs best, followed closely by probability distillation", + "type": "text" + }, + { + "bbox": [ + 386, + 700, + 427, + 710 + ], + "score": 0.83, + "content": "( \\mathsf { s a r - > p } )", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 699, + 505, + 711 + ], + "score": 1.0, + "content": ". In the paragraphs", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "and experiments below, we explore why action distillation outperforms probability distillation.", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "score": 1.0, + "content": "Second, all variants that condition on past probabilities catastrophically fail. 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Values of", + "type": "text" + }, + { + "bbox": [ + 243, + 325, + 283, + 336 + ], + "score": 0.56, + "content": "0 . 3 - 0 . 5", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 325, + 506, + 338 + ], + "score": 1.0, + "content": "perform the best with the value of 0.3 chosen for all", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 336, + 160, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 160, + 348 + ], + "score": 1.0, + "content": "experiments.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 302, + 506, + 348 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 363, + 424, + 376 + ], + "lines": [ + { + "bbox": [ + 105, + 363, + 425, + 377 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 425, + 377 + ], + "score": 1.0, + "content": "L AD NETWORK ARCHITECTURE: TRANSFORMER VS LSTM", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 387, + 505, + 465 + ], + "lines": [ + { + "bbox": [ + 106, + 387, + 506, + 400 + ], + "spans": [ + { + "bbox": [ + 106, + 387, + 506, + 400 + ], + "score": 1.0, + "content": "Here we consider the importance of the Transformer architecture to the success of algorithm dis-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 398, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 506, + 410 + ], + "score": 1.0, + "content": "tillation (AD) by comparing to AD based off of an LSTM (Hochreiter & Schmidhuber, 1997).", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 410, + 505, + 422 + ], + "spans": [ + { + "bbox": [ + 106, + 410, + 371, + 422 + ], + "score": 1.0, + "content": "Specifically, the LSTM receives the concatenated embeddings of", + "type": "text" + }, + { + "bbox": [ + 371, + 410, + 413, + 421 + ], + "score": 0.92, + "content": "\\left( o _ { i } , a _ { i } , r _ { i } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 410, + 505, + 422 + ], + "score": 1.0, + "content": "triplets up to the most", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 419, + 504, + 434 + ], + "spans": [ + { + "bbox": [ + 104, + 419, + 171, + 434 + ], + "score": 1.0, + "content": "recent time step", + "type": "text" + }, + { + "bbox": [ + 172, + 421, + 194, + 431 + ], + "score": 0.84, + "content": "t - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 419, + 494, + 434 + ], + "score": 1.0, + "content": ". The output of the LSTM is then concatenated with the current observation", + "type": "text" + }, + { + "bbox": [ + 495, + 423, + 504, + 432 + ], + "score": 0.81, + "content": "o _ { t }", + "type": "inline_equation" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 431, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 106, + 431, + 505, + 444 + ], + "score": 1.0, + "content": "embedding and both are then fed through a multi-layer perceptron (MLP) policy torso to produce a", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 442, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 106, + 442, + 244, + 455 + ], + "score": 1.0, + "content": "distribution over the present action", + "type": "text" + }, + { + "bbox": [ + 245, + 444, + 254, + 453 + ], + "score": 0.84, + "content": "a _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 442, + 505, + 455 + ], + "score": 1.0, + "content": ". The LSTM hidden size (512), MLP depth (2), and MLP width", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 453, + 444, + 466 + ], + "spans": [ + { + "bbox": [ + 106, + 453, + 444, + 466 + ], + "score": 1.0, + "content": "(256) were swept and tuned by grid search based on downstream reward attainment.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20, + "bbox_fs": [ + 104, + 387, + 506, + 466 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 470, + 505, + 537 + ], + "lines": [ + { + "bbox": [ + 106, + 470, + 505, + 482 + ], + "spans": [ + { + "bbox": [ + 106, + 470, + 505, + 482 + ], + "score": 1.0, + "content": "Comparing Transformer AD and LSTM AD on the Dark Key-to-Door task (Figure 14), we find", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 480, + 506, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 506, + 494 + ], + "score": 1.0, + "content": "that both agents are capable of in-context learning, demonstrating that the success of AD is not", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 492, + 506, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 506, + 505 + ], + "score": 1.0, + "content": "tied to the underlying network architecture. However, we also find that the Transformer variant", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 503, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 505, + 516 + ], + "score": 1.0, + "content": "consistently outperforms the LSTM variant, which is why all other experiments in this paper employ", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 515, + 505, + 526 + ], + "spans": [ + { + "bbox": [ + 106, + 515, + 505, + 526 + ], + "score": 1.0, + "content": "the Transformer variant. This finding is consistent with the recent wider success of Transformer-based", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 525, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 506, + 538 + ], + "score": 1.0, + "content": "architectures over recurrent neural network (RNN)-based architectures in sequence prediction tasks.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 470, + 506, + 538 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 553, + 337, + 565 + ], + "lines": [ + { + "bbox": [ + 105, + 552, + 338, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 338, + 567 + ], + "score": 1.0, + "content": "M DISTILLING ACTIONS VS PROBABILITIES", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 577, + 504, + 610 + ], + "lines": [ + { + "bbox": [ + 105, + 576, + 506, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 506, + 591 + ], + "score": 1.0, + "content": "In the rest of the paper, we use (one-hot) actions taken by the source policy as the prediction target", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 588, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 505, + 600 + ], + "score": 1.0, + "content": "for AD. Here, we compare that choice to predicting the source policy probabilities from which that", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 599, + 429, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 599, + 429, + 612 + ], + "score": 1.0, + "content": "action was sampled. In other words, is it better to distill actions or probabilities?", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 576, + 506, + 612 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 615, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 105, + 615, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 505, + 629 + ], + "score": 1.0, + "content": "Figure 15 compares all combinations of: 1) distilling actions vs probabilities, and 2) conditioning", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 627, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 505, + 640 + ], + "score": 1.0, + "content": "those predictions on past actions, probabilities, or both. The plot labels represent various input/output", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 639, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 181, + 650 + ], + "score": 1.0, + "content": "combinations, e.g.", + "type": "text" + }, + { + "bbox": [ + 182, + 639, + 219, + 649 + ], + "score": 0.33, + "content": "\\mathtt { S a r - } > \\mathtt { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 639, + 505, + 650 + ], + "score": 1.0, + "content": "indicates observing states, actions, and rewards while predicting actions", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 649, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 331, + 662 + ], + "score": 1.0, + "content": "(i.e. the main variant of AD in the rest of the paper), sar", + "type": "text" + }, + { + "bbox": [ + 331, + 650, + 350, + 660 + ], + "score": 0.6, + "content": "- > \\mathrm { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 649, + 505, + 662 + ], + "score": 1.0, + "content": "represents using the same observations", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 659, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 265, + 673 + ], + "score": 1.0, + "content": "but instead predicting probabilities, and", + "type": "text" + }, + { + "bbox": [ + 265, + 662, + 308, + 671 + ], + "score": 0.84, + "content": "\\tt S a p r - > a", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 659, + 506, + 673 + ], + "score": 1.0, + "content": "represents observing states, actions, probabilities,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 670, + 274, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 274, + 684 + ], + "score": 1.0, + "content": "and rewards while predicting actions, etc.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 36.5, + "bbox_fs": [ + 105, + 615, + 506, + 684 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "Two conclusions stand out from this plot. First, the original action distillation variant of AD", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 107, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 107, + 700, + 149, + 710 + ], + "score": 0.59, + "content": "( s \\mathsf { a r } - \\mathsf { > a } )", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 699, + 385, + 711 + ], + "score": 1.0, + "content": "performs best, followed closely by probability distillation", + "type": "text" + }, + { + "bbox": [ + 386, + 700, + 427, + 710 + ], + "score": 0.83, + "content": "( \\mathsf { s a r - > p } )", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 699, + 505, + 711 + ], + "score": 1.0, + "content": ". In the paragraphs", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "and experiments below, we explore why action distillation outperforms probability distillation.", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "score": 1.0, + "content": "Second, all variants that condition on past probabilities catastrophically fail. We speculate on", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 517, + 506, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 506, + 530 + ], + "score": 1.0, + "content": "two somewhat contradictory reasons why this might occur: 1) The AD prediction task involves a", + "type": "text", + "cross_page": true + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 529, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 529, + 505, + 541 + ], + "score": 1.0, + "content": "combination of inferring the current policy, as well as predicting when policy updates will occur. Since", + "type": "text", + "cross_page": true + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 540, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 505, + 552 + ], + "score": 1.0, + "content": "observing policy probabilities provides more information about the current policy than actions do, it is", + "type": "text", + "cross_page": true + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 550, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 506, + 564 + ], + "score": 1.0, + "content": "possible that this leads to AD fully focusing on inferring the current policy and ignoring the prediction", + "type": "text", + "cross_page": true + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 561, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 505, + 574 + ], + "score": 1.0, + "content": "of policy updates. In other words, observing only actions taken may act as a useful information", + "type": "text", + "cross_page": true + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 571, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 104, + 571, + 506, + 585 + ], + "score": 1.0, + "content": "bottleneck. 2) On the other hand, observing probabilities may inadvertently leak information about", + "type": "text", + "cross_page": true + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 584, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 584, + 505, + 595 + ], + "score": 1.0, + "content": "when policy updates occur. If the transformer context includes multiple visits to the same state, AD", + "type": "text", + "cross_page": true + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "could learn to compare the policy probabilities to infer whether a policy update occurred in between", + "type": "text", + "cross_page": true + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 604, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 618 + ], + "score": 1.0, + "content": "the two visits. If AD learned during training to rely on this information, then during autoregressive", + "type": "text", + "cross_page": true + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 616, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 506, + 629 + ], + "score": 1.0, + "content": "evaluation, it may be “waiting“ for a policy update than never occurs. If this is the source of the issue,", + "type": "text", + "cross_page": true + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "then a potential solution would be retraining with explicit policy update tokens and including them", + "type": "text", + "cross_page": true + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 639, + 430, + 651 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 430, + 651 + ], + "score": 1.0, + "content": "between evaluation episodes, however we leave this investigation to future work.", + "type": "text", + "cross_page": true + } + ], + "index": 37 + } + ], + "index": 41.5, + "bbox_fs": [ + 105, + 687, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 218, + 81, + 393, + 218 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 218, + 81, + 393, + 218 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 218, + 81, + 393, + 218 + ], + "spans": [ + { + "bbox": [ + 218, + 81, + 393, + 218 + ], + "score": 0.968, + "type": "image", + "image_path": "454e8467c64103db58787ec5b2c5c859eefc5a040f070dda6d155b0f1e59d8e6.jpg" + } + ] + } + ], + "index": 4.5, + "virtual_lines": [ + { + "bbox": [ + 218, + 81, + 393, + 94.7 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 218, + 94.7, + 393, + 108.4 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 218, + 108.4, + 393, + 122.10000000000001 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 218, + 122.10000000000001, + 393, + 135.8 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 218, + 135.8, + 393, + 149.5 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 218, + 149.5, + 393, + 163.2 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 218, + 163.2, + 393, + 176.89999999999998 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 218, + 176.89999999999998, + 393, + 190.59999999999997 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 218, + 190.59999999999997, + 393, + 204.29999999999995 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 218, + 204.29999999999995, + 393, + 217.99999999999994 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 228, + 506, + 263 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 228, + 506, + 242 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 506, + 242 + ], + "score": 1.0, + "content": "Figure 14: Comparison between algorithm distillation with a Transformer and LSTM architecture on", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 240, + 505, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 209, + 252 + ], + "score": 1.0, + "content": "Dark Key-to-Door. 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Since", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 540, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 505, + 552 + ], + "score": 1.0, + "content": "observing policy probabilities provides more information about the current policy than actions do, it is", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 550, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 506, + 564 + ], + "score": 1.0, + "content": "possible that this leads to AD fully focusing on inferring the current policy and ignoring the prediction", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 561, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 505, + 574 + ], + "score": 1.0, + "content": "of policy updates. In other words, observing only actions taken may act as a useful information", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 571, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 104, + 571, + 506, + 585 + ], + "score": 1.0, + "content": "bottleneck. 2) On the other hand, observing probabilities may inadvertently leak information about", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 584, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 584, + 505, + 595 + ], + "score": 1.0, + "content": "when policy updates occur. If the transformer context includes multiple visits to the same state, AD", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "could learn to compare the policy probabilities to infer whether a policy update occurred in between", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 604, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 618 + ], + "score": 1.0, + "content": "the two visits. If AD learned during training to rely on this information, then during autoregressive", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 616, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 506, + 629 + ], + "score": 1.0, + "content": "evaluation, it may be “waiting“ for a policy update than never occurs. If this is the source of the issue,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "then a potential solution would be retraining with explicit policy update tokens and including them", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 639, + 430, + 651 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 430, + 651 + ], + "score": 1.0, + "content": "between evaluation episodes, however we leave this investigation to future work.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 31.5 + }, + { + "type": "text", + "bbox": [ + 106, + 654, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 322, + 667 + ], + "score": 1.0, + "content": "Now we return to the question of why action distillation", + "type": "text" + }, + { + "bbox": [ + 322, + 656, + 365, + 666 + ], + "score": 0.8, + "content": "\\left( s \\mathsf { a r } - \\mathsf { > a } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 654, + 505, + 667 + ], + "score": 1.0, + "content": "outperforms probability distillation", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 107, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 107, + 667, + 149, + 677 + ], + "score": 0.66, + "content": "( s \\mathsf { a r } \\mathrm { - } \\mathsf { > p } )", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 665, + 505, + 678 + ], + "score": 1.0, + "content": ". Intuitively, distilling one-hot actions will result in a more deterministic policy than", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "distilling probabilities when training on finite data. Indeed, the red curves in the top two plots", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 461, + 701 + ], + "score": 1.0, + "content": "of Figure 16 show that probability distillation (top left) converges to a higher entropy", + "type": "text" + }, + { + "bbox": [ + 461, + 688, + 484, + 699 + ], + "score": 0.79, + "content": "{ \\sim } 1 . 2", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 687, + 506, + 701 + ], + "score": 1.0, + "content": "bits)", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 291, + 712 + ], + "score": 1.0, + "content": "policy than does action distillation (top right,", + "type": "text" + }, + { + "bbox": [ + 291, + 699, + 306, + 709 + ], + "score": 0.83, + "content": "{ \\sim } 1", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "bit). Noting this discrepancy, we speculated that", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "encouraging probability distillation towards a more deterministic policy might lead to increased", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "score": 1.0, + "content": "performance. To do so, we experimented with an entropy penalty regularizer added to the AD loss", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 41 + } + ], + "page_idx": 19, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2023", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 15 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 218, + 81, + 393, + 218 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 218, + 81, + 393, + 218 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 218, + 81, + 393, + 218 + ], + "spans": [ + { + "bbox": [ + 218, + 81, + 393, + 218 + ], + "score": 0.968, + "type": "image", + "image_path": "454e8467c64103db58787ec5b2c5c859eefc5a040f070dda6d155b0f1e59d8e6.jpg" + } + ] + } + ], + "index": 4.5, + "virtual_lines": [ + { + "bbox": [ + 218, + 81, + 393, + 94.7 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 218, + 94.7, + 393, + 108.4 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 218, + 108.4, + 393, + 122.10000000000001 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 218, + 122.10000000000001, + 393, + 135.8 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 218, + 135.8, + 393, + 149.5 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 218, + 149.5, + 393, + 163.2 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 218, + 163.2, + 393, + 176.89999999999998 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 218, + 176.89999999999998, + 393, + 190.59999999999997 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 218, + 190.59999999999997, + 393, + 204.29999999999995 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 218, + 204.29999999999995, + 393, + 217.99999999999994 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 228, + 506, + 263 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 228, + 506, + 242 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 506, + 242 + ], + "score": 1.0, + "content": "Figure 14: Comparison between algorithm distillation with a Transformer and LSTM architecture on", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 240, + 505, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 209, + 252 + ], + "score": 1.0, + "content": "Dark Key-to-Door. 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Noting this discrepancy, we speculated that", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "encouraging probability distillation towards a more deterministic policy might lead to increased", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "score": 1.0, + "content": "performance. 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The bottom two panels show", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 172, + 505, + 185 + ], + "spans": [ + { + "bbox": [ + 106, + 172, + 505, + 185 + ], + "score": 1.0, + "content": "the corresponding changes in evaluation returns. 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Interestingly, this is the amount of regularization that leads probability distillation to have the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 214, + 506, + 230 + ], + "spans": [ + { + "bbox": [ + 104, + 214, + 376, + 230 + ], + "score": 1.0, + "content": "most similar entropy to the unregularized action distillation variant", + "type": "text" + }, + { + "bbox": [ + 376, + 217, + 390, + 227 + ], + "score": 0.72, + "content": "_ { \\sim 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 214, + 506, + 230 + ], + "score": 1.0, + "content": "bit) as well. Further entropy", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 227, + 505, + 240 + ], + "spans": [ + { + "bbox": [ + 105, + 227, + 505, + 240 + ], + "score": 1.0, + "content": "penalization for action distillation, on the other hand, does not lead to increased performance (bottom", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 238, + 506, + 251 + ], + "spans": [ + { + "bbox": [ + 105, + 238, + 506, + 251 + ], + "score": 1.0, + "content": "right). 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Colors", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 586, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 487, + 600 + ], + "score": 1.0, + "content": "indicate the strength of the weight on the entropy penalty regularizer from equation 7. Mean", + "type": "text" + }, + { + "bbox": [ + 487, + 587, + 505, + 597 + ], + "score": 0.73, + "content": "\\pm \\nobreakspace 1 \\nobreakspace", + "type": "inline_equation" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 597, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 487, + 610 + ], + "score": 1.0, + "content": "standard deviation over 5 training seeds and 20 evaluation seeds. 1000 episodes corresponds to", + "type": "text" + }, + { + "bbox": [ + 488, + 598, + 505, + 608 + ], + "score": 0.29, + "content": "5 0 \\mathrm { k }", + "type": "inline_equation" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 608, + 184, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 184, + 622 + ], + "score": 1.0, + "content": "environment steps.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20 + } + ], + "index": 18.0 + }, + { + "type": "text", + "bbox": [ + 106, + 632, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 632, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 646 + ], + "score": 1.0, + "content": "In addition to closing gap between probability and action distillation with an entropy penalty, we", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 321, + 656 + ], + "score": 1.0, + "content": "also explored retuning various hyperparameters for", + "type": "text" + }, + { + "bbox": [ + 321, + 645, + 358, + 655 + ], + "score": 0.87, + "content": "\\mathtt { S a r - > p }", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "rather than reusing those tuned for", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 654, + 507, + 669 + ], + "spans": [ + { + "bbox": [ + 106, + 656, + 144, + 666 + ], + "score": 0.82, + "content": "\\mathtt { S a r - } > \\mathtt { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 654, + 507, + 669 + ], + "score": 1.0, + "content": ". We found that many hyperparameters, such as the dropout rate, attention dropout rate,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "and sequence mask probability, had similar optimal values in the two cases, and so retuning them", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "did not help. However, one hyperparameter that did have an effect was the size of the transformer", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "score": 1.0, + "content": "context window (Figure 17). While action distillation performance increased only up to a context", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "window size of 200 steps (4 episodes) and then plateaued (Figure 17, right), probability distillation", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "score": 1.0, + "content": "performance continued to increase up to a context window size of 300 steps (6 episodes) before", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "plateauing (Figure 17, left). Thus, while a comparison at our default context window size of 200", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 27 + } + ], + "page_idx": 20, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2023", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 310, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 312, + 765 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 312, + 765 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 14 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 108, + 82, + 310, + 94 + ], + "lines": [], + "index": 0, + "bbox_fs": [ + 106, + 81, + 311, + 96 + ], + "lines_deleted": true + }, + { + "type": "interline_equation", + "bbox": [ + 146, + 99, + 464, + 135 + ], + "lines": [ + { + "bbox": [ + 146, + 99, + 464, + 135 + ], + "spans": [ + { + "bbox": [ + 146, + 99, + 464, + 135 + ], + "score": 0.93, + "content": "\\tilde { \\mathcal { L } } ( \\theta , \\alpha ) : = \\sum _ { n = 1 } ^ { N } \\sum _ { t = 1 } ^ { T - 1 } - \\log P _ { \\theta } ( A = a _ { t } ^ { ( n ) } | h _ { t - 1 } ^ { ( n ) } , o _ { t } ^ { ( n ) } ) + \\alpha H \\Big [ P _ { \\theta } ( A | h _ { t - 1 } ^ { ( n ) } , o _ { t } ^ { ( n ) } ) \\Big ] ,", + "type": "interline_equation", + "image_path": "81eed58a4490c319f199c6610f33f8e7ff8e47c1f9af370938c7e2703e38b223.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 146, + 99, + 464, + 111.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 146, + 111.0, + 464, + 123.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 146, + 123.0, + 464, + 135.0 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 138, + 505, + 260 + ], + "lines": [ + { + "bbox": [ + 106, + 139, + 505, + 152 + ], + "spans": [ + { + "bbox": [ + 106, + 139, + 134, + 152 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 139, + 153, + 151 + ], + "score": 0.91, + "content": "H [ \\cdot ]", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 139, + 302, + 152 + ], + "score": 1.0, + "content": "is the Shannon entropy in bits, and", + "type": "text" + }, + { + "bbox": [ + 302, + 142, + 310, + 149 + ], + "score": 0.78, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 310, + 139, + 505, + 152 + ], + "score": 1.0, + "content": "is a regularization weight. The top two panels", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 150, + 506, + 164 + ], + "spans": [ + { + "bbox": [ + 105, + 150, + 284, + 164 + ], + "score": 1.0, + "content": "of Figure 16 show the effect of increasing", + "type": "text" + }, + { + "bbox": [ + 284, + 153, + 291, + 160 + ], + "score": 0.73, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 150, + 506, + 164 + ], + "score": 1.0, + "content": "on decreasing policy entropy for action distillation", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 161, + 506, + 175 + ], + "spans": [ + { + "bbox": [ + 105, + 161, + 506, + 175 + ], + "score": 1.0, + "content": "(top right) and probability distillation (top left), throughout training. The bottom two panels show", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 172, + 505, + 185 + ], + "spans": [ + { + "bbox": [ + 106, + 172, + 505, + 185 + ], + "score": 1.0, + "content": "the corresponding changes in evaluation returns. 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Interestingly, this is the amount of regularization that leads probability distillation to have the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 214, + 506, + 230 + ], + "spans": [ + { + "bbox": [ + 104, + 214, + 376, + 230 + ], + "score": 1.0, + "content": "most similar entropy to the unregularized action distillation variant", + "type": "text" + }, + { + "bbox": [ + 376, + 217, + 390, + 227 + ], + "score": 0.72, + "content": "_ { \\sim 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 214, + 506, + 230 + ], + "score": 1.0, + "content": "bit) as well. Further entropy", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 227, + 505, + 240 + ], + "spans": [ + { + "bbox": [ + 105, + 227, + 505, + 240 + ], + "score": 1.0, + "content": "penalization for action distillation, on the other hand, does not lead to increased performance (bottom", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 238, + 506, + 251 + ], + "spans": [ + { + "bbox": [ + 105, + 238, + 506, + 251 + ], + "score": 1.0, + "content": "right). Together, these results suggest that action distillation naturally leads to the “optimal“ amount", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 249, + 406, + 262 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 406, + 262 + ], + "score": 1.0, + "content": "of entropy regularization on its own, at least in the environments we study.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 9, + "bbox_fs": [ + 104, + 139, + 506, + 262 + ] + }, + { + "type": "image", + "bbox": [ + 125, + 273, + 484, + 554 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 125, + 273, + 484, + 554 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 125, + 273, + 484, + 554 + ], + "spans": [ + { + "bbox": [ + 125, + 273, + 484, + 554 + ], + "score": 0.978, + "type": "image", + "image_path": "1193b4fb8b446f1e1eb9a0060b68dfb52c35743a1269cf2b5dabdd292afbbd8e.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 125, + 273, + 484, + 366.6666666666667 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 125, + 366.6666666666667, + 484, + 460.33333333333337 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 125, + 460.33333333333337, + 484, + 554.0 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 565, + 505, + 619 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 565, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 106, + 565, + 505, + 577 + ], + "score": 1.0, + "content": "Figure 16: Policy entropy during training and reward during evaluation for the action distillation", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 107, + 576, + 504, + 587 + ], + "spans": [ + { + "bbox": [ + 107, + 577, + 150, + 587 + ], + "score": 0.67, + "content": "( s \\mathsf { a r } - \\mathsf { > a } )", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 576, + 266, + 587 + ], + "score": 1.0, + "content": "and probability distillation", + "type": "text" + }, + { + "bbox": [ + 266, + 577, + 309, + 587 + ], + "score": 0.83, + "content": "( \\mathsf { s a r - > p } )", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 576, + 504, + 587 + ], + "score": 1.0, + "content": "variants of AD on Dark Key-to-Door. Colors", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 586, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 487, + 600 + ], + "score": 1.0, + "content": "indicate the strength of the weight on the entropy penalty regularizer from equation 7. Mean", + "type": "text" + }, + { + "bbox": [ + 487, + 587, + 505, + 597 + ], + "score": 0.73, + "content": "\\pm \\nobreakspace 1 \\nobreakspace", + "type": "inline_equation" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 597, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 487, + 610 + ], + "score": 1.0, + "content": "standard deviation over 5 training seeds and 20 evaluation seeds. 1000 episodes corresponds to", + "type": "text" + }, + { + "bbox": [ + 488, + 598, + 505, + 608 + ], + "score": 0.29, + "content": "5 0 \\mathrm { k }", + "type": "inline_equation" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 608, + 184, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 184, + 622 + ], + "score": 1.0, + "content": "environment steps.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20 + } + ], + "index": 18.0 + }, + { + "type": "text", + "bbox": [ + 106, + 632, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 632, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 646 + ], + "score": 1.0, + "content": "In addition to closing gap between probability and action distillation with an entropy penalty, we", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 321, + 656 + ], + "score": 1.0, + "content": "also explored retuning various hyperparameters for", + "type": "text" + }, + { + "bbox": [ + 321, + 645, + 358, + 655 + ], + "score": 0.87, + "content": "\\mathtt { S a r - > p }", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "rather than reusing those tuned for", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 654, + 507, + 669 + ], + "spans": [ + { + "bbox": [ + 106, + 656, + 144, + 666 + ], + "score": 0.82, + "content": "\\mathtt { S a r - } > \\mathtt { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 654, + 507, + 669 + ], + "score": 1.0, + "content": ". We found that many hyperparameters, such as the dropout rate, attention dropout rate,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "and sequence mask probability, had similar optimal values in the two cases, and so retuning them", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "did not help. However, one hyperparameter that did have an effect was the size of the transformer", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "score": 1.0, + "content": "context window (Figure 17). While action distillation performance increased only up to a context", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "window size of 200 steps (4 episodes) and then plateaued (Figure 17, right), probability distillation", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "score": 1.0, + "content": "performance continued to increase up to a context window size of 300 steps (6 episodes) before", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "plateauing (Figure 17, left). 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HyperparameterDark RoomDark Room (Hard)Dark Key-to-DoorWatermaze
Embedding Dim.64
Number of Layers
Number of Heads4
Feedforward Dim.2048
Position EncodingsAbsolute
LayerNorm PlacementPost Norm
Dropout Rate0.1
Context Window600 tokens (200 timesteps)
Attention Dropout Rate00.50.5
Sequence Mask Prob0.5 0.30.50.30.3
Label Smoothing α00.200
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HyperparameterValue
Batch Size128
OptimizerAdam
β0.9
0.99
Gradient Clip Norm Threshold1
Learning Rate ScheduleCosine Decay
Initial Value2e-6
Peak Value3e-4
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LayerHyperparameterValue
Conv Block
ConvChannel128
Kernel5
Stride2
BatchNormDecay Rate0.999
Activationeps1e-5
-ReLU
Max PoolingKernel2
Stride2
DropoutRate0.2
Network
Conv Blocks=3
FinalLinearLayerUnits256
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HyperparameterDark RoomDark Room (Hard)Dark Key-to-DoorWatermaze
Embedding Dim.64
Number of Layers
Number of Heads4
Feedforward Dim.2048
Position EncodingsAbsolute
LayerNorm PlacementPost Norm
Dropout Rate0.1
Context Window600 tokens (200 timesteps)
Attention Dropout Rate00.50.5
Sequence Mask Prob0.5 0.30.50.30.3
Label Smoothing α00.200
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HyperparameterValue
Batch Size128
OptimizerAdam
β0.9
0.99
Gradient Clip Norm Threshold1
Learning Rate ScheduleCosine Decay
Initial Value2e-6
Peak Value3e-4
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LayerHyperparameterValue
Conv Block
ConvChannel128
Kernel5
Stride2
BatchNormDecay Rate0.999
Activationeps1e-5
-ReLU
Max PoolingKernel2
Stride2
DropoutRate0.2
Network
Conv Blocks=3
FinalLinearLayerUnits256
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Hyperparameter Value
Batch Size (Num.Actors)100
入0.95
Agent Discount0.99
Entropy Bonus Weight0.01
MLPLayers3
MLPHidden Dim128
OptimizerAdam
β0.9
β20.999
E1e-6
Learning Rate1e-4
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HyperparameterValue
Batch Size8
Rollout Length40
Rollout Overlap31
Number of Actors16
Reply Buffer Capacity1e5
Offline Data Fraction0.7
入0.75
E0.01
Agent Discount0.9
Target Update Period50
ResNet Channels[32, 64, 64]
ResNetKernels[3,3,3]
ResNet Strides[1, 1, 1]
Pool Kernels[3,3,3]
Pool Strides[2,2,2]
OptimizerAdam
β0.9
阳0.999
E1e-6
Gradient Clip Norm Threshold10
Learning Rate1e-4
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Hyperparameter Value
Batch Size (Num.Actors)100
入0.95
Agent Discount0.99
Entropy Bonus Weight0.01
MLPLayers3
MLPHidden Dim128
OptimizerAdam
β0.9
β20.999
E1e-6
Learning Rate1e-4
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HyperparameterValue
Batch Size8
Rollout Length40
Rollout Overlap31
Number of Actors16
Reply Buffer Capacity1e5
Offline Data Fraction0.7
入0.75
E0.01
Agent Discount0.9
Target Update Period50
ResNet Channels[32, 64, 64]
ResNetKernels[3,3,3]
ResNet Strides[1, 1, 1]
Pool Kernels[3,3,3]
Pool Strides[2,2,2]
OptimizerAdam
β0.9
阳0.999
E1e-6
Gradient Clip Norm Threshold10
Learning Rate1e-4
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HyperparameterValue
RL AlgorithmA3C
Learning Rate3e-4
Batch Size256
Unroll Length20
LSTM Hidden Dim. LSTMNumber ofLayers256
Episodes Per Trial2 10
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HyperparameterValue
RL AlgorithmDQN(Q-入)
Learning Rate1e-4
Batch Size96
Unroll Length40
LSTM Hidden Dim.256
LSTMNumber of Layers1
Episodes Per Trial30
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HyperparameterValue
RL AlgorithmA3C
Learning Rate3e-4
Batch Size256
Unroll Length20
LSTM Hidden Dim. LSTMNumber ofLayers256
Episodes Per Trial2 10
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HyperparameterValue
RL AlgorithmDQN(Q-入)
Learning Rate1e-4
Batch Size96
Unroll Length40
LSTM Hidden Dim.256
LSTMNumber of Layers1
Episodes Per Trial30
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HyperparameterDark RoomDark Room (Hard)Dark Key-to-DoorWatermaze
Embedding Dim.64
Number of Layers
Number of Heads4
Feedforward Dim.2048
Position EncodingsAbsolute
LayerNorm PlacementPost Norm
Dropout Rate0.1
Context Window600 tokens (200 timesteps)
Attention Dropout Rate00.50.5
Sequence Mask Prob0.5 0.30.50.30.3
Label Smoothing α00.200
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HyperparameterValue
Batch Size128
OptimizerAdam
β0.9
0.99
Gradient Clip Norm Threshold1
Learning Rate ScheduleCosine Decay
Initial Value2e-6
Peak Value3e-4
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LayerHyperparameterValue
Conv Block
ConvChannel128
Kernel5
Stride2
BatchNormDecay Rate0.999
Activationeps1e-5
-ReLU
Max PoolingKernel2
Stride2
DropoutRate0.2
Network
Conv Blocks=3
FinalLinearLayerUnits256
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HyperparameterValue
Batch Size8
Rollout Length40
Rollout Overlap31
Number of Actors16
Reply Buffer Capacity1e5
Offline Data Fraction0.7
入0.75
E0.01
Agent Discount0.9
Target Update Period50
ResNet Channels[32, 64, 64]
ResNetKernels[3,3,3]
ResNet Strides[1, 1, 1]
Pool Kernels[3,3,3]
Pool Strides[2,2,2]
OptimizerAdam
β0.9
阳0.999
E1e-6
Gradient Clip Norm Threshold10
Learning Rate1e-4
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Hyperparameter Value
Batch Size (Num.Actors)100
入0.95
Agent Discount0.99
Entropy Bonus Weight0.01
MLPLayers3
MLPHidden Dim128
OptimizerAdam
β0.9
β20.999
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Invariant Quantities + +Anonymous Author(s) +Affiliation +Address +email + +# Abstract + +1 Neural networks have shown promise for modeling dynamical systems from data. +2 Recent models, such as Hamiltonian neural networks, have been designed to +3 ensure known geometric structures of target systems and have shown excellent +4 modeling accuracy. However, in most situations where neural networks learn +5 unknown systems, their underlying structures are also unknown. Even in such +6 cases, one can expect that target systems are associated with first integrals (a.k.a. in +7 variant quantities), which are quantities remaining unchanged over time. First +8 integrals come from the conservation laws of system energy, momentum, and mass, +9 from constraints on states, and from other features of governing equations. By +10 leveraging projection methods and discrete gradient methods, we propose first +11 integral-preserving neural differential equations (FINDE). The proposed FINDE +12 finds and preserves first integrals from data, even in the absence of prior knowl +13 edge about the underlying structures. Experimental results demonstrate that the +14 proposed FINDE is able to predict future states of given systems much longer and +15 find various quantities consistent with well-known first integrals of the systems in +16 a unified manner. + +# 17 1 Introduction + +18 Although neural networks have achieved remarkable results in image and natural language pro +19 cessing [17, 28], they have also been actively investigated for modeling dynamical systems [41]. +20 Target systems include the chemical dynamics to accelerate computer simulations [46], the climate +21 dynamics for climate change prediction and weather forecasting [47, 52], and the physical dynamics +22 of vehicles and robots for optimal control [41]. Their history dates back to at least the 1990s, and +23 many approaches have been proposed so far (see [7, 12, 35, 40, 49, 55] for example). Recently, +24 neural ordinary differential equation (NODE) has redefined neural networks for continuous-time +25 dynamics [8]. A target system is described by an ordinary differential equation (ODE) $\begin{array} { r } { \frac { \mathrm { d } } { \mathrm { d } t } \pmb { u } = \pmb { f } ( t , \pmb { u } ) } \end{array}$ +26 where $\textbf { \em u }$ denotes the system state. Then, a NODE replaces the vector field $f$ with a neural network +27 and employs a numerical integrator to obtain a solution ${ \mathbf { } } { \mathbf { } } { \mathbf { } } { \mathbf { } } { \mathbf { } } { \mathbf { } } { \mathbf { } } { \mathbf { } } { \mathbf { } } { \mathbf { } } { \mathbf { } } \mathbf { } { \mathbf { } } \mathbf { } { \mathbf { } } \mathbf { } { \mathbf { } } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf \Psi \mathbf { } \Psi \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf \Psi { } \mathbf \Psi \mathbf { } \mathbf { } \mathbf \Psi \Psi \mathbf { } \mathbf { } \mathbf \Psi \mathbf { } \mathbf \Psi \mathbf { } \mathbf \Psi \Psi \mathbf { } \mathbf \mathbf { } \mathbf \Psi \mathbf { } \mathbf \mathbf { } \mathbf \mathbf \Psi \Psi \mathbf { } \mathbf \mathbf \Psi \Psi \mathbf { } \mathbf \mathbf \Psi \mathbf { } \mathbf \mathbf \Psi \mathbf \Psi \Psi \mathbf \Psi \mathbf { \mathbf } \mathbf \mathbf \Psi \mathbf \Psi \mathbf \Psi \mathbf \mathbf \mathbf \Psi \mathbf \Psi \mathbf \Psi \mathbf \mathbf \Psi \mathbf \Psi \mathbf \mathbf \Psi \mathbf \Psi \mathbf \mathbf \Psi \mathbf \mathbf \Psi \mathbf \mathbf \mathbf \Psi \mathbf \mathbf \mathbf \Psi \mathbf \mathbf \mathbf \Psi \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \Psi \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \Psi \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf $ . +28 Most real-world systems are associated with first integrals (a.k.a. invariant quantities), which are +29 quantities remaining unchanged over time [27]. If a system has a first integral $V ( { \pmb u } )$ , the solution +30 ${ \mathbf { } } { \mathbf { } } { \mathbf { } } { \mathbf { } } { \mathbf { } } { \mathbf { } } { \mathbf { } } { \mathbf { } } { \mathbf { } } { \mathbf { } } { \mathbf { } } \mathbf { } { \mathbf { } } \mathbf { } { \mathbf { } } \mathbf { } { \mathbf { } } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf \Psi { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf \Psi { } \mathbf \Psi { } \mathbf { } \mathbf \Psi \mathbf { } \mathbf { } \mathbf { } \mathbf \Psi \mathbf { } \mathbf { } \mathbf \Psi \mathbf { } \mathbf { } \mathbf \Psi \mathbf { } \mathbf \Psi \mathbf { } \mathbf \Psi \mathbf { } \mathbf \Psi \mathbf { } \mathbf \mathbf { } \mathbf \mathbf { \Psi } \mathbf \mathbf \Psi \mathbf { } \mathbf \mathbf \Psi \Psi \mathbf { } \mathbf \mathbf \mathbf { \Psi \mathbf } \mathbf \mathbf \Psi \Psi \mathbf \Psi \mathbf { \mathbf } \mathbf \mathbf \mathbf \Psi \mathbf \mathbf \Psi \mathbf \mathbf \Psi \mathbf \Psi \mathbf \Psi \mathbf \Psi \mathbf \Psi \mathbf \mathbf \Psi \mathbf \mathbf \Psi \mathbf \mathbf \Psi \mathbf \mathbf \Psi \mathbf \mathbf \mathbf \Psi \mathbf \mathbf \mathbf \Psi \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \Psi \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \Psi \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf $ for the initial condition $\pmb { u } ( 0 )$ remains at a contour line $V ( \mathbf { \boldsymbol { u } } ( t ) ) = V ( \mathbf { \boldsymbol { u } } ( 0 ) )$ over time. Many +31 previous studies have attempted to learn a target system accurately by incorporating prior knowledge +32 about first integrals. Greydanus et al. [26] proposed Hamiltonian neural network (HNN), which +33 employs a neural network to approximate Hamilton’s equation, thereby conserving the system energy +34 called the Hamiltonian. Finzi et al. [19] proposed neural network architectures that conserve linear +35 and angular momenta by utilizing the graph structure. Finzi et al. [20] also extended HNN to a system +36 with holonomic constraints, which lead to first integrals such as a pendulum length. Matsubara et al. +37 [38] proposed a model that preserves the total mass of a discretized partial differential equation +38 (PDE). These studies have demonstrated that a neural network with more prior knowledge about first +39 integrals predicts the dynamics of the target system more accurately. See Table 1 for comparison. +40 Previous studies have mainly attempted to preserve known first integrals. However, in situations +41 where a neural network learns an unknown target system, it is naturally expected that first integrals +42 associated with the target system are also unknown, and it is not clear which of the above methods are +43 available. Given the above, this study proposes First Integral-preserving Neural Differential Equation +44 (FINDE) to find and preserve first integrals from data. FINDE has the following advantages. +45 Learning First Integrals For modeling continuous-time dynamics with known first integrals, many +46 studies have designed architectures or operations of neural networks [13, 19, 20, 26, 38]. For each +47 type of first integral, one dedicated method was proposed. However, the properties of a target system +48 are generally unknown in practice. In contrast, the proposed FINDE finds various kinds of first +49 integrals from data in a unified manner and preserves them in predictions. A symbolic regression +50 confirms that the learned first integrals are consistent with well-known first integrals of target systems. + +Table 1: Comparison between Related Studies on Preservation of First Integrals. + +
energymonentum massconstraint learning invariants exact conservation
NODE [8]
HNN [26]√
LieConv [19]√厂
DGNet [38]√√
CHNN [20]√√
continuous FINDE (proposed)√√√√√
discrete FINDE (proposed)√厂厂丁
+ +Combination with Known First Integrals The proposed FINDE can be combined with previously proposed neural networks designed to preserve known first integrals, such as HNN. Therefore, FINDE is available in various situations. + +54 Exact Preservation of First Integrals Even if a first integral is associated with a continuous-time +55 system, it is destroyed after the system is discretized in time for computer simulations. This is true +56 even when using a symplectic integrator, which preserves the system energy only approximately [27]. +57 By leveraging discrete gradients [38], the discrete-time version of FINDE preserves first integrals +58 exactly (up to rounding errors) in discrete time and further improves the prediction performance. + +# 59 2 Background and Related Work + +60 First Integrals Let us consider a time-invariant differential system $\begin{array} { r } { \frac { \mathrm { d } } { \mathrm { d } t } \pmb { u } \ = \ f ( \pmb { u } ) } \end{array}$ on an $N$ - +61 dimensional manifold $\mathcal { M }$ , where $\textbf { \em u }$ denotes the system state and $f : \mathcal { M } ^ { } \to \mathcal { T } _ { u } \mathcal { M }$ represents a +62 vector field on the manifold $\mathcal { M }$ . The manifold $\mathcal { M }$ can be $\mathcal { M } = S ^ { 1 } \times \mathbb { R } ^ { 1 }$ for a pendulum. In this +63 paper, we suppose the manifold $\mathcal { M }$ be a Eucleadian space $\mathbb { R } ^ { N }$ for simplicity. +64 Definition 1 (first integral). $A$ quantity $V : { \mathcal { M } } \mathbb { R }$ is referred to as a first integral of a system +65 $\begin{array} { r } { \frac { \mathrm { d } } { \mathrm { d } t } \pmb { u } = f ( \pmb { u } ) } \end{array}$ if it remains constant along with any solution ${ \mathbf { } } { \mathbf { } } { \mathbf { } } { \mathbf { } } { \mathbf { } } { \mathbf { } } { \mathbf { } } { \mathbf { } } { \mathbf { } } { \mathbf { } } \mathbf { } { \mathbf { } } \mathbf { } { \mathbf { } } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf \Psi \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf \mathbf { } \mathbf { } \mathbf { } \mathbf \Psi { } \mathbf \mathbf { } \mathbf { } \mathbf \Psi \mathbf { } \mathbf { } \mathbf \mathbf { } \mathbf \Psi { } \mathbf \mathbf { } \mathbf \mathbf { } \mathbf \mathbf { } \mathbf \Psi \mathbf { } \mathbf \mathbf { } \mathbf \mathbf { } \mathbf \mathbf { } \mathbf \mathbf \Psi \mathbf { } \mathbf \mathbf { } \mathbf \mathbf \mathbf { } \mathbf \mathbf \mathbf { } \mathbf \mathbf \mathbf { \mathbf } \mathbf \mathbf \mathbf \mathbf \Psi \mathbf \Psi \mathbf { } \mathbf \mathbf \mathbf \mathbf \mathbf { } \mathbf \mathbf \mathbf \mathbf \mathbf \Psi \mathbf \mathbf { } \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \Psi \mathbf \mathbf \mathbf \mathbf \mathbf \Psi \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf$ , i.e., $\begin{array} { r } { \frac { \mathrm { d } } { \mathrm { d } t } V ( \pmb { u } ) = 0 } \end{array}$ . + +ntial system $\begin{array} { r } { \frac { \mathrm { d } } { \mathrm { d } t } { \pmb u } = f ( { \pmb u } ) } \end{array}$ is associate ith $K$ functionally independent first integrals $V _ { 1 } , \dots , V _ { K }$ ${ \bf \ddot { u } } ( t )$ given an initial value $\mathbf { \delta } \mathbf { u } _ { 0 }$ stays at the + +$$ +\mathcal { M } ^ { \prime } = \{ \pmb { u } \in \mathcal { M } : V _ { 1 } ( \pmb { u } ) = V _ { 1 } ( \pmb { u } _ { 0 } ) , \allowbreak \dots , V _ { K } ( \pmb { u } ) = V _ { K } ( \pmb { u } _ { 0 } ) \} . +$$ + +69 The tangent space $\mathcal { T } _ { u } \mathcal { M } ^ { \prime } \subset \mathcal { T } _ { u } \mathcal { M }$ of the submanifold $\mathcal { M } ^ { \prime } \subset \mathcal { M }$ at a point $\textbf { \em u }$ is the orthogonal +70 complement to the space spanned by the gradients $\nabla V _ { k } ( { \boldsymbol { \mathbf { \mathit { u } } } } )$ of the first integrals $V _ { k }$ for $k = 1 , \ldots , K$ , +71 that is, + +$$ +\mathcal { T } _ { \boldsymbol { u } } \mathcal { M } ^ { \prime } = \{ \pmb { w } \in \mathcal { T } _ { \boldsymbol { u } } \mathcal { M } : \nabla V _ { k } ( \boldsymbol { u } ) ^ { \top } \pmb { w } = 0 \mathrm { ~ f o r ~ } k = 1 , \dots , K \} +$$ + +If a quantity 72 $V _ { k }$ is a first integral of the system $\begin{array} { r } { \frac { \mathrm { d } } { \mathrm { d } t } \pmb { u } = f ( \pmb { u } ) } \end{array}$ , the time-derivative $f$ at point $\textbf { \em u }$ is on the 73 tangent space $\mathcal { T } _ { u } \mathcal { M } ^ { \prime }$ , being orthogonal to the gradient $\nabla V _ { k }$ of the first integral $V _ { k }$ . Then, it holds that 74 $\begin{array} { r } { \frac { \mathrm { d } } { \mathrm { d } t } \dot { V _ { k } ( \pmb { u } ) } = \nabla V _ { k } ( \pmb { u } ) ^ { \top } \frac { \mathrm { d } } { \mathrm { d } t } \pmb { u } = \bar { \nabla } V _ { k } ( \pmb { u } ) ^ { \top } \pmb { f } ( \bar { \pmb { u } } ) = 0 } \end{array}$ . + +75 One of the most well-known first integrals is the Hamiltonian $H$ , which represents the system energy +76 of a Hamiltonian system. Noether’s theorem states that a continuous symmetry of a system leads to a +77 conservation law (and hence a first integral) [27]; a Hamiltonian system is symmetric to translation +78 in time and conserves the Hamiltonian. Symmetries to translation and rotation in space lead to the +79 conservation of linear and angular momenta. Not all first integrals are related to symmetries. A +80 pendulum can be expressed in Cartesian coordinates, and then the rod length constrains the mass +81 position. This kind of constraint is called a holonomic constraint and leads to a first integral. A model +82 for disease spreading called an susceptible-infected-recovered (SIR) model and the dynamics of +83 chemical reactions have the total mass (population) as a first integral. Also for a system described by +84 a PDE, the total mass is sometimes a first integral [23]. See Appendix A for theoretical classification +85 of dynamics. +86 First Integrals in Numerical Analysis For computer simulations, a differential system is dis +87 cretized in time and solved by numerical integration. Then, the geometric structures of the system +88 are often destroyed, and most first integrals are no longer preserved. A common remedy is a sym +89 plectic integrator, which preserves the symplectic structure and integrates a Hamiltonian system +90 accurately [27]. However, Ge–Marsden theorem states that a symplectic integrator conserves the +91 Hamiltonian only approximately [56]. Hence, many numerical schemes have also been investigated +92 for preserving first integrals exactly, while they cannot preserve the symplectic structure. +93 Let a superscript $s$ denote the state $\pmb { u } ^ { s }$ or time $t ^ { s }$ at $s$ -th time step, and $\Delta t ^ { s } = t ^ { s + 1 } - t ^ { s }$ denote a +94 time step size. A projection method predicts a next state $\tilde { \pmb u } ^ { s + 1 }$ from the current state $\pmb { u } ^ { s }$ using a +95 numerical integrator and projects it onto the submanifold $\mathcal { M } ^ { \prime }$ , obtaining the projected state $\pmb { u } ^ { s + 1 }$ that +96 preserves the first integrals $V _ { k }$ [24] (see also [27, Section IV.4]). In particular, the projected state +97 $\mathbf { \Delta } _ { \mathbf { u } } { } ^ { s + 1 }$ is obtained by solving the optimization problem + +$$ +\boldsymbol { u } ^ { s + 1 } = \operatorname * { a r g m i n } _ { \boldsymbol { u } ^ { \prime } ^ { s + 1 } } | | \boldsymbol { u } ^ { \prime * s + 1 } - \tilde { \boldsymbol { u } } ^ { s + 1 } | | \operatorname * { s u b j e c t } \operatorname { t o } V _ { k } ( \boldsymbol { u } ^ { \prime * s + 1 } ) = V _ { k } ( \boldsymbol { u } ^ { s } ) \operatorname { f o r } k = 1 , \dots , K . +$$ + +98 A local coordinate method defines a coordinate system to the neighborhood of the current state $\pmb { u } ^ { s }$ +99 and integrates a differential equation on it [43] (see also [27, Section IV.5]). A discrete gradient +100 method defines a discrete analogue to a given differential system and integrates it in discrete time [6, +101 23, 25, 29, 44, 45]. This method eliminates numerical errors caused by temporal discretization and is +102 used to preserve the Hamiltonian exactly (up to rounding errors) in discrete time. +103 Except for DGNet, which used discrete gradients to preserve the Hamiltonian [38], all the above +104 methods have never been applied to neural networks due to difficulties that we will introduce later. To +105 our best knowledge, the discrete-time version of FINDE is the first projection method for dynamical +106 systems modeled using neural networks. +107 Preservation of First Integrals by Neural Networks NODE defines an ODE using a neural net +108 work in the most general way with no associated first integrals [8]. NODE is a universal approximator +109 to ODEs [51], and it can approximate any ODE with arbitrary accuracy if there is an infinite amount +110 of training data. In practice, the amount of training data is limited, and prior knowledge about the +111 target system is helpful for learning (see [48] for the case with convolutional neural networks). HNN +112 assumes the target system to be a Hamiltonian system in the canonical form [26]. HNN guarantees +113 various properties of Hamiltonian systems by definition, including the conservation of the energy +114 and the preservation of the symplectic structure in continuous time [27]. Some studies employed a +115 symplectic integrator for HNN to preserve the energy and symplectic structure with smaller numerical +116 errors [10]. LieConv and EMLP-HNN employed neural network architectures with translational +117 and rotational symmetries to preserve momenta [19, 21]. CHNN incorporates a known holonomic +118 constraint in the dynamics [20]. Deep conservation extracts latent dynamics of a PDE system and +119 preserves a quantity of interest by forcing its flux to be zero [34]. $\mathrm { H N N + + }$ also guarantees the +120 conservation of the mass in PDE systems by using a coefficient matrix derived from differential +121 operators [38]. +122 Several studies proposed neural networks to learn Lyapunov functions, which are expected to be +123 non-increasing over time, in contrast to first integrals [37, 50]. If the state moves in the direction of +124 increasing the function, it is projected onto or moved inside the counter line of the gradient of the +125 Lyapunov function. Their idea is similar to the continuous-time version of FINDE but limited to a +126 single non-increasing quantity in continuous time. On the other hand, our proposed FINDE preserves +127 multiple quantities in both continuous and discrete time. +28 Previous studies aimed to preserve known first integrals. Moreover, except for DGNet [38], all +29 the above methods suffer from numerical errors caused by temporal discretization. In contrast, our +30 proposed FINDE learns first integrals from data and can eliminate discretization errors. + +# 131 3 First Integral-Preserving Neural Differential Equation + +132 The main purpose is to find and preserve first integrals from data by neural networks. We suppose +133 that a target system has at least $K$ unknown functionally independent first integrals. Even when +134 a NODE learns the target system, it is not guaranteed to learn these first integrals. Hence, we +135 introduce a neural network with $K$ outputs, each of which is expected to learn one of first integrals +136 expressed as $V _ { k } : \mathbb { R } ^ { N } \mathbb { R }$ for $k = 1 , \dots , K$ . We denote the set of first integrals by a vector +137 $\pmb { V } ( \pmb { u } ) = ( V _ { 1 } ( \pmb { u } ) V _ { 2 } ( \pmb { u } ) \dotsm V _ { K } ( \pmb { u } ) ) ^ { \top }$ . Then, the submanifold $\mathcal { M } ^ { \prime }$ is defined using the neural +138 network $V$ as in Eq. (1). +139 Because there is no way to define local coordinates on such submanifolds, a local coordinate method +140 is not applicable. When using a projection method, the optimization problem in Eq. (3) should +141 be solved at every training iteration as well as in the prediction phase. Optimization problems are +142 computationally expensive, and common libraries for neural networks do not provide backpropagation +143 algorithms for optimization problems [1, 42].1 Until a recent study has proposed an algorithm [38], +144 there was no way to obtain discrete gradients of neural networks. Because of these difficulties, +145 no methods for preserving first integrals have been applied to neural networks. By leveraging a +146 projection method and a discrete gradient method, we propose FINDE as follows. + +# 3.1 Continuous FINDE: Time-Derivative Projection Method + +First, we propose a time-derivative projection method called continuous FINDE (cFINDE) for neural networks, which projects the time-derivative onto the tangent space $\mathcal { T } _ { u } \mathcal { M } ^ { \prime }$ . While it still suffers from numerical errors, it is sufficient to find first integrals from data. + +We suppose that a neural network called a base model defines the time-derivative $\hat { f } : \mathbb { R } ^ { N } \to \mathbb { R } ^ { N }$ Then, we define the time-derivative $f$ of the cFINDE $\begin{array} { r } { \frac { \mathrm { d } } { \mathrm { d } t } \pmb { u } = f ( \pmb { u } ) } \end{array}$ as + +$$ +\begin{array} { r } \boldsymbol { f } ( \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol { \mathbf { \rho } } \boldsymbol \mathbf { \rho } \boldsymbol { \rho } \boldsymbol \end{array} +$$ + +where 153 $\lambda _ { k }$ is a Lagrange multiplier, $\begin{array} { r } { M \mathrm { ~ = ~ } \frac { \partial V } { \partial \mathbf { u } } } \end{array}$ , and $\lambda ( { \pmb u } ) = ( \lambda _ { 1 } ( { \pmb u } ) \lambda _ { 2 } ( { \pmb u } ) \dots \lambda _ { K } ( { \pmb u } ) ) ^ { \top }$ . If $V _ { k }$ + +$$ +\begin{array} { r } { \mathbf { 0 } = \frac { \mathrm { d } } { \mathrm { d } t } { \pmb V } ( { \pmb u } ( t ) ) = M ( \pmb { u } ) \frac { \mathrm { d } } { \mathrm { d } t } { \pmb u } = M ( \pmb { u } ) f ( \pmb { u } ) = M ( \pmb { u } ) ( \hat { f } ( \pmb { u } ) - M ( \pmb { u } ) ^ { \top } \pmb { \lambda } ( \pmb { u } ) ) , } \end{array} +$$ + +where 155 $\mathbf { 0 } = ( 0 \ldots 0 ) ^ { \top }$ . By transforming Eq. (5), we obtain the Lagrange multiplier $\lambda ( { \pmb u } ) =$ 156 $( M ( \pmb { \mathscr { u } } ) M ( \pmb { \mathscr { u } } ) ^ { \top } ) ^ { - 1 } M ( \pmb { \mathscr { u } } ) \hat { f } ( \pmb { \mathscr { u } } )$ . By eliminating it, the cFINDE $\begin{array} { r } { \frac { \mathrm { d } } { \mathrm { d } t } \pmb { u } = f ( \pmb { u } ) } \end{array}$ is given by + +Remark 1 (continuous-time first integral preservation). The cFINDE 57 $\begin{array} { r } { \frac { \mathrm { d } } { \mathrm { d } t } \pmb { u } = f ( \pmb { u } ) } \end{array}$ preserves all first integrals 58 $V _ { k }$ for $k = 1 , \ldots , K$ in continuous time, i.e., $\begin{array} { r } { { \frac { \mathrm { d } } { \mathrm { d } t } } V _ { k } = 0 } \end{array}$ . + +159 The base model $\hat { f }$ can be a NODE, an HNN, or other models depending on available prior knowledge. +160 Also, if a first integral is already known, one can use it directly as one of first integrals $V _ { k }$ instead +161 of learning it using a neural network. Note that even though the base model $\hat { f }$ is an HNN, due to +162 projection, the cFINDE $f$ is no longer a Hamiltonian system in the strict sense. + +Compared to the base model $\hat { f }$ , the cFINDE requires the additional computation of the neural network $V$ , several matrix multiplications, and an inverse operation. The inverse operation needs a computational cost of $O ( K ^ { 3 } )$ , which is not costly if the number $K$ of first integrals is small. For satisfying the constraints and geometric structures, many previous models also need the inverse operation, such as Lagrangian neural network (LNN) [13], neural symplectic form [9], and CHNN [20]. + +69 To eliminate numerical errors caused by temporal discretization, we employ discrete gradients and +70 propose a projection method called discrete FINDE (dFINDE). + +A discrete gradient $\overline { { \nabla } } V$ is a discrete analogue to a gradient $\nabla V$ [6, 23, 25, 29, 44, 45]. Recall that a gradient $\nabla V$ of a function $V : \mathbb { R } ^ { N } \mathbb { R }$ can be regarded as a function $\mathbb { R } ^ { N } \to \mathbb { R } ^ { N }$ that satisfies the chain rule $\begin{array} { r } { \frac { \mathrm { d } } { \mathrm { d } t } V ( \pmb { u } ) = \nabla V ( \pmb { u } ) ^ { \top } \frac { \mathrm { d } } { \mathrm { d } t } \pmb { u } } \end{array}$ . Analogously, a discrete gradient $\overline { \nabla }$ is defined as follows. + +Definition 2 (discrete gradient). A discrete gradient $\overline { { \nabla } } V$ of a function $V : \mathbb { R } ^ { N } \mathbb { R }$ is a function $\mathbb { R } ^ { N } \times \mathbb { R } ^ { N } \xrightarrow [ ] { } \mathbb { R } ^ { N }$ that satisfies + +$$ +V ( \pmb { v } ) - V ( \pmb { u } ) = \overline { { \nabla } } V ( \pmb { v } , \pmb { u } ) ^ { \top } ( \pmb { v } - \pmb { u } ) ~ a n d ~ \overline { { \nabla } } V ( \pmb { u } , \pmb { u } ) = \nabla V ( \pmb { u } ) . +$$ + +176 The first condition is a discrete analogue to the chain rule when replacing the time-derivatives ${ \frac { \mathrm { d } } { \mathrm { d } t } } V$ +177 and ${ \frac { \mathrm { d } } { \mathrm { d } t } } { \pmb u }$ with finite differences $( V ( \pmb { v } ) - V ( \pmb { u } ) )$ and $( { \pmb v } - { \pmb u } )$ , respectively, and the second condition +178 ensures the consistency with the ordinary gradient $\nabla V$ . A discrete gradient $\overline { { \nabla } } V$ is not uniquely +179 determined and has been obtained manually. Recently, the automatic discrete differentiation algorithm +180 (ADDA) has been proposed in [38], which obtains a discrete gradient of a neural network in a similar +181 way to the automatic differentiation algorithm [1, 42]. The discrete gradient is defined in discrete +182 time, and hence a numerical integration using the discrete gradient is free from numerical errors +183 caused by temporal discretization. See Appendix $\mathbf { B }$ and the references [6, 23, 38] for more details. + +Following [11, 15], we introduce a discrete analogue to the tangent space 184 $\mathcal { T } _ { \mathbf { \ b { u } } } \mathcal { M } ^ { \prime }$ called the discrete tangent space 185 $\mathcal { T } _ { ( \pmb { v } , \pmb { u } ) } \mathcal { M } ^ { \prime }$ . In particular, for a pair $( \pmb { v } , \pmb { u } ) \in \mathcal { M } ^ { \prime }$ of points, it is defined as + +$$ +\mathcal { T } _ { ( v , u ) } \mathcal { M } ^ { \prime } = \{ \pmb { w } \in \mathbb { R } ^ { N } : \overline { { \nabla } } V _ { k } ( \pmb { v } , \pmb { u } ) ^ { \top } \pmb { w } = 0 \mathrm { ~ f o r ~ } k = 1 , \ldots , K \} . +$$ + +186 If the finite difference $( \pmb { u } ^ { s + 1 } - \pmb { u } ^ { s } )$ between the predicted and current states is on the discrete +187 tangent space $\mathcal { T } _ { \left( \pmb { u } ^ { s + 1 } , \pmb { u } ^ { s } \right) } \mathcal { M } ^ { \prime }$ , the first integrals $V _ { k }$ are preserved because $V _ { k } ( { \pmb u } ^ { s + 1 } ) - V _ { k } ( { \pmb u } ^ { s } ) =$ +188 $\overline { { \nabla } } V _ { k } ( \pmb { u } ^ { s + 1 } , \pmb { u } ^ { s } ) ^ { \top } ( \pmb { u } ^ { s + 1 } - \pmb { u } ^ { s } ) = 0$ . Note that similar concepts defined in different ways are also +189 referred to as discrete tangent spaces [14, 16]. + +Let ψˆ denote a discrete-time base model that satisfies u˜s+1−us∆ts 190 $\begin{array} { r } { \frac { \tilde { \mathbf { u } } ^ { s + 1 } - \mathbf { u } ^ { s } } { \Delta t ^ { s } } = \hat { \psi } ( \mathbf { u } ^ { s } ; \Delta t ^ { s } ) } \end{array}$ , where $\tilde { { \pmb u } } ^ { s + 1 }$ denotes 191 the predicted state. We assume that the base model $\hat { \psi }$ is composed of a continuous-time base model 192 ˆf and a numerical integrator. Then, the dFINDE us+1−∆ts $\begin{array} { r } { \frac { { \pmb u } ^ { s + 1 } - { \pmb u } ^ { s } } { \Delta t ^ { s } } = \psi ( { \pmb u } ^ { s + 1 } , { \pmb u } ^ { s } ; \Delta t ^ { s } ) } \end{array}$ is given by + +$$ +\psi ( \pmb { u } ^ { s + 1 } , \pmb { u } ^ { s } ; \Delta t ^ { s } ) = \hat { \psi } ( \pmb { u } ^ { s } ; \Delta t ^ { s } ) - \overline { { M } } ( \pmb { u } ^ { s + 1 } , \pmb { u } ^ { s } ) ^ { \top } \lambda ( \pmb { u } ^ { s + 1 } , \pmb { u } ^ { s } ) , +$$ + +where 193 $\overline { { M } } ( { \pmb u } ^ { s + 1 } , { \pmb u } ^ { s } ) = ( \overline { { \nabla } } V _ { 1 } ( { \pmb u } ^ { s + 1 } , { \pmb u } ^ { s } )$ . . . $\overline { { \nabla } } V _ { K } ( \pmb { u } ^ { s + 1 } , \pmb { u } ^ { s } ) ) ^ { \top }$ . As is the case in continuous time, 194 the preservation of the first integrals $V _ { k }$ leads to + +$$ +\begin{array} { r } { \mathbf { 0 } = \frac { V ( u ^ { s + 1 } ) - V ( u ^ { s } ) } { \Delta t ^ { s } } = \overline { { M } } ( u ^ { s + 1 } , u ^ { s } ) \frac { u ^ { s + 1 } - u ^ { s } } { \Delta t ^ { s } } = \overline { { M } } ( u ^ { s + 1 } , u ^ { s } ) \psi ( u ^ { s + 1 } , u ^ { s } ; \Delta t ^ { s } ) . } \end{array} +$$ + +195 Substituting Eq. (9) and eliminating the Lagrange multiplier $\boldsymbol { \lambda }$ , we obtain + +$$ +\psi ( \pmb { u } ^ { s + 1 } , \pmb { u } ^ { s } ; \Delta t ^ { s } ) = ( I - \overline { { Y } } ( \pmb { u } ^ { s + 1 } , \pmb { u } ^ { s } ) ) \hat { \psi } ( \pmb { u } ^ { s } ; \Delta t ^ { s } ) +$$ + +Remark 2 (discrete-time first integral preservation). The dFINDE us+1−us∆ts preserves all first integrals $V _ { k }$ for $k = 1 , \ldots , K$ in discrete time, i.e., $V _ { k } ( { \pmb u } ^ { s + 1 } ) - V _ { k } ( { \pmb u } ^ { s } ) = 0$ $\begin{array} { r } { \frac { { \pmb u } ^ { s + 1 } - { \pmb u } ^ { s } } { \Delta t ^ { s } } = \psi ( { \pmb u } ^ { s + 1 } , { \pmb u } ^ { s } ; \Delta t ^ { s } ) } \end{array}$ + +198 Due to projection, dFINDE can be regarded as a projection method using discrete gradients. For the +199 base model $\hat { \psi }$ , the continuous-time base model $\hat { f }$ can be a NODE, an HNN, or other models, and the +200 numerical integrator can be a Runge–Kutta method, the leapfrog integrator, or others. +201 Because dFINDE is an implicit method, it is computationally expensive for prediction. However, the +202 next state $\boldsymbol { u } ^ { s + 1 }$ is given for training, and the ADDA can explicitly obtain the discrete gradient w.r.t. the +203 pair $( { \pmb u } ^ { s + 1 } , { \pmb u } ^ { s } )$ as well as its computational graph. Thus, dFINDE can be computed explicitly and +204 optimized by standard backpropagation algorithms. Moreover, we suppose that dFINDE projects +205 the finite difference $\hat { \psi }$ only at every time step, whereas cFINDE projects the time-derivative $\hat { f }$ at +206 every substep inside a numerical integrator. Therefore, dFINDE is less computationally expensive +207 than cFINDE for training. In contrast, a typical projection method requires much computational +208 cost to solve an optimization problem for training, and standard backpropagation algorithms are not +209 applicable to it. +10 Remark 3 (trainability). The dFINDE can be trained using the standard backpropagation algorithm, +211 whereas a straightforward application of a projection method cannot. + +Table 2: Datasets, Dynamics, and First Integrals. + +
DatasetDynamicsNFirst Integrals
EnergyMomentumMassConstraint
Two-body problemCanonical Hamiltonian81
Discretized KdV equationNon-canonical Hamiltonian50广1
Double pendulumPoisson8√√
FitzHugh-Nagumo modelDirac4√
+ +# 4 Experiments + +# 4.1 Experimental Settings + +Target Systems We evaluated FINDE and base models using datasets associated with first integrals, summarized in Table 2. A gravitational two-body problem (2-body) on a 2-dimensional configuration space is a typical Hamiltonian system in the canonical form. In addition to the total energy, it has first integrals related to symmetries in space, namely, the linear and angular momenta. The Korteweg–De Vries (KdV) equation is a PDE model of shallow water waves. This is a Hamiltonian system in a non-canonical form and has the Hamiltonian, total mass, and many other quantities as first integrals. A double pendulum (2-pend) is a Hamiltonian system in polar coordinates. However, we transformed it to Cartesian coordinates; it was no longer a Hamiltonian system but a Poisson system. The lengths of two rods work as holonomic constraints and lead to four first integrals. The FitzHugh–Nagumo model is a biological neuron model as an electric circuit, which exhibits a rapid and transient change of voltage called a spike. As an electric circuit, the currents through and voltages applied to the inductor and capacitor can be regarded as system states, and the states are constrained by the circuit topology and Kirchhoff’s current and voltage laws. Then, this system has a state of four elements and two first integrals. Due to energy dissipation in the resistor, the model is not a Poisson system, but one can find a Dirac structure [53]. See Appendix C for more details. + +Implementation We implemented the proposed FINDE and evaluated it under the following settings. We implemented all codes by modifying the officially released codes of HNN [26] 2 and DGNet $[ 3 8 ] ^ { 3 }$ . We used Python v3.8.12 with packages scipy v1.7.3, pytorch v1.10.2, torchdiffeq v0.1.1, functorch v1.10 preview, and gplearn v0.4.2. We used the Dormand–Prince method (dopri5) [18] as the numerical integrator, unless otherwise stated. All experiments were performed on a single NVIDIA A100 provided by (ANONYMOUS PROVIDER). + +Following HNN [26] and DGNet [38], we represented the first integrals $V$ , NODE, and HNN $H$ using fully-connected neural networks with two hidden layers. Each hidden layer had 200 units and preceded a hyperbolic tangent activation function. Each weight matrix was initialized as an orthogonal matrix. The input was the state $\textbf { \em u }$ , and the output represented the first integrals $V$ for FINDE, time-derivative $\hat { f }$ for NODE, and the Hamiltonian $H$ for HNN. For the KdV dataset, we used a 1-dimensional convolutional neural network (CNN), each of whose layers had a kernel size of 3. The double pendulum is a second–order system, implying that the time-derivative ddt q of the position $\pmb q$ is known as the velocity $\textbf { { v } }$ . Hence, we treated only the acceleration $\begin{array} { r } { \frac { \mathrm { d } } { \mathrm { d } t } \pmb { v } } \end{array}$ as the output to learn. This assumption slightly improved the absolute performances but did not change the relative trends. + +We used the $I$ -step error as the loss function to be minimized. In particular, it is the mean squared error (MSE) between the ground truth state $\pmb { u } _ { \mathrm { G T } } ^ { s }$ and the state $\pmb { u } _ { \mathrm { p r e d . } } ^ { s }$ . predicted from the previous step $u _ { \mathrm { G T } } ^ { s - 1 }$ . The base model and FINDE were jointly trained using the Adam optimizer [33] with the parameters $( \beta _ { 1 } , \beta _ { 2 } ) = ( 0 . 9 , 0 . 9 9 9 )$ and a batch size of 200. The learning rate was initialized to $1 0 ^ { - 3 }$ and decayed to zero with a cosine annealing [36]. + +249 Evaluation Metric As an evaluation metric, we used the 1-step error, which is identical to the loss +250 function. We displayed it at the scale of $\times 1 0 ^ { - 9 }$ . The lower this indicator, the better, as emphasized +251 by $\downarrow$ . While several studies used the MSEs of the state or system energy over the whole time +252 series [26, 38], we consider these indicators are misleading, as pointed in several studies [4]. For +253 example, in the case of a periodic orbit, an orbit that is correctly learned except for a slight difference +254 in angular velocity will have the same MSE as an orbit that never moves from its initial position. +255 Instead, we used the valid prediction time $( V P T )$ [4, 32, 54]. VPT denotes the time point $s$ divided by +256 the length $S$ of time series at which the MSE of the predicted state $\pmb { u } _ { \mathrm { p r e d . } } ^ { s }$ . exceeds a given threshold $\theta$ +257 for the first time in an initial value problem, that is, + +$$ +\begin{array} { r } { V P T ( u _ { \mathrm { p r e d . } } ; u _ { \mathrm { G T } } ) = \frac { 1 } { S } \arg \operatorname* { m a x } _ { s _ { f } } \{ s _ { f } | \mathrm { M S E } ( u _ { \mathrm { p r e d . } } ^ { s } , u _ { \mathrm { G T } } ^ { s } ) < \theta \mathrm { f o r } \mathrm { a l l } s \leq s _ { f } \} . } \end{array} +$$ + +258 To obtain VPTs, we normalized each element of state to have zero mean and unit variance in the +259 training data and set $\theta$ to 0.01. The higher this indicator, the better, as emphasized by $\uparrow$ . Because of +260 the “spiking” behavior of the FitzHugh–Nagumo model, a small error in phase is regarded as a large +261 error in state. To measure the qualitative performance, we calculated VPTs by allowing for a delay +262 and advance of up to 5 steps. + +# 4.2 First Integral Preservation for Hamiltonian System + +Before learning first integrals from data, we first evaluated FINDE as a numerical integrator using a known mass-spring system. The system has the state $\mathbf { \boldsymbol { \mathscr { u } } } \doteq ( q v ) ^ { \top }$ , the dynamics ddt q = v and d $\begin{array} { r } { { \frac { \mathrm { d } } { \mathrm { d } t } } v \ = - q } \end{array}$ , and the system energy $E ( q , v ) =$ ${ \scriptstyle { \frac { 1 } { 2 } } } ( q ^ { 2 } + v ^ { 2 } )$ . Using the initial value $( 1 . 0 \ 0 . 0 ) ^ { \top }$ and the time step size $\Delta t = 0 . 2$ , we solved the initial value problem of the true ODE using the leapfrog integrator. We applied FINDE with the true system energy $E$ as the first integral $V$ . Note that no neural networks nor training were involved. + +![](images/e62087da4d5eea6ebb9b01535f06401e481e011b62a41af256ef1f7e2091ccfd.jpg) +Figure 1: Integration of a known mass-spring system. + +273 The results with the analytical solution are shown in Fig. 1. The upper panel shows that the time series +274 predicted by comparison methods overlap each other and are apparently almost identical. However, +275 the lower panel shows that the energy obtained from the states predicted by the leapfrog integrator is +276 fluctuating. The same is true for the case with the cFINDE. This is because the symplectic integrator +277 and the cFINDE suffer from numerical errors caused by temporal discretization. In contrast, the +278 dFINDE preserves the energy accurately. This is because, at every step, the dFINDE projects the state +279 $( q v ) ^ { \top }$ onto the discrete tangent space $\mathcal { T } _ { ( \pmb { v } , \pmb { u } ) } \mathcal { M } ^ { \prime }$ . Although a smaller step size reduces numerical +280 errors, this result demonstrates the advantage of dFINDE. + +# 281 4.3 Learning First Integrals from Data of Hamiltonian System + +82 We evaluated FINDE on learning from the 2-body dataset. We used HNN as the base model $\hat { f }$ . We +83 found that the FINDE got better performances if it did not treat the Hamiltonian $H$ of the HNN +84 as one of first integrals $V _ { k }$ . The medians and standard deviations of 5 trials are summarized in the +85 leftmost column of Table 3. The cFINDE achieved better VPTs than the vanilla HNN with $K = 1$ +86 to 2, and the performance was suddenly degraded for $K = 3$ . The dFINDE showed a similar trend +7 with slightly better performances. The HNN with FINDE found two first integrals in addition to the +88 Hamiltonian $H$ of the HNN. Even though a two-body problem is a Hamiltonian system that HNN +89 can learn, the prior knowledge that there exist first integrals other than the Hamiltonian $H$ can be +90 a clue to better learning. The HNN with FINDE got worse 1-step errors, suggesting that without +FINDE, HNN overfitted short-term change and had difficulty predicting long-term dynamics. + +We performed a symbolic regression of first integrals $V$ learned by the neural network. For $K = 2$ , the learned first integrals $V$ were identical to the linear momenta in the $x$ - and $y$ -directions up to affine transformation in most cases. See Appendix D.1 for more details. + +We depict example results in Fig. 2. In the absence of FINDE, the mass positions $( x _ { 1 } , y _ { 1 } ) , ( x _ { 2 } , y _ { 2 } )$ became inaccurate in a short time and the center-of-gravity position $\textstyle ( x _ { c } , y _ { c } ) = ( { \frac { x _ { 1 } + x _ { 2 } } { 2 } } , { \frac { y _ { 1 } + y _ { 2 } } { 2 } } )$ deviated rapidly. The HNN with cFINDE accurately predicted the state for a longer period. Even after errors in the mass positions became non-negligible, errors in the center-of-gravity position were still small. We show the absolute errors averaged over all trials in Fig. 3. In each of $x$ - and $y$ -directions, + +Table 3: Results of FINDE. + +
2-body + HNNKdV2-pendFitzHugh-Nagumo
ModelK1-step↓VPT个1-step↓VPT个1-step↓VPT个1-step↓VPT个
base model -5.17 ±0.57 0.362 ±0.0265.59 ±0.300.339 ±0.0380.82 ±0.020.110±0.03573.66 ±12.59 0.236 ±0.053
+ cFINDE17.10 ±1.250.374 ±0.0366.24 ±0.440.371 ±0.0880.75 ±0.040.156±0.04254.18 ±8.120.127 ±0.148
27.78 ±1.390.450 ±0.0522.59 ±0.110.608 ±0.0850.73±0.050.198 ±0.08837.03 ±3.810.437 ±0.084
3>1030.147 ±0.146*3.19 ±0.370.730 ±0.0910.69 ±0.030.411 ±0.093>1060.007 ±0.007*
4>1030.101 ±0.0053.65 ±0.300.641 ±0.0710.77 ±0.070.395 ±0.083一
5>1030.080±0.0144.68 ±0.430.601 ±0.0690.80±0.070.585 ±0.097
6>10³0.070 ±0.0197.79 ±0.510.425 ±0.06712.53±0.000.005 ±0.000*
+ dFINDE17.01 ±1.060.379 ±0.04011.61 ±6.600.288 ±0.0830.75 ±0.100.152 ±0.01747.07 ±8.030.117 ±0.122
27.03 ±1.000.475 ±0.0222.70 ±0.260.598 ±0.0590.74±0.050.271 ±0.11133.24 ±3.400.455 ±0.032
3 54.78 ±36.39 0.309 ±0.0243.78±0.270.636 ±0.0240.69±0.050.447 ±0.081319.70 ±91.11 0.049 ±0.007
4>100.102 ±0.0153.48±0.320.780 ±0.0590.71 ±0.030.454 ±0.060
5>1030.086±0.011*5.26 ±0.150.718±0.0380.86 ±0.090.591 ±0.087
6>1030.059 ±0.0179.60 ±3.610.573 ±0.121 58.88 ±22.98 0.037 ±0.039
+ +A standard deviation follows $\pm$ symbol. Underlines indicate results better than the base models’ results, and bolded fonts indicate the best results. ∗ denotes that some trials failed in training because of the underflow of the step size. A dash denotes a case we did not try. + +![](images/89d03c9a7890406db2fbecc814f8b6b34cdc584df01fb3eb8c8146f7deede677.jpg) +Figure 2: Example results of the 2-body dataset. + +![](images/468b3482f1aadb8120b8263d0dcca0d63d619e797bf1f63eb5d4a25bb716d4d7.jpg) +Figure 3: Mean absolute errors of states for the 2-body dataset with or without cFINDE. + +300 the HNN without FINDE produced errors in the center-of-gravity position $x _ { c }$ (or $y _ { c }$ ) and those in the +301 mass positions $x _ { 1 } , x _ { 2 }$ (or $y _ { 1 } , y _ { 2 } )$ at almost the same level. In contrast, when the cFINDE is present, +302 errors in the center-of-gravity position were much smaller than those in the mass positions, implying +303 that errors in one mass position canceled out errors in the other mass position. + +Therefore, we conclude that FINDE not only had better prediction accuracy but also found and preserved linear momenta (which are related to symmetries in space) more accurately despite not having prior knowledge about symmetries. + +# 307 4.4 Learning First Integrals from Data of Unknown Systems + +308 It is often unclear whether a target system is a Hamiltonian system or not, but one can expect that the +309 target system has several first integrals. We evaluated FINDE using NODE as the base model. We +310 summarized the results in Table 3. +311 For the KdV dataset, the NODE with FINDE got much better 1-step errors and VPTs for a wide +312 range of $K$ . Figure 4 shows an example result. The top panels show that the prediction results were +313 apparently similar. The bottom panels summarize mean absolute errors in states $\textbf { \em u }$ , total mass $\textstyle \sum _ { k } u _ { k }$ +314 and energy. In the absence of FINDE, the NODE increased all of its errors in proportion to time. With +315 the cFINDE, the error in total mass increased at the point where the two solitons collided but then +316 returned to the original level. Although the calculation is slightly inaccurate, the cFINDE learned +317 to preserve the total mass. The rightmost panel shows that the error in energy continued to increase +318 for $K = 2$ , but it stayed within a small range for $K = 3$ . These results suggest that the first or +319 second quantity learned by the cFINDE was total mass, the third quantity was system energy, and the +320 remaining quantity may correspond to one of the many first integrals of the KdV equation. +321 For the 2-pend dataset, the NODE with FINDE got better 1-step errors and VPTs for $K = 1$ to 5 +322 except for the 1-step error of the dFINDE with $K = 5$ . In addition to the system energy, the double +323 pendulum has two holonomic constraints on the position, which lead to two additional constraints +324 involving the velocity (see Appendix C for details). Thus, it is reasonable that the NODE with FINDE +325 got the best VPTs for $K = 5$ first integrals and totally failed when assuming $K > 5$ first integrals. +326 As exemplified in Fig. 5, the NODE without FINDE did not preserve the lengths of rods, making +327 the states deviate gradually. See Appendix D.2 for the case when actual constraints are known. For +328 the FitzHugh–Nagumo dataset, the NODE with FINDE got much better 1-step errors and VPTs for +329 $K = 2$ . As exemplified in Fig. 6, the ground truth state converged to a periodic orbit, and only the +330 NODE with cFINDE for $K = 2$ reproduced such dynamics. On the other hand, the state did not +331 stay at a limited region without FINDE and converged to a wrong equilibrium with the cFINDE for +332 $K = 1$ . For $K = 1$ , the sole quantity $V _ { 1 }$ may have tried to learn both of the two first integrals and +333 remained under-trained. In these two cases, FINDE found all first integrals; $K = 5$ for the 2-pend +334 dataset and $K = 2$ for the FitzHugh–Nagumo dataset. + +![](images/be5e28ffd5b113697cd23b60e955f60ccb12a63a936fb9ef9c08be823a75a887.jpg) +Figure 4: Example results of the KdV dataset. (top) Predicted states. Red belts denote moving solitons. (bottom) Mean absolute errors. + +![](images/8556882ad46dc0590252f5c7531273f1da140556e98cc6d80d3d2b4923a070e9.jpg) +Figure 5: Example results of the 2-pend dataset for 2,000 steps. + +![](images/7787089d38eb6cba10618809e53bcab45a879c57163f3cfcbb6474ad1a9cd456.jpg) +Figure 6: Example results of the FitzHugh–Nagumo dataset. + +# 335 5 Conclusion + +This study proposed first integral-preserving neural differential equation (FINDE). FINDE projects the time evolution onto the submanifold defined using the (discrete) gradients of first integrals represented by a neural network. With an appropriate number of assumed first integrals, FINDE predicted future states more accurately than base models. Not only that, FINDE found and preserved the system energy and the total mass as first integrals, first integrals related to symmetries in space, and first integrals led by constraints in a unified manner. Therefore, FINDE has the potential to make a scientific discovery by revealing unknown properties of target dynamical systems. + +The 1-step errors were on the order of $1 0 ^ { - 5 }$ to $1 0 ^ { - 4 }$ in absolute error, being much larger than the numerical error tolerance of $1 0 ^ { - 9 }$ used in the experiments; numerical errors were negligible compared to modeling errors. However, the dFINDE tended to get VPTs better than the cFINDE despite the fact that its advantage is to eliminate numerical errors caused by temporal discretization. This result suggests that a method leading to smaller numerical errors results in a model with smaller modeling errors. Similar tendencies have been observed in previous works [10, 38], and these results may form a new frontier for integrating numerical and modeling errors. + +# References + +[1] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G. 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[N/A] No societal impact is supposed. +(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [N/A] We have read the guidelines carefully, but no ethical impact is supposed. + +2. If you are including theoretical results... + +(a) Did you state the full set of assumptions of all theoretical results? [Yes] We have introduced the background of the proposed method and provided the full set of assumptions in Section 2. Even though a slight modification may make the proposed method available on a general manifold, we have clearly stated that our theoretical and experimental results were limited to the finite-dimensional Eucleadian spaces. +(b) Did you include complete proofs of all theoretical results? [Yes] We have provided a proof just before each of Remarks 1 and 2. + +3. If you ran experiments... + +(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] We have enclosed the source code to reproduce all experiments in supplemental materials. +(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] We have provided the software and hardware environment, network architectures, and hyperparameters in Section 4.1. We have also provided detailed hyperparameters to generate datasets in Appendix C. +(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] We have summarized the standard deviations over five trials in Table 3. +(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] We have provided the hardware environment in Section 4.1, but we have anonymized the cloud service providers to avoid a potential violation of the double-blind policy. + +4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... + +(a) If your work uses existing assets, did you cite the creators? [Yes] We have made the source code for all experiments by modifying the source codes of HNN [26] and DGNet [38]. We have cited these references and added links to respective repositories in the footnotes. +(b) Did you mention the license of the assets? [Yes] We have verified that the source codes of HNN [26] and DGNet [38] are provided in Apache-2.0 License and MIT License, respectively. We have clearly stated these facts in the footnotes. +(c) Did you include any new assets either in the supplemental material or as a URL? [Yes] We have enclosed the source code to generate datasets in supplemental materials. +(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A] +(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A] + +5. If you used crowdsourcing or conducted research with human subjects... + +(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A] + +534 (b) Did you describe any potential participant risks, with links to Institutional Review +535 Board (IRB) approvals, if applicable? [N/A] +536 (c) Did you include the estimated hourly wage paid to participants and the total amount +537 spent on participant compensation? [N/A] \ No newline at end of file diff --git a/parse/dev/xDaoT2zlJ0r/xDaoT2zlJ0r_content_list.json b/parse/dev/xDaoT2zlJ0r/xDaoT2zlJ0r_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..9af8f920d3dcfaddede8d8ab821f55978c405aaa --- /dev/null +++ b/parse/dev/xDaoT2zlJ0r/xDaoT2zlJ0r_content_list.json @@ -0,0 +1,1288 @@ +[ + { + "type": "text", + "text": "FINDE: Neural Differential Equations for Finding and Preserving Invariant Quantities ", + "text_level": 1, + "bbox": [ + 209, + 122, + 789, + 172 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Anonymous Author(s) \nAffiliation \nAddress \nemail ", + "bbox": [ + 423, + 226, + 578, + 281 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Abstract ", + "text_level": 1, + "bbox": [ + 462, + 318, + 535, + 334 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 Neural networks have shown promise for modeling dynamical systems from data. \n2 Recent models, such as Hamiltonian neural networks, have been designed to \n3 ensure known geometric structures of target systems and have shown excellent \n4 modeling accuracy. However, in most situations where neural networks learn \n5 unknown systems, their underlying structures are also unknown. Even in such \n6 cases, one can expect that target systems are associated with first integrals (a.k.a. in \n7 variant quantities), which are quantities remaining unchanged over time. First \n8 integrals come from the conservation laws of system energy, momentum, and mass, \n9 from constraints on states, and from other features of governing equations. By \n10 leveraging projection methods and discrete gradient methods, we propose first \n11 integral-preserving neural differential equations (FINDE). The proposed FINDE \n12 finds and preserves first integrals from data, even in the absence of prior knowl \n13 edge about the underlying structures. Experimental results demonstrate that the \n14 proposed FINDE is able to predict future states of given systems much longer and \n15 find various quantities consistent with well-known first integrals of the systems in \n16 a unified manner. ", + "bbox": [ + 148, + 349, + 766, + 570 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "17 1 Introduction ", + "text_level": 1, + "bbox": [ + 148, + 598, + 312, + 614 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "18 Although neural networks have achieved remarkable results in image and natural language pro \n19 cessing [17, 28], they have also been actively investigated for modeling dynamical systems [41]. \n20 Target systems include the chemical dynamics to accelerate computer simulations [46], the climate \n21 dynamics for climate change prediction and weather forecasting [47, 52], and the physical dynamics \n22 of vehicles and robots for optimal control [41]. Their history dates back to at least the 1990s, and \n23 many approaches have been proposed so far (see [7, 12, 35, 40, 49, 55] for example). Recently, \n24 neural ordinary differential equation (NODE) has redefined neural networks for continuous-time \n25 dynamics [8]. A target system is described by an ordinary differential equation (ODE) $\\begin{array} { r } { \\frac { \\mathrm { d } } { \\mathrm { d } t } \\pmb { u } = \\pmb { f } ( t , \\pmb { u } ) } \\end{array}$ \n26 where $\\textbf { \\em u }$ denotes the system state. Then, a NODE replaces the vector field $f$ with a neural network \n27 and employs a numerical integrator to obtain a solution ${ \\mathbf { } } { \\mathbf { } } { \\mathbf { } } { \\mathbf { } } { \\mathbf { } } { \\mathbf { } } { \\mathbf { } } { \\mathbf { } } { \\mathbf { } } { \\mathbf { } } { \\mathbf { } } \\mathbf { } { \\mathbf { } } \\mathbf { } { \\mathbf { } } \\mathbf { } { \\mathbf { } } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf \\Psi \\mathbf { } \\Psi \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf \\Psi { } \\mathbf \\Psi \\mathbf { } \\mathbf { } \\mathbf \\Psi \\Psi \\mathbf { } \\mathbf { } \\mathbf \\Psi \\mathbf { } \\mathbf \\Psi \\mathbf { } \\mathbf \\Psi \\Psi \\mathbf { } \\mathbf \\mathbf { } \\mathbf \\Psi \\mathbf { } \\mathbf \\mathbf { } \\mathbf \\mathbf \\Psi \\Psi \\mathbf { } \\mathbf \\mathbf \\Psi \\Psi \\mathbf { } \\mathbf \\mathbf \\Psi \\mathbf { } \\mathbf \\mathbf \\Psi \\mathbf \\Psi \\Psi \\mathbf \\Psi \\mathbf { \\mathbf } \\mathbf \\mathbf \\Psi \\mathbf \\Psi \\mathbf \\Psi \\mathbf \\mathbf \\mathbf \\Psi \\mathbf \\Psi \\mathbf \\Psi \\mathbf \\mathbf \\Psi \\mathbf \\Psi \\mathbf \\mathbf \\Psi \\mathbf \\Psi \\mathbf \\mathbf \\Psi \\mathbf \\mathbf \\Psi \\mathbf \\mathbf \\mathbf \\Psi \\mathbf \\mathbf \\mathbf \\Psi \\mathbf \\mathbf \\mathbf \\Psi \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\Psi \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\Psi \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf $ . \n28 Most real-world systems are associated with first integrals (a.k.a. invariant quantities), which are \n29 quantities remaining unchanged over time [27]. If a system has a first integral $V ( { \\pmb u } )$ , the solution \n30 ${ \\mathbf { } } { \\mathbf { } } { \\mathbf { } } { \\mathbf { } } { \\mathbf { } } { \\mathbf { } } { \\mathbf { } } { \\mathbf { } } { \\mathbf { } } { \\mathbf { } } { \\mathbf { } } \\mathbf { } { \\mathbf { } } \\mathbf { } { \\mathbf { } } \\mathbf { } { \\mathbf { } } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf \\Psi { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf \\Psi { } \\mathbf \\Psi { } \\mathbf { } \\mathbf \\Psi \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf \\Psi \\mathbf { } \\mathbf { } \\mathbf \\Psi \\mathbf { } \\mathbf { } \\mathbf \\Psi \\mathbf { } \\mathbf \\Psi \\mathbf { } \\mathbf \\Psi \\mathbf { } \\mathbf \\Psi \\mathbf { } \\mathbf \\mathbf { } \\mathbf \\mathbf { \\Psi } \\mathbf \\mathbf \\Psi \\mathbf { } \\mathbf \\mathbf \\Psi \\Psi \\mathbf { } \\mathbf \\mathbf \\mathbf { \\Psi \\mathbf } \\mathbf \\mathbf \\Psi \\Psi \\mathbf \\Psi \\mathbf { \\mathbf } \\mathbf \\mathbf \\mathbf \\Psi \\mathbf \\mathbf \\Psi \\mathbf \\mathbf \\Psi \\mathbf \\Psi \\mathbf \\Psi \\mathbf \\Psi \\mathbf \\Psi \\mathbf \\mathbf \\Psi \\mathbf \\mathbf \\Psi \\mathbf \\mathbf \\Psi \\mathbf \\mathbf \\Psi \\mathbf \\mathbf \\mathbf \\Psi \\mathbf \\mathbf \\mathbf \\Psi \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\Psi \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\Psi \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf $ for the initial condition $\\pmb { u } ( 0 )$ remains at a contour line $V ( \\mathbf { \\boldsymbol { u } } ( t ) ) = V ( \\mathbf { \\boldsymbol { u } } ( 0 ) )$ over time. Many \n31 previous studies have attempted to learn a target system accurately by incorporating prior knowledge \n32 about first integrals. Greydanus et al. [26] proposed Hamiltonian neural network (HNN), which \n33 employs a neural network to approximate Hamilton’s equation, thereby conserving the system energy \n34 called the Hamiltonian. Finzi et al. [19] proposed neural network architectures that conserve linear \n35 and angular momenta by utilizing the graph structure. Finzi et al. [20] also extended HNN to a system \n36 with holonomic constraints, which lead to first integrals such as a pendulum length. Matsubara et al. \n37 [38] proposed a model that preserves the total mass of a discretized partial differential equation \n38 (PDE). These studies have demonstrated that a neural network with more prior knowledge about first \n39 integrals predicts the dynamics of the target system more accurately. See Table 1 for comparison. \n40 Previous studies have mainly attempted to preserve known first integrals. However, in situations \n41 where a neural network learns an unknown target system, it is naturally expected that first integrals \n42 associated with the target system are also unknown, and it is not clear which of the above methods are \n43 available. Given the above, this study proposes First Integral-preserving Neural Differential Equation \n44 (FINDE) to find and preserve first integrals from data. FINDE has the following advantages. \n45 Learning First Integrals For modeling continuous-time dynamics with known first integrals, many \n46 studies have designed architectures or operations of neural networks [13, 19, 20, 26, 38]. For each \n47 type of first integral, one dedicated method was proposed. However, the properties of a target system \n48 are generally unknown in practice. In contrast, the proposed FINDE finds various kinds of first \n49 integrals from data in a unified manner and preserves them in predictions. A symbolic regression \n50 confirms that the learned first integrals are consistent with well-known first integrals of target systems. ", + "bbox": [ + 147, + 630, + 825, + 768 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 145, + 775, + 825, + 900 + ], + "page_idx": 0 + }, + { + "type": "table", + "img_path": "images/b68f4c36f0f8436ffa368325e26671ba061972753e3b18687305a48494f583fa.jpg", + "table_caption": [ + "Table 1: Comparison between Related Studies on Preservation of First Integrals. " + ], + "table_footnote": [], + "table_body": "
energymonentum massconstraint learning invariants exact conservation
NODE [8]
HNN [26]√
LieConv [19]√厂
DGNet [38]√√
CHNN [20]√√
continuous FINDE (proposed)√√√√√
discrete FINDE (proposed)√厂厂丁
", + "bbox": [ + 217, + 113, + 779, + 260 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 147, + 285, + 825, + 327 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 147, + 333, + 825, + 402 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 147, + 416, + 825, + 501 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Combination with Known First Integrals The proposed FINDE can be combined with previously proposed neural networks designed to preserve known first integrals, such as HNN. Therefore, FINDE is available in various situations. ", + "bbox": [ + 163, + 515, + 823, + 556 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "54 Exact Preservation of First Integrals Even if a first integral is associated with a continuous-time \n55 system, it is destroyed after the system is discretized in time for computer simulations. This is true \n56 even when using a symplectic integrator, which preserves the system energy only approximately [27]. \n57 By leveraging discrete gradients [38], the discrete-time version of FINDE preserves first integrals \n58 exactly (up to rounding errors) in discrete time and further improves the prediction performance. ", + "bbox": [ + 147, + 571, + 826, + 641 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "59 2 Background and Related Work ", + "text_level": 1, + "bbox": [ + 147, + 660, + 465, + 678 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "60 First Integrals Let us consider a time-invariant differential system $\\begin{array} { r } { \\frac { \\mathrm { d } } { \\mathrm { d } t } \\pmb { u } \\ = \\ f ( \\pmb { u } ) } \\end{array}$ on an $N$ - \n61 dimensional manifold $\\mathcal { M }$ , where $\\textbf { \\em u }$ denotes the system state and $f : \\mathcal { M } ^ { } \\to \\mathcal { T } _ { u } \\mathcal { M }$ represents a \n62 vector field on the manifold $\\mathcal { M }$ . The manifold $\\mathcal { M }$ can be $\\mathcal { M } = S ^ { 1 } \\times \\mathbb { R } ^ { 1 }$ for a pendulum. In this \n63 paper, we suppose the manifold $\\mathcal { M }$ be a Eucleadian space $\\mathbb { R } ^ { N }$ for simplicity. \n64 Definition 1 (first integral). $A$ quantity $V : { \\mathcal { M } } \\mathbb { R }$ is referred to as a first integral of a system \n65 $\\begin{array} { r } { \\frac { \\mathrm { d } } { \\mathrm { d } t } \\pmb { u } = f ( \\pmb { u } ) } \\end{array}$ if it remains constant along with any solution ${ \\mathbf { } } { \\mathbf { } } { \\mathbf { } } { \\mathbf { } } { \\mathbf { } } { \\mathbf { } } { \\mathbf { } } { \\mathbf { } } { \\mathbf { } } { \\mathbf { } } \\mathbf { } { \\mathbf { } } \\mathbf { } { \\mathbf { } } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf \\Psi \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf \\Psi { } \\mathbf \\mathbf { } \\mathbf { } \\mathbf \\Psi \\mathbf { } \\mathbf { } \\mathbf \\mathbf { } \\mathbf \\Psi { } \\mathbf \\mathbf { } \\mathbf \\mathbf { } \\mathbf \\mathbf { } \\mathbf \\Psi \\mathbf { } \\mathbf \\mathbf { } \\mathbf \\mathbf { } \\mathbf \\mathbf { } \\mathbf \\mathbf \\Psi \\mathbf { } \\mathbf \\mathbf { } \\mathbf \\mathbf \\mathbf { } \\mathbf \\mathbf \\mathbf { } \\mathbf \\mathbf \\mathbf { \\mathbf } \\mathbf \\mathbf \\mathbf \\mathbf \\Psi \\mathbf \\Psi \\mathbf { } \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf { } \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\Psi \\mathbf \\mathbf { } \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\Psi \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\Psi \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf$ , i.e., $\\begin{array} { r } { \\frac { \\mathrm { d } } { \\mathrm { d } t } V ( \\pmb { u } ) = 0 } \\end{array}$ . ", + "bbox": [ + 148, + 691, + 826, + 750 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 155, + 752, + 823, + 785 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "ntial system $\\begin{array} { r } { \\frac { \\mathrm { d } } { \\mathrm { d } t } { \\pmb u } = f ( { \\pmb u } ) } \\end{array}$ is associate ith $K$ functionally independent first integrals $V _ { 1 } , \\dots , V _ { K }$ ${ \\bf \\ddot { u } } ( t )$ given an initial value $\\mathbf { \\delta } \\mathbf { u } _ { 0 }$ stays at the ", + "bbox": [ + 158, + 794, + 823, + 825 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/4fda744cad7a5884b3bb06dbd3ef5c5d88f008ecb07ec9d2a6e8ca4da3156a71.jpg", + "text": "$$\n\\mathcal { M } ^ { \\prime } = \\{ \\pmb { u } \\in \\mathcal { M } : V _ { 1 } ( \\pmb { u } ) = V _ { 1 } ( \\pmb { u } _ { 0 } ) , \\allowbreak \\dots , V _ { K } ( \\pmb { u } ) = V _ { K } ( \\pmb { u } _ { 0 } ) \\} .\n$$", + "text_format": "latex", + "bbox": [ + 297, + 835, + 699, + 853 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "69 The tangent space $\\mathcal { T } _ { u } \\mathcal { M } ^ { \\prime } \\subset \\mathcal { T } _ { u } \\mathcal { M }$ of the submanifold $\\mathcal { M } ^ { \\prime } \\subset \\mathcal { M }$ at a point $\\textbf { \\em u }$ is the orthogonal \n70 complement to the space spanned by the gradients $\\nabla V _ { k } ( { \\boldsymbol { \\mathbf { \\mathit { u } } } } )$ of the first integrals $V _ { k }$ for $k = 1 , \\ldots , K$ , \n71 that is, ", + "bbox": [ + 143, + 854, + 826, + 896 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/7ddeaa6911b38c8b1572e33538f92f0f8bcecebf0496bb90944eb978a90dba82.jpg", + "text": "$$\n\\mathcal { T } _ { \\boldsymbol { u } } \\mathcal { M } ^ { \\prime } = \\{ \\pmb { w } \\in \\mathcal { T } _ { \\boldsymbol { u } } \\mathcal { M } : \\nabla V _ { k } ( \\boldsymbol { u } ) ^ { \\top } \\pmb { w } = 0 \\mathrm { ~ f o r ~ } k = 1 , \\dots , K \\}\n$$", + "text_format": "latex", + "bbox": [ + 299, + 893, + 697, + 912 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "If a quantity 72 $V _ { k }$ is a first integral of the system $\\begin{array} { r } { \\frac { \\mathrm { d } } { \\mathrm { d } t } \\pmb { u } = f ( \\pmb { u } ) } \\end{array}$ , the time-derivative $f$ at point $\\textbf { \\em u }$ is on the 73 tangent space $\\mathcal { T } _ { u } \\mathcal { M } ^ { \\prime }$ , being orthogonal to the gradient $\\nabla V _ { k }$ of the first integral $V _ { k }$ . Then, it holds that 74 $\\begin{array} { r } { \\frac { \\mathrm { d } } { \\mathrm { d } t } \\dot { V _ { k } ( \\pmb { u } ) } = \\nabla V _ { k } ( \\pmb { u } ) ^ { \\top } \\frac { \\mathrm { d } } { \\mathrm { d } t } \\pmb { u } = \\bar { \\nabla } V _ { k } ( \\pmb { u } ) ^ { \\top } \\pmb { f } ( \\bar { \\pmb { u } } ) = 0 } \\end{array}$ . ", + "bbox": [ + 147, + 90, + 825, + 138 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "75 One of the most well-known first integrals is the Hamiltonian $H$ , which represents the system energy \n76 of a Hamiltonian system. Noether’s theorem states that a continuous symmetry of a system leads to a \n77 conservation law (and hence a first integral) [27]; a Hamiltonian system is symmetric to translation \n78 in time and conserves the Hamiltonian. Symmetries to translation and rotation in space lead to the \n79 conservation of linear and angular momenta. Not all first integrals are related to symmetries. A \n80 pendulum can be expressed in Cartesian coordinates, and then the rod length constrains the mass \n81 position. This kind of constraint is called a holonomic constraint and leads to a first integral. A model \n82 for disease spreading called an susceptible-infected-recovered (SIR) model and the dynamics of \n83 chemical reactions have the total mass (population) as a first integral. Also for a system described by \n84 a PDE, the total mass is sometimes a first integral [23]. See Appendix A for theoretical classification \n85 of dynamics. \n86 First Integrals in Numerical Analysis For computer simulations, a differential system is dis \n87 cretized in time and solved by numerical integration. Then, the geometric structures of the system \n88 are often destroyed, and most first integrals are no longer preserved. A common remedy is a sym \n89 plectic integrator, which preserves the symplectic structure and integrates a Hamiltonian system \n90 accurately [27]. However, Ge–Marsden theorem states that a symplectic integrator conserves the \n91 Hamiltonian only approximately [56]. Hence, many numerical schemes have also been investigated \n92 for preserving first integrals exactly, while they cannot preserve the symplectic structure. \n93 Let a superscript $s$ denote the state $\\pmb { u } ^ { s }$ or time $t ^ { s }$ at $s$ -th time step, and $\\Delta t ^ { s } = t ^ { s + 1 } - t ^ { s }$ denote a \n94 time step size. A projection method predicts a next state $\\tilde { \\pmb u } ^ { s + 1 }$ from the current state $\\pmb { u } ^ { s }$ using a \n95 numerical integrator and projects it onto the submanifold $\\mathcal { M } ^ { \\prime }$ , obtaining the projected state $\\pmb { u } ^ { s + 1 }$ that \n96 preserves the first integrals $V _ { k }$ [24] (see also [27, Section IV.4]). In particular, the projected state \n97 $\\mathbf { \\Delta } _ { \\mathbf { u } } { } ^ { s + 1 }$ is obtained by solving the optimization problem ", + "bbox": [ + 145, + 142, + 825, + 295 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 147, + 310, + 825, + 409 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 145, + 414, + 825, + 484 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/3879da5325cf97dc85691dd8ea4bb6d737ade345d2e91cf7216e13310995d06b.jpg", + "text": "$$\n\\boldsymbol { u } ^ { s + 1 } = \\operatorname * { a r g m i n } _ { \\boldsymbol { u } ^ { \\prime } ^ { s + 1 } } | | \\boldsymbol { u } ^ { \\prime * s + 1 } - \\tilde { \\boldsymbol { u } } ^ { s + 1 } | | \\operatorname * { s u b j e c t } \\operatorname { t o } V _ { k } ( \\boldsymbol { u } ^ { \\prime * s + 1 } ) = V _ { k } ( \\boldsymbol { u } ^ { s } ) \\operatorname { f o r } k = 1 , \\dots , K .\n$$", + "text_format": "latex", + "bbox": [ + 217, + 489, + 781, + 515 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "98 A local coordinate method defines a coordinate system to the neighborhood of the current state $\\pmb { u } ^ { s }$ \n99 and integrates a differential equation on it [43] (see also [27, Section IV.5]). A discrete gradient \n100 method defines a discrete analogue to a given differential system and integrates it in discrete time [6, \n101 23, 25, 29, 44, 45]. This method eliminates numerical errors caused by temporal discretization and is \n102 used to preserve the Hamiltonian exactly (up to rounding errors) in discrete time. \n103 Except for DGNet, which used discrete gradients to preserve the Hamiltonian [38], all the above \n104 methods have never been applied to neural networks due to difficulties that we will introduce later. To \n105 our best knowledge, the discrete-time version of FINDE is the first projection method for dynamical \n106 systems modeled using neural networks. \n107 Preservation of First Integrals by Neural Networks NODE defines an ODE using a neural net \n108 work in the most general way with no associated first integrals [8]. NODE is a universal approximator \n109 to ODEs [51], and it can approximate any ODE with arbitrary accuracy if there is an infinite amount \n110 of training data. In practice, the amount of training data is limited, and prior knowledge about the \n111 target system is helpful for learning (see [48] for the case with convolutional neural networks). HNN \n112 assumes the target system to be a Hamiltonian system in the canonical form [26]. HNN guarantees \n113 various properties of Hamiltonian systems by definition, including the conservation of the energy \n114 and the preservation of the symplectic structure in continuous time [27]. Some studies employed a \n115 symplectic integrator for HNN to preserve the energy and symplectic structure with smaller numerical \n116 errors [10]. LieConv and EMLP-HNN employed neural network architectures with translational \n117 and rotational symmetries to preserve momenta [19, 21]. CHNN incorporates a known holonomic \n118 constraint in the dynamics [20]. Deep conservation extracts latent dynamics of a PDE system and \n119 preserves a quantity of interest by forcing its flux to be zero [34]. $\\mathrm { H N N + + }$ also guarantees the \n120 conservation of the mass in PDE systems by using a coefficient matrix derived from differential \n121 operators [38]. \n122 Several studies proposed neural networks to learn Lyapunov functions, which are expected to be \n123 non-increasing over time, in contrast to first integrals [37, 50]. If the state moves in the direction of \n124 increasing the function, it is projected onto or moved inside the counter line of the gradient of the \n125 Lyapunov function. Their idea is similar to the continuous-time version of FINDE but limited to a \n126 single non-increasing quantity in continuous time. On the other hand, our proposed FINDE preserves \n127 multiple quantities in both continuous and discrete time. \n28 Previous studies aimed to preserve known first integrals. Moreover, except for DGNet [38], all \n29 the above methods suffer from numerical errors caused by temporal discretization. In contrast, our \n30 proposed FINDE learns first integrals from data and can eliminate discretization errors. ", + "bbox": [ + 143, + 522, + 825, + 592 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 142, + 598, + 825, + 654 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 670, + 825, + 877 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 145, + 883, + 825, + 911 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 90, + 825, + 147 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 151, + 154, + 828, + 195 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "131 3 First Integral-Preserving Neural Differential Equation ", + "text_level": 1, + "bbox": [ + 150, + 213, + 660, + 231 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "132 The main purpose is to find and preserve first integrals from data by neural networks. We suppose \n133 that a target system has at least $K$ unknown functionally independent first integrals. Even when \n134 a NODE learns the target system, it is not guaranteed to learn these first integrals. Hence, we \n135 introduce a neural network with $K$ outputs, each of which is expected to learn one of first integrals \n136 expressed as $V _ { k } : \\mathbb { R } ^ { N } \\mathbb { R }$ for $k = 1 , \\dots , K$ . We denote the set of first integrals by a vector \n137 $\\pmb { V } ( \\pmb { u } ) = ( V _ { 1 } ( \\pmb { u } ) V _ { 2 } ( \\pmb { u } ) \\dotsm V _ { K } ( \\pmb { u } ) ) ^ { \\top }$ . Then, the submanifold $\\mathcal { M } ^ { \\prime }$ is defined using the neural \n138 network $V$ as in Eq. (1). \n139 Because there is no way to define local coordinates on such submanifolds, a local coordinate method \n140 is not applicable. When using a projection method, the optimization problem in Eq. (3) should \n141 be solved at every training iteration as well as in the prediction phase. Optimization problems are \n142 computationally expensive, and common libraries for neural networks do not provide backpropagation \n143 algorithms for optimization problems [1, 42].1 Until a recent study has proposed an algorithm [38], \n144 there was no way to obtain discrete gradients of neural networks. Because of these difficulties, \n145 no methods for preserving first integrals have been applied to neural networks. By leveraging a \n146 projection method and a discrete gradient method, we propose FINDE as follows. ", + "bbox": [ + 140, + 243, + 825, + 342 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 347, + 826, + 458 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.1 Continuous FINDE: Time-Derivative Projection Method ", + "text_level": 1, + "bbox": [ + 161, + 472, + 604, + 488 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "First, we propose a time-derivative projection method called continuous FINDE (cFINDE) for neural networks, which projects the time-derivative onto the tangent space $\\mathcal { T } _ { u } \\mathcal { M } ^ { \\prime }$ . While it still suffers from numerical errors, it is sufficient to find first integrals from data. ", + "bbox": [ + 160, + 498, + 825, + 540 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We suppose that a neural network called a base model defines the time-derivative $\\hat { f } : \\mathbb { R } ^ { N } \\to \\mathbb { R } ^ { N }$ Then, we define the time-derivative $f$ of the cFINDE $\\begin{array} { r } { \\frac { \\mathrm { d } } { \\mathrm { d } t } \\pmb { u } = f ( \\pmb { u } ) } \\end{array}$ as ", + "bbox": [ + 158, + 547, + 823, + 579 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/8f8b79adb26b9ef479e91536ac45479b6c800405868d7256ba6e077ccedb198f.jpg", + "text": "$$\n\\begin{array} { r } \\boldsymbol { f } ( \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol \\mathbf { \\rho } \\boldsymbol { \\rho } \\boldsymbol \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 300, + 582, + 694, + 602 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where 153 $\\lambda _ { k }$ is a Lagrange multiplier, $\\begin{array} { r } { M \\mathrm { ~ = ~ } \\frac { \\partial V } { \\partial \\mathbf { u } } } \\end{array}$ , and $\\lambda ( { \\pmb u } ) = ( \\lambda _ { 1 } ( { \\pmb u } ) \\lambda _ { 2 } ( { \\pmb u } ) \\dots \\lambda _ { K } ( { \\pmb u } ) ) ^ { \\top }$ . If $V _ { k }$ ", + "bbox": [ + 151, + 602, + 825, + 632 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/9c2e9d3f26afcb1d2909b43b5c98d078e94dbe3a47d5ab0465bdde96a7dd7e0f.jpg", + "text": "$$\n\\begin{array} { r } { \\mathbf { 0 } = \\frac { \\mathrm { d } } { \\mathrm { d } t } { \\pmb V } ( { \\pmb u } ( t ) ) = M ( \\pmb { u } ) \\frac { \\mathrm { d } } { \\mathrm { d } t } { \\pmb u } = M ( \\pmb { u } ) f ( \\pmb { u } ) = M ( \\pmb { u } ) ( \\hat { f } ( \\pmb { u } ) - M ( \\pmb { u } ) ^ { \\top } \\pmb { \\lambda } ( \\pmb { u } ) ) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 240, + 632, + 754, + 652 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where 155 $\\mathbf { 0 } = ( 0 \\ldots 0 ) ^ { \\top }$ . By transforming Eq. (5), we obtain the Lagrange multiplier $\\lambda ( { \\pmb u } ) =$ 156 $( M ( \\pmb { \\mathscr { u } } ) M ( \\pmb { \\mathscr { u } } ) ^ { \\top } ) ^ { - 1 } M ( \\pmb { \\mathscr { u } } ) \\hat { f } ( \\pmb { \\mathscr { u } } )$ . By eliminating it, the cFINDE $\\begin{array} { r } { \\frac { \\mathrm { d } } { \\mathrm { d } t } \\pmb { u } = f ( \\pmb { u } ) } \\end{array}$ is given by ", + "bbox": [ + 150, + 655, + 825, + 686 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Remark 1 (continuous-time first integral preservation). The cFINDE 57 $\\begin{array} { r } { \\frac { \\mathrm { d } } { \\mathrm { d } t } \\pmb { u } = f ( \\pmb { u } ) } \\end{array}$ preserves all first integrals 58 $V _ { k }$ for $k = 1 , \\ldots , K$ in continuous time, i.e., $\\begin{array} { r } { { \\frac { \\mathrm { d } } { \\mathrm { d } t } } V _ { k } = 0 } \\end{array}$ . ", + "bbox": [ + 155, + 709, + 825, + 742 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "159 The base model $\\hat { f }$ can be a NODE, an HNN, or other models depending on available prior knowledge. \n160 Also, if a first integral is already known, one can use it directly as one of first integrals $V _ { k }$ instead \n161 of learning it using a neural network. Note that even though the base model $\\hat { f }$ is an HNN, due to \n162 projection, the cFINDE $f$ is no longer a Hamiltonian system in the strict sense. ", + "bbox": [ + 143, + 752, + 825, + 813 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Compared to the base model $\\hat { f }$ , the cFINDE requires the additional computation of the neural network $V$ , several matrix multiplications, and an inverse operation. The inverse operation needs a computational cost of $O ( K ^ { 3 } )$ , which is not costly if the number $K$ of first integrals is small. For satisfying the constraints and geometric structures, many previous models also need the inverse operation, such as Lagrangian neural network (LNN) [13], neural symplectic form [9], and CHNN [20]. ", + "bbox": [ + 173, + 819, + 825, + 890 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "69 To eliminate numerical errors caused by temporal discretization, we employ discrete gradients and \n70 propose a projection method called discrete FINDE (dFINDE). ", + "bbox": [ + 156, + 116, + 825, + 145 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "A discrete gradient $\\overline { { \\nabla } } V$ is a discrete analogue to a gradient $\\nabla V$ [6, 23, 25, 29, 44, 45]. Recall that a gradient $\\nabla V$ of a function $V : \\mathbb { R } ^ { N } \\mathbb { R }$ can be regarded as a function $\\mathbb { R } ^ { N } \\to \\mathbb { R } ^ { N }$ that satisfies the chain rule $\\begin{array} { r } { \\frac { \\mathrm { d } } { \\mathrm { d } t } V ( \\pmb { u } ) = \\nabla V ( \\pmb { u } ) ^ { \\top } \\frac { \\mathrm { d } } { \\mathrm { d } t } \\pmb { u } } \\end{array}$ . Analogously, a discrete gradient $\\overline { \\nabla }$ is defined as follows. ", + "bbox": [ + 160, + 151, + 825, + 196 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Definition 2 (discrete gradient). A discrete gradient $\\overline { { \\nabla } } V$ of a function $V : \\mathbb { R } ^ { N } \\mathbb { R }$ is a function $\\mathbb { R } ^ { N } \\times \\mathbb { R } ^ { N } \\xrightarrow [ ] { } \\mathbb { R } ^ { N }$ that satisfies ", + "bbox": [ + 160, + 200, + 826, + 229 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/a1e6840231973b85c64c34a00b362183503d5ede99d36f77a752d512eb641e6d.jpg", + "text": "$$\nV ( \\pmb { v } ) - V ( \\pmb { u } ) = \\overline { { \\nabla } } V ( \\pmb { v } , \\pmb { u } ) ^ { \\top } ( \\pmb { v } - \\pmb { u } ) ~ a n d ~ \\overline { { \\nabla } } V ( \\pmb { u } , \\pmb { u } ) = \\nabla V ( \\pmb { u } ) .\n$$", + "text_format": "latex", + "bbox": [ + 281, + 233, + 717, + 252 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "176 The first condition is a discrete analogue to the chain rule when replacing the time-derivatives ${ \\frac { \\mathrm { d } } { \\mathrm { d } t } } V$ \n177 and ${ \\frac { \\mathrm { d } } { \\mathrm { d } t } } { \\pmb u }$ with finite differences $( V ( \\pmb { v } ) - V ( \\pmb { u } ) )$ and $( { \\pmb v } - { \\pmb u } )$ , respectively, and the second condition \n178 ensures the consistency with the ordinary gradient $\\nabla V$ . A discrete gradient $\\overline { { \\nabla } } V$ is not uniquely \n179 determined and has been obtained manually. Recently, the automatic discrete differentiation algorithm \n180 (ADDA) has been proposed in [38], which obtains a discrete gradient of a neural network in a similar \n181 way to the automatic differentiation algorithm [1, 42]. The discrete gradient is defined in discrete \n182 time, and hence a numerical integration using the discrete gradient is free from numerical errors \n183 caused by temporal discretization. See Appendix $\\mathbf { B }$ and the references [6, 23, 38] for more details. ", + "bbox": [ + 140, + 262, + 825, + 381 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Following [11, 15], we introduce a discrete analogue to the tangent space 184 $\\mathcal { T } _ { \\mathbf { \\ b { u } } } \\mathcal { M } ^ { \\prime }$ called the discrete tangent space 185 $\\mathcal { T } _ { ( \\pmb { v } , \\pmb { u } ) } \\mathcal { M } ^ { \\prime }$ . In particular, for a pair $( \\pmb { v } , \\pmb { u } ) \\in \\mathcal { M } ^ { \\prime }$ of points, it is defined as ", + "bbox": [ + 150, + 386, + 825, + 416 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/d15c5c5b37f8223f9e78ac890b92d60d5645de00be83ce74944e8ad884d45cf9.jpg", + "text": "$$\n\\mathcal { T } _ { ( v , u ) } \\mathcal { M } ^ { \\prime } = \\{ \\pmb { w } \\in \\mathbb { R } ^ { N } : \\overline { { \\nabla } } V _ { k } ( \\pmb { v } , \\pmb { u } ) ^ { \\top } \\pmb { w } = 0 \\mathrm { ~ f o r ~ } k = 1 , \\ldots , K \\} .\n$$", + "text_format": "latex", + "bbox": [ + 284, + 420, + 712, + 439 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "186 If the finite difference $( \\pmb { u } ^ { s + 1 } - \\pmb { u } ^ { s } )$ between the predicted and current states is on the discrete \n187 tangent space $\\mathcal { T } _ { \\left( \\pmb { u } ^ { s + 1 } , \\pmb { u } ^ { s } \\right) } \\mathcal { M } ^ { \\prime }$ , the first integrals $V _ { k }$ are preserved because $V _ { k } ( { \\pmb u } ^ { s + 1 } ) - V _ { k } ( { \\pmb u } ^ { s } ) =$ \n188 $\\overline { { \\nabla } } V _ { k } ( \\pmb { u } ^ { s + 1 } , \\pmb { u } ^ { s } ) ^ { \\top } ( \\pmb { u } ^ { s + 1 } - \\pmb { u } ^ { s } ) = 0$ . Note that similar concepts defined in different ways are also \n189 referred to as discrete tangent spaces [14, 16]. ", + "bbox": [ + 140, + 443, + 825, + 503 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Let ψˆ denote a discrete-time base model that satisfies u˜s+1−us∆ts 190 $\\begin{array} { r } { \\frac { \\tilde { \\mathbf { u } } ^ { s + 1 } - \\mathbf { u } ^ { s } } { \\Delta t ^ { s } } = \\hat { \\psi } ( \\mathbf { u } ^ { s } ; \\Delta t ^ { s } ) } \\end{array}$ , where $\\tilde { { \\pmb u } } ^ { s + 1 }$ denotes 191 the predicted state. We assume that the base model $\\hat { \\psi }$ is composed of a continuous-time base model 192 ˆf and a numerical integrator. Then, the dFINDE us+1−∆ts $\\begin{array} { r } { \\frac { { \\pmb u } ^ { s + 1 } - { \\pmb u } ^ { s } } { \\Delta t ^ { s } } = \\psi ( { \\pmb u } ^ { s + 1 } , { \\pmb u } ^ { s } ; \\Delta t ^ { s } ) } \\end{array}$ is given by ", + "bbox": [ + 142, + 508, + 825, + 563 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/d1a21837891a0816a8d90a934c2bd058b77755c9a71f9d7120f2e96416ef165f.jpg", + "text": "$$\n\\psi ( \\pmb { u } ^ { s + 1 } , \\pmb { u } ^ { s } ; \\Delta t ^ { s } ) = \\hat { \\psi } ( \\pmb { u } ^ { s } ; \\Delta t ^ { s } ) - \\overline { { M } } ( \\pmb { u } ^ { s + 1 } , \\pmb { u } ^ { s } ) ^ { \\top } \\lambda ( \\pmb { u } ^ { s + 1 } , \\pmb { u } ^ { s } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 289, + 566, + 705, + 585 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where 193 $\\overline { { M } } ( { \\pmb u } ^ { s + 1 } , { \\pmb u } ^ { s } ) = ( \\overline { { \\nabla } } V _ { 1 } ( { \\pmb u } ^ { s + 1 } , { \\pmb u } ^ { s } )$ . . . $\\overline { { \\nabla } } V _ { K } ( \\pmb { u } ^ { s + 1 } , \\pmb { u } ^ { s } ) ) ^ { \\top }$ . As is the case in continuous time, 194 the preservation of the first integrals $V _ { k }$ leads to ", + "bbox": [ + 145, + 589, + 826, + 619 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/c57172cf644c028312037ccb9029fe77af4e4c29c8dac75574ca1e6b7a921bd3.jpg", + "text": "$$\n\\begin{array} { r } { \\mathbf { 0 } = \\frac { V ( u ^ { s + 1 } ) - V ( u ^ { s } ) } { \\Delta t ^ { s } } = \\overline { { M } } ( u ^ { s + 1 } , u ^ { s } ) \\frac { u ^ { s + 1 } - u ^ { s } } { \\Delta t ^ { s } } = \\overline { { M } } ( u ^ { s + 1 } , u ^ { s } ) \\psi ( u ^ { s + 1 } , u ^ { s } ; \\Delta t ^ { s } ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 205, + 622, + 746, + 645 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "195 Substituting Eq. (9) and eliminating the Lagrange multiplier $\\boldsymbol { \\lambda }$ , we obtain ", + "bbox": [ + 140, + 647, + 658, + 662 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/5cf3eab1461c710ba916e91303d63747b554722245d42fa5ae31f91a77890e74.jpg", + "text": "$$\n\\psi ( \\pmb { u } ^ { s + 1 } , \\pmb { u } ^ { s } ; \\Delta t ^ { s } ) = ( I - \\overline { { Y } } ( \\pmb { u } ^ { s + 1 } , \\pmb { u } ^ { s } ) ) \\hat { \\psi } ( \\pmb { u } ^ { s } ; \\Delta t ^ { s } )\n$$", + "text_format": "latex", + "bbox": [ + 209, + 667, + 547, + 686 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Remark 2 (discrete-time first integral preservation). The dFINDE us+1−us∆ts preserves all first integrals $V _ { k }$ for $k = 1 , \\ldots , K$ in discrete time, i.e., $V _ { k } ( { \\pmb u } ^ { s + 1 } ) - V _ { k } ( { \\pmb u } ^ { s } ) = 0$ $\\begin{array} { r } { \\frac { { \\pmb u } ^ { s + 1 } - { \\pmb u } ^ { s } } { \\Delta t ^ { s } } = \\psi ( { \\pmb u } ^ { s + 1 } , { \\pmb u } ^ { s } ; \\Delta t ^ { s } ) } \\end{array}$ ", + "bbox": [ + 158, + 690, + 828, + 726 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "198 Due to projection, dFINDE can be regarded as a projection method using discrete gradients. For the \n199 base model $\\hat { \\psi }$ , the continuous-time base model $\\hat { f }$ can be a NODE, an HNN, or other models, and the \n200 numerical integrator can be a Runge–Kutta method, the leapfrog integrator, or others. \n201 Because dFINDE is an implicit method, it is computationally expensive for prediction. However, the \n202 next state $\\boldsymbol { u } ^ { s + 1 }$ is given for training, and the ADDA can explicitly obtain the discrete gradient w.r.t. the \n203 pair $( { \\pmb u } ^ { s + 1 } , { \\pmb u } ^ { s } )$ as well as its computational graph. Thus, dFINDE can be computed explicitly and \n204 optimized by standard backpropagation algorithms. Moreover, we suppose that dFINDE projects \n205 the finite difference $\\hat { \\psi }$ only at every time step, whereas cFINDE projects the time-derivative $\\hat { f }$ at \n206 every substep inside a numerical integrator. Therefore, dFINDE is less computationally expensive \n207 than cFINDE for training. In contrast, a typical projection method requires much computational \n208 cost to solve an optimization problem for training, and standard backpropagation algorithms are not \n209 applicable to it. \n10 Remark 3 (trainability). The dFINDE can be trained using the standard backpropagation algorithm, \n211 whereas a straightforward application of a projection method cannot. ", + "bbox": [ + 147, + 733, + 825, + 779 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 143, + 784, + 825, + 912 + ], + "page_idx": 4 + }, + { + "type": "table", + "img_path": "images/9532b26093d205096da34a1d05959c144853a2f40d48bde0cd43c48af1534f6f.jpg", + "table_caption": [ + "Table 2: Datasets, Dynamics, and First Integrals. " + ], + "table_footnote": [], + "table_body": "
DatasetDynamicsNFirst Integrals
EnergyMomentumMassConstraint
Two-body problemCanonical Hamiltonian81
Discretized KdV equationNon-canonical Hamiltonian50广1
Double pendulumPoisson8√√
FitzHugh-Nagumo modelDirac4√
", + "bbox": [ + 173, + 112, + 825, + 210 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 150, + 236, + 825, + 265 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4 Experiments ", + "text_level": 1, + "bbox": [ + 171, + 290, + 312, + 306 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.1 Experimental Settings ", + "text_level": 1, + "bbox": [ + 166, + 321, + 366, + 337 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Target Systems We evaluated FINDE and base models using datasets associated with first integrals, summarized in Table 2. A gravitational two-body problem (2-body) on a 2-dimensional configuration space is a typical Hamiltonian system in the canonical form. In addition to the total energy, it has first integrals related to symmetries in space, namely, the linear and angular momenta. The Korteweg–De Vries (KdV) equation is a PDE model of shallow water waves. This is a Hamiltonian system in a non-canonical form and has the Hamiltonian, total mass, and many other quantities as first integrals. A double pendulum (2-pend) is a Hamiltonian system in polar coordinates. However, we transformed it to Cartesian coordinates; it was no longer a Hamiltonian system but a Poisson system. The lengths of two rods work as holonomic constraints and lead to four first integrals. The FitzHugh–Nagumo model is a biological neuron model as an electric circuit, which exhibits a rapid and transient change of voltage called a spike. As an electric circuit, the currents through and voltages applied to the inductor and capacitor can be regarded as system states, and the states are constrained by the circuit topology and Kirchhoff’s current and voltage laws. Then, this system has a state of four elements and two first integrals. Due to energy dissipation in the resistor, the model is not a Poisson system, but one can find a Dirac structure [53]. See Appendix C for more details. ", + "bbox": [ + 171, + 347, + 825, + 554 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Implementation We implemented the proposed FINDE and evaluated it under the following settings. We implemented all codes by modifying the officially released codes of HNN [26] 2 and DGNet $[ 3 8 ] ^ { 3 }$ . We used Python v3.8.12 with packages scipy v1.7.3, pytorch v1.10.2, torchdiffeq v0.1.1, functorch v1.10 preview, and gplearn v0.4.2. We used the Dormand–Prince method (dopri5) [18] as the numerical integrator, unless otherwise stated. All experiments were performed on a single NVIDIA A100 provided by (ANONYMOUS PROVIDER). ", + "bbox": [ + 173, + 570, + 825, + 654 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Following HNN [26] and DGNet [38], we represented the first integrals $V$ , NODE, and HNN $H$ using fully-connected neural networks with two hidden layers. Each hidden layer had 200 units and preceded a hyperbolic tangent activation function. Each weight matrix was initialized as an orthogonal matrix. The input was the state $\\textbf { \\em u }$ , and the output represented the first integrals $V$ for FINDE, time-derivative $\\hat { f }$ for NODE, and the Hamiltonian $H$ for HNN. For the KdV dataset, we used a 1-dimensional convolutional neural network (CNN), each of whose layers had a kernel size of 3. The double pendulum is a second–order system, implying that the time-derivative ddt q of the position $\\pmb q$ is known as the velocity $\\textbf { { v } }$ . Hence, we treated only the acceleration $\\begin{array} { r } { \\frac { \\mathrm { d } } { \\mathrm { d } t } \\pmb { v } } \\end{array}$ as the output to learn. This assumption slightly improved the absolute performances but did not change the relative trends. ", + "bbox": [ + 173, + 660, + 825, + 791 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We used the $I$ -step error as the loss function to be minimized. In particular, it is the mean squared error (MSE) between the ground truth state $\\pmb { u } _ { \\mathrm { G T } } ^ { s }$ and the state $\\pmb { u } _ { \\mathrm { p r e d . } } ^ { s }$ . predicted from the previous step $u _ { \\mathrm { G T } } ^ { s - 1 }$ . The base model and FINDE were jointly trained using the Adam optimizer [33] with the parameters $( \\beta _ { 1 } , \\beta _ { 2 } ) = ( 0 . 9 , 0 . 9 9 9 )$ and a batch size of 200. The learning rate was initialized to $1 0 ^ { - 3 }$ and decayed to zero with a cosine annealing [36]. ", + "bbox": [ + 173, + 797, + 825, + 872 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "249 Evaluation Metric As an evaluation metric, we used the 1-step error, which is identical to the loss \n250 function. We displayed it at the scale of $\\times 1 0 ^ { - 9 }$ . The lower this indicator, the better, as emphasized \n251 by $\\downarrow$ . While several studies used the MSEs of the state or system energy over the whole time \n252 series [26, 38], we consider these indicators are misleading, as pointed in several studies [4]. For \n253 example, in the case of a periodic orbit, an orbit that is correctly learned except for a slight difference \n254 in angular velocity will have the same MSE as an orbit that never moves from its initial position. \n255 Instead, we used the valid prediction time $( V P T )$ [4, 32, 54]. VPT denotes the time point $s$ divided by \n256 the length $S$ of time series at which the MSE of the predicted state $\\pmb { u } _ { \\mathrm { p r e d . } } ^ { s }$ . exceeds a given threshold $\\theta$ \n257 for the first time in an initial value problem, that is, ", + "bbox": [ + 138, + 90, + 825, + 217 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/811dd5f21e7701a7467a228c4cae451c6d1973664d4fe839e7bf95ebdae7362c.jpg", + "text": "$$\n\\begin{array} { r } { V P T ( u _ { \\mathrm { p r e d . } } ; u _ { \\mathrm { G T } } ) = \\frac { 1 } { S } \\arg \\operatorname* { m a x } _ { s _ { f } } \\{ s _ { f } | \\mathrm { M S E } ( u _ { \\mathrm { p r e d . } } ^ { s } , u _ { \\mathrm { G T } } ^ { s } ) < \\theta \\mathrm { f o r } \\mathrm { a l l } s \\leq s _ { f } \\} . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 241, + 222, + 754, + 241 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "258 To obtain VPTs, we normalized each element of state to have zero mean and unit variance in the \n259 training data and set $\\theta$ to 0.01. The higher this indicator, the better, as emphasized by $\\uparrow$ . Because of \n260 the “spiking” behavior of the FitzHugh–Nagumo model, a small error in phase is regarded as a large \n261 error in state. To measure the qualitative performance, we calculated VPTs by allowing for a delay \n262 and advance of up to 5 steps. ", + "bbox": [ + 140, + 247, + 825, + 316 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.2 First Integral Preservation for Hamiltonian System ", + "text_level": 1, + "bbox": [ + 171, + 332, + 568, + 347 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Before learning first integrals from data, we first evaluated FINDE as a numerical integrator using a known mass-spring system. The system has the state $\\mathbf { \\boldsymbol { \\mathscr { u } } } \\doteq ( q v ) ^ { \\top }$ , the dynamics ddt q = v and d $\\begin{array} { r } { { \\frac { \\mathrm { d } } { \\mathrm { d } t } } v \\ = - q } \\end{array}$ , and the system energy $E ( q , v ) =$ ${ \\scriptstyle { \\frac { 1 } { 2 } } } ( q ^ { 2 } + v ^ { 2 } )$ . Using the initial value $( 1 . 0 \\ 0 . 0 ) ^ { \\top }$ and the time step size $\\Delta t = 0 . 2$ , we solved the initial value problem of the true ODE using the leapfrog integrator. We applied FINDE with the true system energy $E$ as the first integral $V$ . Note that no neural networks nor training were involved. ", + "bbox": [ + 171, + 357, + 584, + 487 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/e62087da4d5eea6ebb9b01535f06401e481e011b62a41af256ef1f7e2091ccfd.jpg", + "image_caption": [ + "Figure 1: Integration of a known mass-spring system. " + ], + "image_footnote": [], + "bbox": [ + 596, + 315, + 815, + 459 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "273 The results with the analytical solution are shown in Fig. 1. The upper panel shows that the time series \n274 predicted by comparison methods overlap each other and are apparently almost identical. However, \n275 the lower panel shows that the energy obtained from the states predicted by the leapfrog integrator is \n276 fluctuating. The same is true for the case with the cFINDE. This is because the symplectic integrator \n277 and the cFINDE suffer from numerical errors caused by temporal discretization. In contrast, the \n278 dFINDE preserves the energy accurately. This is because, at every step, the dFINDE projects the state \n279 $( q v ) ^ { \\top }$ onto the discrete tangent space $\\mathcal { T } _ { ( \\pmb { v } , \\pmb { u } ) } \\mathcal { M } ^ { \\prime }$ . Although a smaller step size reduces numerical \n280 errors, this result demonstrates the advantage of dFINDE. ", + "bbox": [ + 147, + 492, + 825, + 604 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "281 4.3 Learning First Integrals from Data of Hamiltonian System ", + "text_level": 1, + "bbox": [ + 151, + 619, + 619, + 635 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "82 We evaluated FINDE on learning from the 2-body dataset. We used HNN as the base model $\\hat { f }$ . We \n83 found that the FINDE got better performances if it did not treat the Hamiltonian $H$ of the HNN \n84 as one of first integrals $V _ { k }$ . The medians and standard deviations of 5 trials are summarized in the \n85 leftmost column of Table 3. The cFINDE achieved better VPTs than the vanilla HNN with $K = 1$ \n86 to 2, and the performance was suddenly degraded for $K = 3$ . The dFINDE showed a similar trend \n7 with slightly better performances. The HNN with FINDE found two first integrals in addition to the \n88 Hamiltonian $H$ of the HNN. Even though a two-body problem is a Hamiltonian system that HNN \n89 can learn, the prior knowledge that there exist first integrals other than the Hamiltonian $H$ can be \n90 a clue to better learning. The HNN with FINDE got worse 1-step errors, suggesting that without \nFINDE, HNN overfitted short-term change and had difficulty predicting long-term dynamics. ", + "bbox": [ + 155, + 647, + 825, + 786 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We performed a symbolic regression of first integrals $V$ learned by the neural network. For $K = 2$ , the learned first integrals $V$ were identical to the linear momenta in the $x$ - and $y$ -directions up to affine transformation in most cases. See Appendix D.1 for more details. ", + "bbox": [ + 163, + 791, + 825, + 834 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We depict example results in Fig. 2. In the absence of FINDE, the mass positions $( x _ { 1 } , y _ { 1 } ) , ( x _ { 2 } , y _ { 2 } )$ became inaccurate in a short time and the center-of-gravity position $\\textstyle ( x _ { c } , y _ { c } ) = ( { \\frac { x _ { 1 } + x _ { 2 } } { 2 } } , { \\frac { y _ { 1 } + y _ { 2 } } { 2 } } )$ deviated rapidly. The HNN with cFINDE accurately predicted the state for a longer period. Even after errors in the mass positions became non-negligible, errors in the center-of-gravity position were still small. We show the absolute errors averaged over all trials in Fig. 3. In each of $x$ - and $y$ -directions, ", + "bbox": [ + 158, + 840, + 825, + 911 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/e6b3cd7b17a1d2790ac4925eed911dd13f929f98dbd945f672f955bbceabeefd.jpg", + "table_caption": [ + "Table 3: Results of FINDE. " + ], + "table_footnote": [ + "A standard deviation follows $\\pm$ symbol. Underlines indicate results better than the base models’ results, and bolded fonts indicate the best results. ∗ denotes that some trials failed in training because of the underflow of the step size. A dash denotes a case we did not try. " + ], + "table_body": "
2-body + HNNKdV2-pendFitzHugh-Nagumo
ModelK1-step↓VPT个1-step↓VPT个1-step↓VPT个1-step↓VPT个
base model -5.17 ±0.57 0.362 ±0.0265.59 ±0.300.339 ±0.0380.82 ±0.020.110±0.03573.66 ±12.59 0.236 ±0.053
+ cFINDE17.10 ±1.250.374 ±0.0366.24 ±0.440.371 ±0.0880.75 ±0.040.156±0.04254.18 ±8.120.127 ±0.148
27.78 ±1.390.450 ±0.0522.59 ±0.110.608 ±0.0850.73±0.050.198 ±0.08837.03 ±3.810.437 ±0.084
3>1030.147 ±0.146*3.19 ±0.370.730 ±0.0910.69 ±0.030.411 ±0.093>1060.007 ±0.007*
4>1030.101 ±0.0053.65 ±0.300.641 ±0.0710.77 ±0.070.395 ±0.083一
5>1030.080±0.0144.68 ±0.430.601 ±0.0690.80±0.070.585 ±0.097
6>10³0.070 ±0.0197.79 ±0.510.425 ±0.06712.53±0.000.005 ±0.000*
+ dFINDE17.01 ±1.060.379 ±0.04011.61 ±6.600.288 ±0.0830.75 ±0.100.152 ±0.01747.07 ±8.030.117 ±0.122
27.03 ±1.000.475 ±0.0222.70 ±0.260.598 ±0.0590.74±0.050.271 ±0.11133.24 ±3.400.455 ±0.032
3 54.78 ±36.39 0.309 ±0.0243.78±0.270.636 ±0.0240.69±0.050.447 ±0.081319.70 ±91.11 0.049 ±0.007
4>100.102 ±0.0153.48±0.320.780 ±0.0590.71 ±0.030.454 ±0.060
5>1030.086±0.011*5.26 ±0.150.718±0.0380.86 ±0.090.591 ±0.087
6>1030.059 ±0.0179.60 ±3.610.573 ±0.121 58.88 ±22.98 0.037 ±0.039
", + "bbox": [ + 192, + 112, + 807, + 333 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/89d03c9a7890406db2fbecc814f8b6b34cdc584df01fb3eb8c8146f7deede677.jpg", + "image_caption": [ + "Figure 2: Example results of the 2-body dataset. " + ], + "image_footnote": [], + "bbox": [ + 174, + 390, + 524, + 531 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/468b3482f1aadb8120b8263d0dcca0d63d619e797bf1f63eb5d4a25bb716d4d7.jpg", + "image_caption": [ + "Figure 3: Mean absolute errors of states for the 2-body dataset with or without cFINDE. " + ], + "image_footnote": [], + "bbox": [ + 529, + 386, + 812, + 522 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "300 the HNN without FINDE produced errors in the center-of-gravity position $x _ { c }$ (or $y _ { c }$ ) and those in the \n301 mass positions $x _ { 1 } , x _ { 2 }$ (or $y _ { 1 } , y _ { 2 } )$ at almost the same level. In contrast, when the cFINDE is present, \n302 errors in the center-of-gravity position were much smaller than those in the mass positions, implying \n303 that errors in one mass position canceled out errors in the other mass position. ", + "bbox": [ + 142, + 569, + 825, + 625 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Therefore, we conclude that FINDE not only had better prediction accuracy but also found and preserved linear momenta (which are related to symmetries in space) more accurately despite not having prior knowledge about symmetries. ", + "bbox": [ + 166, + 631, + 823, + 672 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "307 4.4 Learning First Integrals from Data of Unknown Systems ", + "text_level": 1, + "bbox": [ + 150, + 695, + 606, + 712 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "308 It is often unclear whether a target system is a Hamiltonian system or not, but one can expect that the \n309 target system has several first integrals. We evaluated FINDE using NODE as the base model. We \n310 summarized the results in Table 3. \n311 For the KdV dataset, the NODE with FINDE got much better 1-step errors and VPTs for a wide \n312 range of $K$ . Figure 4 shows an example result. The top panels show that the prediction results were \n313 apparently similar. The bottom panels summarize mean absolute errors in states $\\textbf { \\em u }$ , total mass $\\textstyle \\sum _ { k } u _ { k }$ \n314 and energy. In the absence of FINDE, the NODE increased all of its errors in proportion to time. With \n315 the cFINDE, the error in total mass increased at the point where the two solitons collided but then \n316 returned to the original level. Although the calculation is slightly inaccurate, the cFINDE learned \n317 to preserve the total mass. The rightmost panel shows that the error in energy continued to increase \n318 for $K = 2$ , but it stayed within a small range for $K = 3$ . These results suggest that the first or \n319 second quantity learned by the cFINDE was total mass, the third quantity was system energy, and the \n320 remaining quantity may correspond to one of the many first integrals of the KdV equation. \n321 For the 2-pend dataset, the NODE with FINDE got better 1-step errors and VPTs for $K = 1$ to 5 \n322 except for the 1-step error of the dFINDE with $K = 5$ . In addition to the system energy, the double \n323 pendulum has two holonomic constraints on the position, which lead to two additional constraints \n324 involving the velocity (see Appendix C for details). Thus, it is reasonable that the NODE with FINDE \n325 got the best VPTs for $K = 5$ first integrals and totally failed when assuming $K > 5$ first integrals. \n326 As exemplified in Fig. 5, the NODE without FINDE did not preserve the lengths of rods, making \n327 the states deviate gradually. See Appendix D.2 for the case when actual constraints are known. For \n328 the FitzHugh–Nagumo dataset, the NODE with FINDE got much better 1-step errors and VPTs for \n329 $K = 2$ . As exemplified in Fig. 6, the ground truth state converged to a periodic orbit, and only the \n330 NODE with cFINDE for $K = 2$ reproduced such dynamics. On the other hand, the state did not \n331 stay at a limited region without FINDE and converged to a wrong equilibrium with the cFINDE for \n332 $K = 1$ . For $K = 1$ , the sole quantity $V _ { 1 }$ may have tried to learn both of the two first integrals and \n333 remained under-trained. In these two cases, FINDE found all first integrals; $K = 5$ for the 2-pend \n334 dataset and $K = 2$ for the FitzHugh–Nagumo dataset. ", + "bbox": [ + 142, + 724, + 825, + 766 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 143, + 772, + 825, + 911 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/be5e28ffd5b113697cd23b60e955f60ccb12a63a936fb9ef9c08be823a75a887.jpg", + "image_caption": [ + "Figure 4: Example results of the KdV dataset. (top) Predicted states. Red belts denote moving solitons. (bottom) Mean absolute errors. " + ], + "image_footnote": [], + "bbox": [ + 176, + 87, + 777, + 246 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/8556882ad46dc0590252f5c7531273f1da140556e98cc6d80d3d2b4923a070e9.jpg", + "image_caption": [ + "Figure 5: Example results of the 2-pend dataset for 2,000 steps. " + ], + "image_footnote": [], + "bbox": [ + 174, + 286, + 405, + 397 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/7787089d38eb6cba10618809e53bcab45a879c57163f3cfcbb6474ad1a9cd456.jpg", + "image_caption": [ + "Figure 6: Example results of the FitzHugh–Nagumo dataset. " + ], + "image_footnote": [], + "bbox": [ + 415, + 282, + 794, + 415 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 138, + 462, + 825, + 656 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "335 5 Conclusion ", + "text_level": 1, + "bbox": [ + 151, + 678, + 299, + 694 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "This study proposed first integral-preserving neural differential equation (FINDE). FINDE projects the time evolution onto the submanifold defined using the (discrete) gradients of first integrals represented by a neural network. With an appropriate number of assumed first integrals, FINDE predicted future states more accurately than base models. Not only that, FINDE found and preserved the system energy and the total mass as first integrals, first integrals related to symmetries in space, and first integrals led by constraints in a unified manner. Therefore, FINDE has the potential to make a scientific discovery by revealing unknown properties of target dynamical systems. ", + "bbox": [ + 169, + 710, + 825, + 809 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "The 1-step errors were on the order of $1 0 ^ { - 5 }$ to $1 0 ^ { - 4 }$ in absolute error, being much larger than the numerical error tolerance of $1 0 ^ { - 9 }$ used in the experiments; numerical errors were negligible compared to modeling errors. However, the dFINDE tended to get VPTs better than the cFINDE despite the fact that its advantage is to eliminate numerical errors caused by temporal discretization. This result suggests that a method leading to smaller numerical errors results in a model with smaller modeling errors. Similar tendencies have been observed in previous works [10, 38], and these results may form a new frontier for integrating numerical and modeling errors. 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", + "bbox": [ + 147, + 85, + 826, + 669 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "1. For all authors... ", + "bbox": [ + 214, + 116, + 339, + 130 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes] The theoretical contributions are summarized in Remarks 1 and 2. The performance improvements were validated numerically in Table 3 and visually in Figs. 1–6. \n(b) Did you describe the limitations of your work? [Yes] We have discussed an increase in computational complexity at the bottoms of Sections 3.1 and 3.2. We also presented the limitations in Appendix D.2 while their situations were originally outside the scope of the proposed method. \n(c) Did you discuss any potential negative societal impacts of your work? [N/A] No societal impact is supposed. \n(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [N/A] We have read the guidelines carefully, but no ethical impact is supposed. ", + "bbox": [ + 235, + 133, + 825, + 306 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "2. If you are including theoretical results... ", + "bbox": [ + 215, + 310, + 493, + 325 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "(a) Did you state the full set of assumptions of all theoretical results? [Yes] We have introduced the background of the proposed method and provided the full set of assumptions in Section 2. Even though a slight modification may make the proposed method available on a general manifold, we have clearly stated that our theoretical and experimental results were limited to the finite-dimensional Eucleadian spaces. \n(b) Did you include complete proofs of all theoretical results? [Yes] We have provided a proof just before each of Remarks 1 and 2. ", + "bbox": [ + 233, + 329, + 825, + 428 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "3. If you ran experiments... ", + "bbox": [ + 214, + 433, + 393, + 446 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] We have enclosed the source code to reproduce all experiments in supplemental materials. \n(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] We have provided the software and hardware environment, network architectures, and hyperparameters in Section 4.1. We have also provided detailed hyperparameters to generate datasets in Appendix C. \n(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] We have summarized the standard deviations over five trials in Table 3. \n(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] We have provided the hardware environment in Section 4.1, but we have anonymized the cloud service providers to avoid a potential violation of the double-blind policy. ", + "bbox": [ + 236, + 450, + 826, + 650 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... ", + "bbox": [ + 215, + 655, + 823, + 669 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "(a) If your work uses existing assets, did you cite the creators? [Yes] We have made the source code for all experiments by modifying the source codes of HNN [26] and DGNet [38]. We have cited these references and added links to respective repositories in the footnotes. \n(b) Did you mention the license of the assets? [Yes] We have verified that the source codes of HNN [26] and DGNet [38] are provided in Apache-2.0 License and MIT License, respectively. We have clearly stated these facts in the footnotes. \n(c) Did you include any new assets either in the supplemental material or as a URL? [Yes] We have enclosed the source code to generate datasets in supplemental materials. \n(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A] \n(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A] ", + "bbox": [ + 238, + 672, + 825, + 861 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "5. If you used crowdsourcing or conducted research with human subjects... ", + "bbox": [ + 214, + 864, + 707, + 880 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A] ", + "bbox": [ + 235, + 883, + 826, + 912 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "534 (b) Did you describe any potential participant risks, with links to Institutional Review \n535 Board (IRB) approvals, if applicable? [N/A] \n536 (c) Did you include the estimated hourly wage paid to participants and the total amount \n537 spent on participant compensation? [N/A] ", + "bbox": [ + 138, + 90, + 826, + 151 + ], + "page_idx": 13 + } +] \ No newline at end of file diff --git a/parse/dev/xDaoT2zlJ0r/xDaoT2zlJ0r_middle.json b/parse/dev/xDaoT2zlJ0r/xDaoT2zlJ0r_middle.json new file mode 100644 index 0000000000000000000000000000000000000000..3367db6a493e98bba2923a4be5d9e87322728230 --- /dev/null +++ b/parse/dev/xDaoT2zlJ0r/xDaoT2zlJ0r_middle.json @@ -0,0 +1,48798 @@ +{ + "pdf_info": [ + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 128, + 97, + 483, + 137 + ], + "lines": [ + { + "bbox": [ + 162, + 95, + 449, + 118 + ], + "spans": [ + { + "bbox": [ + 162, + 95, + 449, + 118 + ], + "score": 1.0, + "content": "FINDE: Neural Differential Equations", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 126, + 116, + 483, + 140 + ], + "spans": [ + { + "bbox": [ + 126, + 116, + 483, + 140 + ], + "score": 1.0, + "content": "for Finding and Preserving Invariant Quantities", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 259, + 179, + 354, + 223 + ], + "lines": [ + { + "bbox": [ + 258, + 178, + 356, + 191 + ], + "spans": [ + { + "bbox": [ + 258, + 178, + 356, + 191 + ], + "score": 1.0, + "content": "Anonymous Author(s)", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 283, + 189, + 328, + 202 + ], + "spans": [ + { + "bbox": [ + 283, + 189, + 328, + 202 + ], + "score": 1.0, + "content": "Affiliation", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 286, + 200, + 324, + 213 + ], + "spans": [ + { + "bbox": [ + 286, + 200, + 324, + 213 + ], + "score": 1.0, + "content": "Address", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 290, + 213, + 320, + 222 + ], + "spans": [ + { + "bbox": [ + 290, + 213, + 320, + 222 + ], + "score": 1.0, + "content": "email", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3.5 + }, + { + "type": "title", + "bbox": [ + 283, + 252, + 328, + 265 + ], + "lines": [ + { + "bbox": [ + 281, + 251, + 331, + 267 + ], + "spans": [ + { + "bbox": [ + 281, + 251, + 331, + 267 + ], + "score": 1.0, + "content": "Abstract", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 91, + 277, + 469, + 452 + ], + "lines": [ + { + "bbox": [ + 93, + 277, + 470, + 290 + ], + "spans": [ + { + "bbox": [ + 93, + 280, + 99, + 289 + ], + "score": 1.0, + "content": "1", + "type": "text" + }, + { + "bbox": [ + 141, + 277, + 470, + 290 + ], + "score": 1.0, + "content": "Neural networks have shown promise for modeling dynamical systems from data.", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 93, + 289, + 469, + 300 + ], + "spans": [ + { + "bbox": [ + 93, + 291, + 99, + 299 + ], + "score": 1.0, + "content": "2", + "type": "text" + }, + { + "bbox": [ + 141, + 289, + 469, + 300 + ], + "score": 1.0, + "content": "Recent models, such as Hamiltonian neural networks, have been designed to", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 92, + 300, + 470, + 311 + ], + "spans": [ + { + "bbox": [ + 92, + 302, + 99, + 311 + ], + "score": 1.0, + "content": "3", + "type": "text" + }, + { + "bbox": [ + 141, + 300, + 470, + 311 + ], + "score": 1.0, + "content": "ensure known geometric structures of target systems and have shown excellent", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 93, + 311, + 470, + 322 + ], + "spans": [ + { + "bbox": [ + 93, + 313, + 99, + 321 + ], + "score": 1.0, + "content": "4", + "type": "text" + }, + { + "bbox": [ + 141, + 311, + 470, + 322 + ], + "score": 1.0, + "content": "modeling accuracy. 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Many", + "type": "text" + } + ], + "index": 36, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 648, + 506, + 660 + ], + "spans": [ + { + "bbox": [ + 88, + 649, + 100, + 660 + ], + "score": 1.0, + "content": "31", + "type": "text" + }, + { + "bbox": [ + 105, + 648, + 506, + 660 + ], + "score": 1.0, + "content": "previous studies have attempted to learn a target system accurately by incorporating prior knowledge", + "type": "text" + } + ], + "index": 37, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 658, + 506, + 671 + ], + "spans": [ + { + "bbox": [ + 88, + 659, + 100, + 670 + ], + "score": 1.0, + "content": "32", + "type": "text" + }, + { + "bbox": [ + 105, + 658, + 506, + 671 + ], + "score": 1.0, + "content": "about first integrals. Greydanus et al. [26] proposed Hamiltonian neural network (HNN), which", + "type": "text" + } + ], + "index": 38, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 668, + 506, + 684 + ], + "spans": [ + { + "bbox": [ + 88, + 671, + 100, + 681 + ], + "score": 1.0, + "content": "33", + "type": "text" + }, + { + "bbox": [ + 105, + 668, + 506, + 684 + ], + "score": 1.0, + "content": "employs a neural network to approximate Hamilton’s equation, thereby conserving the system energy", + "type": "text" + } + ], + "index": 39, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 679, + 506, + 693 + ], + "spans": [ + { + "bbox": [ + 88, + 682, + 100, + 692 + ], + "score": 1.0, + "content": "34", + "type": "text" + }, + { + "bbox": [ + 105, + 679, + 506, + 693 + ], + "score": 1.0, + "content": "called the Hamiltonian. Finzi et al. [19] proposed neural network architectures that conserve linear", + "type": "text" + } + ], + "index": 40, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 691, + 506, + 704 + ], + "spans": [ + { + "bbox": [ + 88, + 693, + 99, + 703 + ], + "score": 1.0, + "content": "35", + "type": "text" + }, + { + "bbox": [ + 105, + 691, + 506, + 704 + ], + "score": 1.0, + "content": "and angular momenta by utilizing the graph structure. Finzi et al. [20] also extended HNN to a system", + "type": "text" + } + ], + "index": 41, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 702, + 506, + 714 + ], + "spans": [ + { + "bbox": [ + 89, + 704, + 100, + 713 + ], + "score": 1.0, + "content": "36", + "type": "text" + }, + { + "bbox": [ + 105, + 702, + 506, + 714 + ], + "score": 1.0, + "content": "with holonomic constraints, which lead to first integrals such as a pendulum length. Matsubara et al.", + "type": "text" + } + ], + "index": 42, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 226, + 505, + 238 + ], + "spans": [ + { + "bbox": [ + 89, + 227, + 100, + 237 + ], + "score": 1.0, + "content": "37", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 226, + 505, + 238 + ], + "score": 1.0, + "content": "[38] proposed a model that preserves the total mass of a discretized partial differential equation", + "type": "text", + "cross_page": true + } + ], + "index": 4, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 237, + 506, + 249 + ], + "spans": [ + { + "bbox": [ + 89, + 239, + 100, + 248 + ], + "score": 1.0, + "content": "38", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 237, + 506, + 249 + ], + "score": 1.0, + "content": "(PDE). These studies have demonstrated that a neural network with more prior knowledge about first", + "type": "text", + "cross_page": true + } + ], + "index": 5, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 248, + 496, + 261 + ], + "spans": [ + { + "bbox": [ + 89, + 250, + 100, + 260 + ], + "score": 1.0, + "content": "39", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 248, + 496, + 261 + ], + "score": 1.0, + "content": "integrals predicts the dynamics of the target system more accurately. See Table 1 for comparison.", + "type": "text", + "cross_page": true + } + ], + "index": 6, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 263, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 89, + 266, + 100, + 275 + ], + "score": 1.0, + "content": "40", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 263, + 505, + 277 + ], + "score": 1.0, + "content": "Previous studies have mainly attempted to preserve known first integrals. However, in situations", + "type": "text", + "cross_page": true + } + ], + "index": 7, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 275, + 505, + 287 + ], + "spans": [ + { + "bbox": [ + 89, + 277, + 99, + 286 + ], + "score": 1.0, + "content": "41", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 275, + 505, + 287 + ], + "score": 1.0, + "content": "where a neural network learns an unknown target system, it is naturally expected that first integrals", + "type": "text", + "cross_page": true + } + ], + "index": 8, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 285, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 89, + 288, + 100, + 297 + ], + "score": 1.0, + "content": "42", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 285, + 505, + 299 + ], + "score": 1.0, + "content": "associated with the target system are also unknown, and it is not clear which of the above methods are", + "type": "text", + "cross_page": true + } + ], + "index": 9, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 296, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 89, + 299, + 100, + 309 + ], + "score": 1.0, + "content": "43", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 296, + 505, + 309 + ], + "score": 1.0, + "content": "available. Given the above, this study proposes First Integral-preserving Neural Differential Equation", + "type": "text", + "cross_page": true + } + ], + "index": 10, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 306, + 477, + 321 + ], + "spans": [ + { + "bbox": [ + 89, + 309, + 100, + 319 + ], + "score": 1.0, + "content": "44", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 306, + 477, + 321 + ], + "score": 1.0, + "content": "(FINDE) to find and preserve first integrals from data. FINDE has the following advantages.", + "type": "text", + "cross_page": true + } + ], + "index": 11, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 329, + 505, + 344 + ], + "spans": [ + { + "bbox": [ + 89, + 333, + 100, + 342 + ], + "score": 1.0, + "content": "45", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 104, + 329, + 505, + 344 + ], + "score": 1.0, + "content": "Learning First Integrals For modeling continuous-time dynamics with known first integrals, many", + "type": "text", + "cross_page": true + } + ], + "index": 12, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 342, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 89, + 344, + 100, + 353 + ], + "score": 1.0, + "content": "46", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 342, + 505, + 353 + ], + "score": 1.0, + "content": "studies have designed architectures or operations of neural networks [13, 19, 20, 26, 38]. 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A symbolic regression", + "type": "text", + "cross_page": true + } + ], + "index": 16, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 384, + 507, + 398 + ], + "spans": [ + { + "bbox": [ + 89, + 388, + 100, + 397 + ], + "score": 1.0, + "content": "50", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 104, + 384, + 507, + 398 + ], + "score": 1.0, + "content": "confirms that the learned first integrals are consistent with well-known first integrals of target systems.", + "type": "text", + "cross_page": true + } + ], + "index": 17, + "is_list_start_line": true + } + ], + "index": 28.5, + "bbox_fs": [ + 89, + 498, + 506, + 611 + ] + }, + { + "type": "index", + "bbox": [ + 89, + 614, + 505, + 713 + ], + "lines": [], + "index": 38, + "bbox_fs": [ + 88, + 615, + 506, + 714 + ], + "lines_deleted": true + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 133, + 90, + 477, + 206 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 142, + 78, + 467, + 88 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 143, + 76, + 467, + 91 + ], + "spans": [ + { + "bbox": [ + 143, + 76, + 467, + 91 + ], + "score": 1.0, + "content": "Table 1: Comparison between Related Studies on Preservation of First Integrals.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 133, + 90, + 477, + 206 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 133, + 90, + 477, + 206 + ], + "spans": [ + { + "bbox": [ + 133, + 90, + 477, + 206 + ], + "score": 0.965, + "html": "
energymonentum massconstraint learning invariants exact conservation
NODE [8]
HNN [26]√
LieConv [19]√厂
DGNet [38]√√
CHNN [20]√√
continuous FINDE (proposed)√√√√√
discrete FINDE (proposed)√厂厂丁
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energymonentum massconstraint learning invariants exact conservation
NODE [8]
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LieConv [19]√厂
DGNet [38]√√
CHNN [20]√√
continuous FINDE (proposed)√√√√√
discrete FINDE (proposed)√厂厂丁
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This method eliminates numerical errors caused by temporal discretization and is", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 86, + 458, + 432, + 470 + ], + "spans": [ + { + "bbox": [ + 86, + 460, + 100, + 469 + ], + "score": 1.0, + "content": "102", + "type": "text" + }, + { + "bbox": [ + 106, + 458, + 432, + 470 + ], + "score": 1.0, + "content": "used to preserve the Hamiltonian exactly (up to rounding errors) in discrete time.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 87, + 474, + 505, + 518 + ], + "lines": [ + { + "bbox": [ + 86, + 474, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 86, + 476, + 99, + 486 + ], + "score": 1.0, + "content": "103", + "type": "text" + }, + { + "bbox": [ + 106, + 474, + 505, + 487 + ], + "score": 1.0, + "content": "Except for DGNet, which used discrete gradients to preserve the Hamiltonian [38], all the above", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 86, + 484, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 86, + 487, + 99, + 496 + ], + "score": 1.0, + "content": "104", + "type": "text" + }, + { + "bbox": [ + 106, + 484, + 505, + 497 + ], + "score": 1.0, + "content": "methods have never been applied to neural networks due to difficulties that we will introduce later. To", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 86, + 496, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 86, + 498, + 99, + 507 + ], + "score": 1.0, + "content": "105", + "type": "text" + }, + { + "bbox": [ + 106, + 496, + 505, + 508 + ], + "score": 1.0, + "content": "our best knowledge, the discrete-time version of FINDE is the first projection method for dynamical", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 86, + 507, + 270, + 520 + ], + "spans": [ + { + "bbox": [ + 86, + 509, + 99, + 518 + ], + "score": 1.0, + "content": "106", + "type": "text" + }, + { + "bbox": [ + 106, + 507, + 270, + 520 + ], + "score": 1.0, + "content": "systems modeled using neural networks.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33.5 + }, + { + "type": "text", + "bbox": [ + 86, + 531, + 505, + 695 + ], + "lines": [ + { + "bbox": [ + 86, + 530, + 507, + 544 + ], + "spans": [ + { + "bbox": [ + 86, + 533, + 99, + 542 + ], + "score": 1.0, + "content": "107", + "type": "text" + }, + { + "bbox": [ + 104, + 530, + 507, + 544 + ], + "score": 1.0, + "content": "Preservation of First Integrals by Neural Networks NODE defines an ODE using a neural net-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 86, + 542, + 506, + 554 + ], + "spans": [ + { + "bbox": [ + 86, + 543, + 100, + 554 + ], + "score": 1.0, + "content": "108", + "type": "text" + }, + { + "bbox": [ + 105, + 542, + 506, + 554 + ], + "score": 1.0, + "content": "work in the most general way with no associated first integrals [8]. NODE is a universal approximator", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 86, + 552, + 506, + 566 + ], + "spans": [ + { + "bbox": [ + 86, + 555, + 99, + 564 + ], + "score": 1.0, + "content": "109", + "type": "text" + }, + { + "bbox": [ + 105, + 552, + 506, + 566 + ], + "score": 1.0, + "content": "to ODEs [51], and it can approximate any ODE with arbitrary accuracy if there is an infinite amount", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 86, + 563, + 506, + 577 + ], + "spans": [ + { + "bbox": [ + 86, + 566, + 99, + 575 + ], + "score": 1.0, + "content": "110", + "type": "text" + }, + { + "bbox": [ + 105, + 563, + 506, + 577 + ], + "score": 1.0, + "content": "of training data. In practice, the amount of training data is limited, and prior knowledge about the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 86, + 574, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 86, + 577, + 99, + 586 + ], + "score": 1.0, + "content": "111", + "type": "text" + }, + { + "bbox": [ + 105, + 574, + 506, + 587 + ], + "score": 1.0, + "content": "target system is helpful for learning (see [48] for the case with convolutional neural networks). HNN", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 86, + 585, + 506, + 598 + ], + "spans": [ + { + "bbox": [ + 86, + 587, + 100, + 597 + ], + "score": 1.0, + "content": "112", + "type": "text" + }, + { + "bbox": [ + 105, + 585, + 506, + 598 + ], + "score": 1.0, + "content": "assumes the target system to be a Hamiltonian system in the canonical form [26]. HNN guarantees", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 86, + 594, + 506, + 611 + ], + "spans": [ + { + "bbox": [ + 86, + 598, + 100, + 609 + ], + "score": 1.0, + "content": "113", + "type": "text" + }, + { + "bbox": [ + 105, + 594, + 506, + 611 + ], + "score": 1.0, + "content": "various properties of Hamiltonian systems by definition, including the conservation of the energy", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 86, + 607, + 506, + 620 + ], + "spans": [ + { + "bbox": [ + 86, + 610, + 99, + 618 + ], + "score": 1.0, + "content": "114", + "type": "text" + }, + { + "bbox": [ + 105, + 607, + 506, + 620 + ], + "score": 1.0, + "content": "and the preservation of the symplectic structure in continuous time [27]. Some studies employed a", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 86, + 618, + 506, + 630 + ], + "spans": [ + { + "bbox": [ + 86, + 620, + 100, + 630 + ], + "score": 1.0, + "content": "115", + "type": "text" + }, + { + "bbox": [ + 104, + 618, + 506, + 630 + ], + "score": 1.0, + "content": "symplectic integrator for HNN to preserve the energy and symplectic structure with smaller numerical", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 86, + 628, + 506, + 641 + ], + "spans": [ + { + "bbox": [ + 86, + 631, + 100, + 641 + ], + "score": 1.0, + "content": "116", + "type": "text" + }, + { + "bbox": [ + 104, + 628, + 506, + 641 + ], + "score": 1.0, + "content": "errors [10]. 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To", + "type": "text" + } + ], + "index": 33, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 496, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 86, + 498, + 99, + 507 + ], + "score": 1.0, + "content": "105", + "type": "text" + }, + { + "bbox": [ + 106, + 496, + 505, + 508 + ], + "score": 1.0, + "content": "our best knowledge, the discrete-time version of FINDE is the first projection method for dynamical", + "type": "text" + } + ], + "index": 34, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 507, + 270, + 520 + ], + "spans": [ + { + "bbox": [ + 86, + 509, + 99, + 518 + ], + "score": 1.0, + "content": "106", + "type": "text" + }, + { + "bbox": [ + 106, + 507, + 270, + 520 + ], + "score": 1.0, + "content": "systems modeled using neural networks.", + "type": "text" + } + ], + "index": 35, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 530, + 507, + 544 + ], + "spans": [ + { + "bbox": [ + 86, + 533, + 99, + 542 + ], + "score": 1.0, + "content": "107", + "type": "text" + }, + { + "bbox": [ + 104, + 530, + 507, + 544 + ], + "score": 1.0, + "content": "Preservation of First Integrals by Neural Networks NODE defines an ODE using a neural net-", + "type": "text" + } + ], + "index": 36, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 542, + 506, + 554 + ], + "spans": [ + { + "bbox": [ + 86, + 543, + 100, + 554 + ], + "score": 1.0, + "content": "108", + "type": "text" + }, + { + "bbox": [ + 105, + 542, + 506, + 554 + ], + "score": 1.0, + "content": "work in the most general way with no associated first integrals [8]. NODE is a universal approximator", + "type": "text" + } + ], + "index": 37, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 552, + 506, + 566 + ], + "spans": [ + { + "bbox": [ + 86, + 555, + 99, + 564 + ], + "score": 1.0, + "content": "109", + "type": "text" + }, + { + "bbox": [ + 105, + 552, + 506, + 566 + ], + "score": 1.0, + "content": "to ODEs [51], and it can approximate any ODE with arbitrary accuracy if there is an infinite amount", + "type": "text" + } + ], + "index": 38, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 563, + 506, + 577 + ], + "spans": [ + { + "bbox": [ + 86, + 566, + 99, + 575 + ], + "score": 1.0, + "content": "110", + "type": "text" + }, + { + "bbox": [ + 105, + 563, + 506, + 577 + ], + "score": 1.0, + "content": "of training data. In practice, the amount of training data is limited, and prior knowledge about the", + "type": "text" + } + ], + "index": 39, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 574, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 86, + 577, + 99, + 586 + ], + "score": 1.0, + "content": "111", + "type": "text" + }, + { + "bbox": [ + 105, + 574, + 506, + 587 + ], + "score": 1.0, + "content": "target system is helpful for learning (see [48] for the case with convolutional neural networks). HNN", + "type": "text" + } + ], + "index": 40, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 585, + 506, + 598 + ], + "spans": [ + { + "bbox": [ + 86, + 587, + 100, + 597 + ], + "score": 1.0, + "content": "112", + "type": "text" + }, + { + "bbox": [ + 105, + 585, + 506, + 598 + ], + "score": 1.0, + "content": "assumes the target system to be a Hamiltonian system in the canonical form [26]. HNN guarantees", + "type": "text" + } + ], + "index": 41, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 594, + 506, + 611 + ], + "spans": [ + { + "bbox": [ + 86, + 598, + 100, + 609 + ], + "score": 1.0, + "content": "113", + "type": "text" + }, + { + "bbox": [ + 105, + 594, + 506, + 611 + ], + "score": 1.0, + "content": "various properties of Hamiltonian systems by definition, including the conservation of the energy", + "type": "text" + } + ], + "index": 42, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 607, + 506, + 620 + ], + "spans": [ + { + "bbox": [ + 86, + 610, + 99, + 618 + ], + "score": 1.0, + "content": "114", + "type": "text" + }, + { + "bbox": [ + 105, + 607, + 506, + 620 + ], + "score": 1.0, + "content": "and the preservation of the symplectic structure in continuous time [27]. 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Deep conservation extracts latent dynamics of a PDE system and", + "type": "text" + } + ], + "index": 47, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 662, + 506, + 675 + ], + "spans": [ + { + "bbox": [ + 86, + 664, + 100, + 673 + ], + "score": 1.0, + "content": "119", + "type": "text" + }, + { + "bbox": [ + 104, + 662, + 386, + 675 + ], + "score": 1.0, + "content": "preserves a quantity of interest by forcing its flux to be zero [34].", + "type": "text" + }, + { + "bbox": [ + 387, + 662, + 422, + 672 + ], + "score": 0.8, + "content": "\\mathrm { H N N + + }", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 662, + 506, + 675 + ], + "score": 1.0, + "content": "also guarantees the", + "type": "text" + } + ], + "index": 48, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 672, + 506, + 685 + ], + "spans": [ + { + "bbox": [ + 86, + 674, + 100, + 685 + ], + "score": 1.0, + "content": "120", + "type": "text" + }, + { + "bbox": [ + 104, + 672, + 506, + 685 + ], + "score": 1.0, + "content": "conservation of the mass in PDE systems by using a coefficient matrix derived from differential", + "type": "text" + } + ], + "index": 49, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 683, + 168, + 697 + ], + "spans": [ + { + "bbox": [ + 86, + 686, + 99, + 695 + ], + "score": 1.0, + "content": "121", + "type": "text" + }, + { + "bbox": [ + 105, + 683, + 168, + 697 + ], + "score": 1.0, + "content": "operators [38].", + "type": "text" + } + ], + "index": 50, + "is_list_start_line": true + }, + { + "bbox": [ + 87, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 87, + 702, + 100, + 712 + ], + "score": 1.0, + "content": "122", + "type": "text" + }, + { + "bbox": [ + 105, + 700, + 505, + 712 + ], + "score": 1.0, + "content": "Several studies proposed neural networks to learn Lyapunov functions, which are expected to be", + "type": "text" + } + ], + "index": 51, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 711, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 85, + 711, + 506, + 722 + ], + "score": 1.0, + "content": "123 non-increasing over time, in contrast to first integrals [37, 50]. If the state moves in the direction of", + "type": "text" + } + ], + "index": 52, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 73, + 505, + 86 + ], + "spans": [ + { + "bbox": [ + 86, + 75, + 100, + 84 + ], + "score": 1.0, + "content": "124", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 73, + 505, + 86 + ], + "score": 1.0, + "content": "increasing the function, it is projected onto or moved inside the counter line of the gradient of the", + "type": "text", + "cross_page": true + } + ], + "index": 0, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 83, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 86, + 86, + 100, + 95 + ], + "score": 1.0, + "content": "125", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 83, + 505, + 96 + ], + "score": 1.0, + "content": "Lyapunov function. Their idea is similar to the continuous-time version of FINDE but limited to a", + "type": "text", + "cross_page": true + } + ], + "index": 1, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 94, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 85, + 96, + 100, + 107 + ], + "score": 1.0, + "content": "126", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 94, + 505, + 107 + ], + "score": 1.0, + "content": "single non-increasing quantity in continuous time. On the other hand, our proposed FINDE preserves", + "type": "text", + "cross_page": true + } + ], + "index": 2, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 106, + 333, + 118 + ], + "spans": [ + { + "bbox": [ + 86, + 108, + 100, + 117 + ], + "score": 1.0, + "content": "127", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 106, + 333, + 118 + ], + "score": 1.0, + "content": "multiple quantities in both continuous and discrete time.", + "type": "text", + "cross_page": true + } + ], + "index": 3, + "is_list_start_line": true + }, + { + "bbox": [ + 92, + 122, + 505, + 134 + ], + "spans": [ + { + "bbox": [ + 92, + 124, + 100, + 132 + ], + "score": 1.0, + "content": "28", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 104, + 122, + 505, + 134 + ], + "score": 1.0, + "content": "Previous studies aimed to preserve known first integrals. Moreover, except for DGNet [38], all", + "type": "text", + "cross_page": true + } + ], + "index": 4, + "is_list_start_line": true + }, + { + "bbox": [ + 92, + 133, + 506, + 145 + ], + "spans": [ + { + "bbox": [ + 92, + 136, + 100, + 143 + ], + "score": 1.0, + "content": "29", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 104, + 133, + 506, + 145 + ], + "score": 1.0, + "content": "the above methods suffer from numerical errors caused by temporal discretization. In contrast, our", + "type": "text", + "cross_page": true + } + ], + "index": 5, + "is_list_start_line": true + }, + { + "bbox": [ + 92, + 144, + 456, + 156 + ], + "spans": [ + { + "bbox": [ + 92, + 147, + 100, + 155 + ], + "score": 1.0, + "content": "30", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 144, + 456, + 156 + ], + "score": 1.0, + "content": "proposed FINDE learns first integrals from data and can eliminate discretization errors.", + "type": "text", + "cross_page": true + } + ], + "index": 6, + "is_list_start_line": true + } + ], + "index": 29, + "bbox_fs": [ + 86, + 414, + 506, + 470 + ] + }, + { + "type": "index", + "bbox": [ + 87, + 474, + 505, + 518 + ], + "lines": [], + "index": 33.5, + "bbox_fs": [ + 86, + 474, + 505, + 520 + ], + "lines_deleted": true + }, + { + "type": "index", + "bbox": [ + 86, + 531, + 505, + 695 + ], + "lines": [], + "index": 43, + "bbox_fs": [ + 86, + 530, + 507, + 697 + ], + "lines_deleted": true + }, + { + "type": "index", + "bbox": [ + 89, + 700, + 505, + 722 + ], + "lines": [], + "index": 51.5, + "bbox_fs": [ + 85, + 700, + 506, + 722 + ], + "lines_deleted": true + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 86, + 72, + 505, + 117 + ], + "lines": [ + { + "bbox": [ + 86, + 73, + 505, + 86 + ], + "spans": [ + { + "bbox": [ + 86, + 75, + 100, + 84 + ], + "score": 1.0, + "content": "124", + "type": "text" + }, + { + "bbox": [ + 106, + 73, + 505, + 86 + ], + "score": 1.0, + "content": "increasing the function, it is projected onto or moved inside the counter line of the gradient of the", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 86, + 83, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 86, + 86, + 100, + 95 + ], + "score": 1.0, + "content": "125", + "type": "text" + }, + { + "bbox": [ + 105, + 83, + 505, + 96 + ], + "score": 1.0, + "content": "Lyapunov function. Their idea is similar to the continuous-time version of FINDE but limited to a", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 85, + 94, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 85, + 96, + 100, + 107 + ], + "score": 1.0, + "content": "126", + "type": "text" + }, + { + "bbox": [ + 105, + 94, + 505, + 107 + ], + "score": 1.0, + "content": "single non-increasing quantity in continuous time. On the other hand, our proposed FINDE preserves", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 86, + 106, + 333, + 118 + ], + "spans": [ + { + "bbox": [ + 86, + 108, + 100, + 117 + ], + "score": 1.0, + "content": "127", + "type": "text" + }, + { + "bbox": [ + 106, + 106, + 333, + 118 + ], + "score": 1.0, + "content": "multiple quantities in both continuous and discrete time.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 93, + 122, + 507, + 155 + ], + "lines": [ + { + "bbox": [ + 92, + 122, + 505, + 134 + ], + "spans": [ + { + "bbox": [ + 92, + 124, + 100, + 132 + ], + "score": 1.0, + "content": "28", + "type": "text" + }, + { + "bbox": [ + 104, + 122, + 505, + 134 + ], + "score": 1.0, + "content": "Previous studies aimed to preserve known first integrals. Moreover, except for DGNet [38], all", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 92, + 133, + 506, + 145 + ], + "spans": [ + { + "bbox": [ + 92, + 136, + 100, + 143 + ], + "score": 1.0, + "content": "29", + "type": "text" + }, + { + "bbox": [ + 104, + 133, + 506, + 145 + ], + "score": 1.0, + "content": "the above methods suffer from numerical errors caused by temporal discretization. In contrast, our", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 92, + 144, + 456, + 156 + ], + "spans": [ + { + "bbox": [ + 92, + 147, + 100, + 155 + ], + "score": 1.0, + "content": "30", + "type": "text" + }, + { + "bbox": [ + 105, + 144, + 456, + 156 + ], + "score": 1.0, + "content": "proposed FINDE learns first integrals from data and can eliminate discretization errors.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5 + }, + { + "type": "title", + "bbox": [ + 92, + 169, + 404, + 183 + ], + "lines": [ + { + "bbox": [ + 87, + 168, + 405, + 187 + ], + "spans": [ + { + "bbox": [ + 87, + 168, + 405, + 187 + ], + "score": 1.0, + "content": "131 3 First Integral-Preserving Neural Differential Equation", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 86, + 193, + 505, + 271 + ], + "lines": [ + { + "bbox": [ + 86, + 192, + 506, + 207 + ], + "spans": [ + { + "bbox": [ + 86, + 195, + 100, + 205 + ], + "score": 1.0, + "content": "132", + "type": "text" + }, + { + "bbox": [ + 105, + 192, + 506, + 207 + ], + "score": 1.0, + "content": "The main purpose is to find and preserve first integrals from data by neural networks. We suppose", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 86, + 204, + 506, + 217 + ], + "spans": [ + { + "bbox": [ + 86, + 206, + 100, + 216 + ], + "score": 1.0, + "content": "133", + "type": "text" + }, + { + "bbox": [ + 105, + 204, + 238, + 217 + ], + "score": 1.0, + "content": "that a target system has at least", + "type": "text" + }, + { + "bbox": [ + 238, + 205, + 249, + 214 + ], + "score": 0.81, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 204, + 506, + 217 + ], + "score": 1.0, + "content": "unknown functionally independent first integrals. Even when", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 86, + 214, + 506, + 228 + ], + "spans": [ + { + "bbox": [ + 86, + 217, + 100, + 227 + ], + "score": 1.0, + "content": "134", + "type": "text" + }, + { + "bbox": [ + 105, + 214, + 506, + 228 + ], + "score": 1.0, + "content": "a NODE learns the target system, it is not guaranteed to learn these first integrals. Hence, we", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 86, + 226, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 86, + 228, + 100, + 238 + ], + "score": 1.0, + "content": "135", + "type": "text" + }, + { + "bbox": [ + 105, + 226, + 237, + 239 + ], + "score": 1.0, + "content": "introduce a neural network with", + "type": "text" + }, + { + "bbox": [ + 237, + 227, + 248, + 236 + ], + "score": 0.81, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 226, + 505, + 239 + ], + "score": 1.0, + "content": "outputs, each of which is expected to learn one of first integrals", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 86, + 235, + 507, + 250 + ], + "spans": [ + { + "bbox": [ + 86, + 239, + 100, + 249 + ], + "score": 1.0, + "content": "136", + "type": "text" + }, + { + "bbox": [ + 105, + 235, + 161, + 250 + ], + "score": 1.0, + "content": "expressed as", + "type": "text" + }, + { + "bbox": [ + 162, + 236, + 226, + 248 + ], + "score": 0.91, + "content": "V _ { k } : \\mathbb { R } ^ { N } \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 235, + 243, + 250 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 243, + 237, + 302, + 248 + ], + "score": 0.92, + "content": "k = 1 , \\dots , K", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 235, + 507, + 250 + ], + "score": 1.0, + "content": ". We denote the set of first integrals by a vector", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 86, + 246, + 506, + 261 + ], + "spans": [ + { + "bbox": [ + 86, + 250, + 100, + 259 + ], + "score": 1.0, + "content": "137", + "type": "text" + }, + { + "bbox": [ + 106, + 248, + 267, + 260 + ], + "score": 0.72, + "content": "\\pmb { V } ( \\pmb { u } ) = ( V _ { 1 } ( \\pmb { u } ) V _ { 2 } ( \\pmb { u } ) \\dotsm V _ { K } ( \\pmb { u } ) ) ^ { \\top }", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 246, + 373, + 261 + ], + "score": 1.0, + "content": ". Then, the submanifold", + "type": "text" + }, + { + "bbox": [ + 373, + 248, + 389, + 258 + ], + "score": 0.88, + "content": "\\mathcal { M } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 246, + 506, + 261 + ], + "score": 1.0, + "content": "is defined using the neural", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 86, + 258, + 208, + 272 + ], + "spans": [ + { + "bbox": [ + 86, + 261, + 100, + 270 + ], + "score": 1.0, + "content": "138", + "type": "text" + }, + { + "bbox": [ + 105, + 258, + 141, + 272 + ], + "score": 1.0, + "content": "network", + "type": "text" + }, + { + "bbox": [ + 142, + 260, + 152, + 269 + ], + "score": 0.74, + "content": "V", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 258, + 208, + 272 + ], + "score": 1.0, + "content": "as in Eq. (1).", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 86, + 275, + 506, + 363 + ], + "lines": [ + { + "bbox": [ + 86, + 275, + 505, + 287 + ], + "spans": [ + { + "bbox": [ + 86, + 277, + 99, + 286 + ], + "score": 1.0, + "content": "139", + "type": "text" + }, + { + "bbox": [ + 105, + 275, + 505, + 287 + ], + "score": 1.0, + "content": "Because there is no way to define local coordinates on such submanifolds, a local coordinate method", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 86, + 286, + 506, + 299 + ], + "spans": [ + { + "bbox": [ + 86, + 288, + 100, + 299 + ], + "score": 1.0, + "content": "140", + "type": "text" + }, + { + "bbox": [ + 105, + 286, + 506, + 299 + ], + "score": 1.0, + "content": "is not applicable. When using a projection method, the optimization problem in Eq. (3) should", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 86, + 298, + 505, + 310 + ], + "spans": [ + { + "bbox": [ + 86, + 299, + 100, + 309 + ], + "score": 1.0, + "content": "141", + "type": "text" + }, + { + "bbox": [ + 105, + 298, + 505, + 310 + ], + "score": 1.0, + "content": "be solved at every training iteration as well as in the prediction phase. Optimization problems are", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 86, + 308, + 506, + 321 + ], + "spans": [ + { + "bbox": [ + 86, + 310, + 100, + 320 + ], + "score": 1.0, + "content": "142", + "type": "text" + }, + { + "bbox": [ + 105, + 308, + 506, + 321 + ], + "score": 1.0, + "content": "computationally expensive, and common libraries for neural networks do not provide backpropagation", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 86, + 318, + 506, + 332 + ], + "spans": [ + { + "bbox": [ + 86, + 321, + 100, + 331 + ], + "score": 1.0, + "content": "143", + "type": "text" + }, + { + "bbox": [ + 105, + 318, + 506, + 332 + ], + "score": 1.0, + "content": "algorithms for optimization problems [1, 42].1 Until a recent study has proposed an algorithm [38],", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 86, + 329, + 507, + 343 + ], + "spans": [ + { + "bbox": [ + 86, + 332, + 100, + 342 + ], + "score": 1.0, + "content": "144", + "type": "text" + }, + { + "bbox": [ + 105, + 329, + 507, + 343 + ], + "score": 1.0, + "content": "there was no way to obtain discrete gradients of neural networks. Because of these difficulties,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 86, + 339, + 507, + 355 + ], + "spans": [ + { + "bbox": [ + 86, + 343, + 100, + 352 + ], + "score": 1.0, + "content": "145", + "type": "text" + }, + { + "bbox": [ + 104, + 339, + 507, + 355 + ], + "score": 1.0, + "content": "no methods for preserving first integrals have been applied to neural networks. By leveraging a", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 86, + 352, + 435, + 365 + ], + "spans": [ + { + "bbox": [ + 86, + 354, + 100, + 363 + ], + "score": 1.0, + "content": "146", + "type": "text" + }, + { + "bbox": [ + 105, + 352, + 435, + 365 + ], + "score": 1.0, + "content": "projection method and a discrete gradient method, we propose FINDE as follows.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 18.5 + }, + { + "type": "title", + "bbox": [ + 99, + 374, + 370, + 387 + ], + "lines": [ + { + "bbox": [ + 105, + 373, + 372, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 372, + 389 + ], + "score": 1.0, + "content": "3.1 Continuous FINDE: Time-Derivative Projection Method", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 98, + 395, + 505, + 428 + ], + "lines": [ + { + "bbox": [ + 106, + 395, + 505, + 407 + ], + "spans": [ + { + "bbox": [ + 106, + 395, + 505, + 407 + ], + "score": 1.0, + "content": "First, we propose a time-derivative projection method called continuous FINDE (cFINDE) for neural", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 406, + 505, + 418 + ], + "spans": [ + { + "bbox": [ + 106, + 406, + 371, + 418 + ], + "score": 1.0, + "content": "networks, which projects the time-derivative onto the tangent space", + "type": "text" + }, + { + "bbox": [ + 372, + 406, + 399, + 417 + ], + "score": 0.92, + "content": "\\mathcal { T } _ { u } \\mathcal { M } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 406, + 505, + 418 + ], + "score": 1.0, + "content": ". While it still suffers from", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 417, + 360, + 429 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 360, + 429 + ], + "score": 1.0, + "content": "numerical errors, it is sufficient to find first integrals from data.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 97, + 434, + 504, + 459 + ], + "lines": [ + { + "bbox": [ + 104, + 433, + 503, + 447 + ], + "spans": [ + { + "bbox": [ + 104, + 434, + 441, + 447 + ], + "score": 1.0, + "content": "We suppose that a neural network called a base model defines the time-derivative", + "type": "text" + }, + { + "bbox": [ + 441, + 433, + 503, + 447 + ], + "score": 0.92, + "content": "\\hat { f } : \\mathbb { R } ^ { N } \\to \\mathbb { R } ^ { N }", + "type": "inline_equation" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 443, + 384, + 461 + ], + "spans": [ + { + "bbox": [ + 104, + 443, + 249, + 461 + ], + "score": 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\\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol { \\mathbf { \\rho } } \\boldsymbol \\mathbf { \\rho } \\boldsymbol { \\rho } \\boldsymbol \\end{array}", + "type": "interline_equation", + "image_path": "8f8b79adb26b9ef479e91536ac45479b6c800405868d7256ba6e077ccedb198f.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 184, + 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For satisfying", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 683, + 505, + 695 + ], + "spans": [ + { + "bbox": [ + 106, + 683, + 505, + 695 + ], + "score": 1.0, + "content": "the constraints and geometric structures, many previous models also need the inverse operation, such", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 694, + 459, + 707 + ], + "spans": [ + { + "bbox": [ + 105, + 694, + 459, + 707 + ], + "score": 1.0, + "content": "as Lagrangian neural network (LNN) [13], neural symplectic form [9], and CHNN [20].", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 43 + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 119, + 711, + 421, + 722 + ], + "lines": [ + { + "bbox": [ + 119, + 709, + 421, + 725 + ], + "spans": [ + { + "bbox": [ + 119, + 709, + 421, + 725 + ], + "score": 1.0, + "content": "1The algorithm proposed in [2] might work, but it is outside the scope of this paper.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 741, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 740, + 310, + 752 + ], + "spans": [ + { + "bbox": [ + 301, + 740, + 310, + 752 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 12, + "width": 9 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "index", + "bbox": [ + 86, + 72, + 505, + 117 + ], + "lines": [], + "index": 1.5, + "bbox_fs": [ + 85, + 73, + 505, + 118 + ], + "lines_deleted": true + }, + { + "type": "index", + "bbox": [ + 93, + 122, + 507, + 155 + ], + "lines": [], + "index": 5, + "bbox_fs": [ + 92, + 122, + 506, + 156 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 92, + 169, + 404, + 183 + ], + "lines": [ + { + "bbox": [ + 87, + 168, + 405, + 187 + ], + "spans": [ + { + "bbox": [ + 87, + 168, + 405, + 187 + ], + "score": 1.0, + "content": "131 3 First Integral-Preserving Neural Differential Equation", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "index", + "bbox": [ + 86, + 193, + 505, + 271 + ], + "lines": [ + { + "bbox": [ + 86, + 192, + 506, + 207 + ], + "spans": [ + { + "bbox": [ + 86, + 195, + 100, + 205 + ], + "score": 1.0, + "content": "132", + "type": "text" + }, + { + "bbox": [ + 105, + 192, + 506, + 207 + ], + "score": 1.0, + "content": "The main purpose is to find and preserve first integrals from data by neural networks. We suppose", + "type": "text" + } + ], + "index": 8, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 204, + 506, + 217 + ], + "spans": [ + { + "bbox": [ + 86, + 206, + 100, + 216 + ], + "score": 1.0, + "content": "133", + "type": "text" + }, + { + "bbox": [ + 105, + 204, + 238, + 217 + ], + "score": 1.0, + "content": "that a target system has at least", + "type": "text" + }, + { + "bbox": [ + 238, + 205, + 249, + 214 + ], + "score": 0.81, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 204, + 506, + 217 + ], + "score": 1.0, + "content": "unknown functionally independent first integrals. 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Because of these difficulties,", + "type": "text" + } + ], + "index": 20, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 339, + 507, + 355 + ], + "spans": [ + { + "bbox": [ + 86, + 343, + 100, + 352 + ], + "score": 1.0, + "content": "145", + "type": "text" + }, + { + "bbox": [ + 104, + 339, + 507, + 355 + ], + "score": 1.0, + "content": "no methods for preserving first integrals have been applied to neural networks. By leveraging a", + "type": "text" + } + ], + "index": 21, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 352, + 435, + 365 + ], + "spans": [ + { + "bbox": [ + 86, + 354, + 100, + 363 + ], + "score": 1.0, + "content": "146", + "type": "text" + }, + { + "bbox": [ + 105, + 352, + 435, + 365 + ], + "score": 1.0, + "content": "projection method and a discrete gradient method, we propose FINDE as follows.", + "type": "text" + } + ], + "index": 22, + "is_list_start_line": true + } + ], + "index": 11, + "bbox_fs": [ + 86, + 192, + 507, + 272 + ] + }, + { + "type": "index", + "bbox": [ + 86, + 275, + 506, + 363 + ], + "lines": [], + "index": 18.5, + "bbox_fs": [ + 86, + 275, + 507, + 365 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 99, + 374, + 370, + 387 + ], + "lines": [ + { + "bbox": [ + 105, + 373, + 372, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 372, + 389 + ], + "score": 1.0, + "content": "3.1 Continuous FINDE: Time-Derivative Projection Method", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 98, + 395, + 505, + 428 + ], + "lines": [ + { + "bbox": [ + 106, + 395, + 505, + 407 + ], + "spans": [ + { + "bbox": [ + 106, + 395, + 505, + 407 + ], + "score": 1.0, + "content": "First, we propose a time-derivative projection method called continuous FINDE (cFINDE) for neural", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 406, + 505, + 418 + ], + "spans": [ + { + "bbox": [ + 106, + 406, + 371, + 418 + ], + "score": 1.0, + "content": "networks, which projects the time-derivative onto the tangent space", + "type": "text" + }, + { + "bbox": [ + 372, + 406, + 399, + 417 + ], + "score": 0.92, + "content": "\\mathcal { T } _ { u } \\mathcal { M } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 406, + 505, + 418 + ], + "score": 1.0, + "content": ". While it still suffers from", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 417, + 360, + 429 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 360, + 429 + ], + "score": 1.0, + "content": "numerical errors, it is sufficient to find first integrals from data.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25, + "bbox_fs": [ + 106, + 395, + 505, + 429 + ] + }, + { + "type": "text", + "bbox": [ + 97, + 434, + 504, + 459 + ], + "lines": [ + { + "bbox": [ + 104, + 433, + 503, + 447 + ], + "spans": [ + { + "bbox": [ + 104, + 434, + 441, + 447 + ], + "score": 1.0, + "content": "We suppose that a neural network called a base model defines the time-derivative", + "type": "text" + }, + { + "bbox": [ + 441, + 433, + 503, + 447 + ], + "score": 0.92, + "content": "\\hat { f } : \\mathbb { R } ^ { N } \\to \\mathbb { R } ^ { N }", + "type": "inline_equation" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 443, + 384, + 461 + ], + "spans": [ + { + "bbox": 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See Appendix", + "type": "text" + }, + { + "bbox": [ + 304, + 291, + 312, + 300 + ], + "score": 0.31, + "content": "\\mathbf { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 290, + 501, + 302 + ], + "score": 1.0, + "content": "and the references [6, 23, 38] for more details.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 92, + 306, + 505, + 330 + ], + "lines": [ + { + "bbox": [ + 88, + 306, + 506, + 319 + ], + "spans": [ + { + "bbox": [ + 88, + 306, + 401, + 319 + ], + "score": 1.0, + "content": "Following [11, 15], we introduce a discrete analogue to the tangent space 184", + "type": "text" + }, + { + "bbox": [ + 402, + 306, + 429, + 317 + ], + "score": 0.92, + "content": "\\mathcal { T } _ { \\mathbf { \\ b { u } } } \\mathcal { M } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 306, + 506, + 319 + ], + "score": 1.0, + "content": "called the discrete", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 87, + 316, + 456, + 330 + ], + "spans": [ + { + "bbox": [ + 87, + 316, + 163, + 330 + ], + "score": 1.0, + "content": "tangent space 185", + "type": "text" + }, + { + "bbox": [ + 163, + 317, + 204, + 330 + ], + "score": 0.94, + "content": "\\mathcal { T } _ { ( \\pmb { v } , \\pmb { u } ) } \\mathcal { M } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 316, + 301, + 330 + ], + "score": 1.0, + "content": ". 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Note that similar concepts defined in different ways are also", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 86, + 387, + 291, + 399 + ], + "spans": [ + { + "bbox": [ + 86, + 389, + 100, + 398 + ], + "score": 1.0, + "content": "189", + "type": "text" + }, + { + "bbox": [ + 105, + 387, + 291, + 399 + ], + "score": 1.0, + "content": "referred to as discrete tangent spaces [14, 16].", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 87, + 403, + 505, + 446 + ], + "lines": [ + { + "bbox": [ + 83, + 401, + 506, + 421 + ], + "spans": [ + { + "bbox": [ + 83, + 401, + 363, + 421 + ], + "score": 1.0, + "content": "Let ψˆ denote a discrete-time base model that satisfies u˜s+1−us∆ts 190", + "type": "text" + }, + { + "bbox": [ + 322, + 403, + 418, + 419 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\frac { \\tilde { \\mathbf { u } } ^ { s + 1 } - \\mathbf { u } ^ { s } } { \\Delta t ^ { s } } = \\hat { \\psi } ( \\mathbf { u } ^ { s } ; \\Delta t ^ { s } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 402, + 448, + 419 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 448, + 405, + 471, + 416 + ], + "score": 0.89, + "content": "\\tilde { { \\pmb u } } ^ { s + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 402, + 506, + 419 + ], + "score": 1.0, + "content": "denotes", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 86, + 418, + 505, + 432 + ], + "spans": [ + { + "bbox": [ + 86, + 422, + 100, + 431 + ], + "score": 1.0, + "content": "191", + "type": "text" + }, + { + "bbox": [ + 105, + 420, + 311, + 432 + ], + "score": 1.0, + "content": "the predicted state. We assume that the base model", + "type": "text" + }, + { + "bbox": [ + 311, + 418, + 320, + 432 + ], + "score": 0.84, + "content": "\\hat { \\psi }", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 420, + 505, + 432 + ], + "score": 1.0, + "content": "is composed of a continuous-time base model", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 86, + 428, + 471, + 449 + ], + "spans": [ + { + "bbox": [ + 86, + 434, + 101, + 446 + ], + "score": 1.0, + "content": "192", + "type": "text" + }, + { + "bbox": [ + 99, + 428, + 327, + 449 + ], + "score": 1.0, + "content": "ˆf and a numerical integrator. Then, the dFINDE us+1−∆ts", + "type": "text" + }, + { + "bbox": [ + 303, + 431, + 423, + 446 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\frac { { \\pmb u } ^ { s + 1 } - { \\pmb u } ^ { s } } { \\Delta t ^ { s } } = \\psi ( { \\pmb u } ^ { s + 1 } , { \\pmb u } ^ { s } ; \\Delta t ^ { s } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 430, + 471, + 449 + ], + "score": 1.0, + "content": "is given by", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24 + }, + { + "type": "interline_equation", + "bbox": [ + 177, + 449, + 432, + 464 + ], + "lines": [ + { + "bbox": [ + 177, + 449, + 432, + 464 + ], + "spans": [ + { + "bbox": [ + 177, + 449, + 432, + 464 + ], + "score": 0.88, + "content": "\\psi ( \\pmb { u } ^ { s + 1 } , \\pmb { u } ^ { s } ; \\Delta t ^ { s } ) = \\hat { \\psi } ( \\pmb { u } ^ { s } ; \\Delta t ^ { s } ) - \\overline { { M } } ( \\pmb { u } ^ { s + 1 } , \\pmb { u } ^ { s } ) ^ { \\top } \\lambda ( \\pmb { u } ^ { s + 1 } , \\pmb { u } ^ { s } ) ,", + "type": "interline_equation", + "image_path": "d1a21837891a0816a8d90a934c2bd058b77755c9a71f9d7120f2e96416ef165f.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 177, + 449, + 432, + 464 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 89, + 467, + 506, + 491 + ], + "lines": [ + { + "bbox": [ + 84, + 465, + 507, + 482 + ], + "spans": [ + { + "bbox": [ + 84, + 465, + 134, + 482 + ], + "score": 1.0, + "content": "where 193", + "type": "text" + }, + { + "bbox": [ + 134, + 467, + 265, + 480 + ], + "score": 0.8, + "content": "\\overline { { M } } ( { \\pmb u } ^ { s + 1 } , { \\pmb u } ^ { s } ) = ( \\overline { { \\nabla } } V _ { 1 } ( { \\pmb u } ^ { s + 1 } , { \\pmb u } ^ { s } )", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 465, + 285, + 482 + ], + "score": 1.0, + "content": ". . .", + "type": "text" + }, + { + "bbox": [ + 285, + 466, + 359, + 480 + ], + "score": 0.87, + "content": "\\overline { { \\nabla } } V _ { K } ( \\pmb { u } ^ { s + 1 } , \\pmb { u } ^ { s } ) ) ^ { \\top }", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 465, + 507, + 482 + ], + "score": 1.0, + "content": ". As is the case in continuous time,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 85, + 478, + 300, + 492 + ], + "spans": [ + { + "bbox": [ + 85, + 478, + 253, + 492 + ], + "score": 1.0, + "content": "194 the preservation of the first integrals", + "type": "text" + }, + { + "bbox": [ + 253, + 480, + 264, + 490 + ], + "score": 0.86, + "content": "V _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 478, + 300, + 492 + ], + "score": 1.0, + "content": "leads to", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5 + }, + { + "type": "interline_equation", + "bbox": [ + 126, + 493, + 457, + 511 + ], + "lines": [ + { + "bbox": [ + 126, + 493, + 457, + 511 + ], + "spans": [ + { + "bbox": [ + 126, + 493, + 457, + 511 + ], + "score": 0.87, + "content": "\\begin{array} { r } { \\mathbf { 0 } = \\frac { V ( u ^ { s + 1 } ) - V ( u ^ { s } ) } { \\Delta t ^ { s } } = \\overline { { M } } ( u ^ { s + 1 } , u ^ { s } ) \\frac { u ^ { s + 1 } - u ^ { s } } { \\Delta t ^ { s } } = \\overline { { M } } ( u ^ { s + 1 } , u ^ { s } ) \\psi ( u ^ { s + 1 } , u ^ { s } ; \\Delta t ^ { s } ) . } \\end{array}", + "type": "interline_equation", + "image_path": "c57172cf644c028312037ccb9029fe77af4e4c29c8dac75574ca1e6b7a921bd3.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 126, + 493, + 457, + 511 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 86, + 513, + 403, + 525 + ], + "lines": [ + { + "bbox": [ + 85, + 513, + 402, + 527 + ], + "spans": [ + { + "bbox": [ + 85, + 513, + 348, + 527 + ], + "score": 1.0, + "content": "195 Substituting Eq. (9) and eliminating the Lagrange multiplier", + "type": "text" + }, + { + "bbox": [ + 349, + 514, + 356, + 523 + ], + "score": 0.79, + "content": "\\boldsymbol { \\lambda }", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 513, + 402, + 527 + ], + "score": 1.0, + "content": ", we obtain", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "interline_equation", + "bbox": [ + 128, + 529, + 335, + 544 + ], + "lines": [ + { + "bbox": [ + 128, + 529, + 335, + 544 + ], + "spans": [ + { + "bbox": [ + 128, + 529, + 335, + 544 + ], + "score": 0.85, + "content": "\\psi ( \\pmb { u } ^ { s + 1 } , \\pmb { u } ^ { s } ; \\Delta t ^ { s } ) = ( I - \\overline { { Y } } ( \\pmb { u } ^ { s + 1 } , \\pmb { u } ^ { s } ) ) \\hat { \\psi } ( \\pmb { u } ^ { s } ; \\Delta t ^ { s } )", + "type": "interline_equation", + "image_path": "5cf3eab1461c710ba916e91303d63747b554722245d42fa5ae31f91a77890e74.jpg" + } + ] + } + ], + "index": 31, + "virtual_lines": [ + { + "bbox": [ + 128, + 529, + 335, + 544 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 97, + 547, + 507, + 575 + ], + "lines": [ + { + "bbox": [ + 103, + 544, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 103, + 544, + 425, + 566 + ], + "score": 1.0, + "content": "Remark 2 (discrete-time first integral preservation). The dFINDE us+1−us∆ts", + "type": "text" + }, + { + "bbox": [ + 103, + 547, + 215, + 577 + ], + "score": 1.0, + "content": "preserves all first integrals", + "type": "text" + }, + { + "bbox": [ + 215, + 563, + 227, + 574 + ], + "score": 0.84, + "content": "V _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 547, + 242, + 577 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 243, + 562, + 299, + 574 + ], + "score": 0.89, + "content": "k = 1 , \\ldots , K", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 547, + 383, + 577 + ], + "score": 1.0, + "content": "in discrete time, i.e.,", + "type": "text" + }, + { + "bbox": [ + 383, + 563, + 484, + 574 + ], + "score": 0.88, + "content": "V _ { k } ( { \\pmb u } ^ { s + 1 } ) - V _ { k } ( { \\pmb u } ^ { s } ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 547, + 505, + 562 + ], + "score": 0.89, + "content": "\\begin{array} { r } { \\frac { { \\pmb u } ^ { s + 1 } - { \\pmb u } ^ { s } } { \\Delta t ^ { s } } = \\psi ( { \\pmb u } ^ { s + 1 } , { \\pmb u } ^ { s } ; \\Delta t ^ { s } ) } \\end{array}", + "type": "inline_equation" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 90, + 581, + 505, + 617 + ], + "lines": [ + { + "bbox": [ + 89, + 582, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 89, + 584, + 100, + 593 + ], + "score": 1.0, + "content": "198", + "type": "text" + }, + { + "bbox": [ + 105, + 582, + 505, + 594 + ], + "score": 1.0, + "content": "Due to projection, dFINDE can be regarded as a projection method using discrete gradients. For the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 89, + 592, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 89, + 597, + 100, + 605 + ], + "score": 1.0, + "content": "199", + "type": "text" + }, + { + "bbox": [ + 105, + 594, + 153, + 607 + ], + "score": 1.0, + "content": "base model", + "type": "text" + }, + { + "bbox": [ + 154, + 592, + 162, + 606 + ], + "score": 0.86, + "content": "\\hat { \\psi }", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 594, + 293, + 607 + ], + "score": 1.0, + "content": ", the continuous-time base model", + "type": "text" + }, + { + "bbox": [ + 294, + 592, + 301, + 606 + ], + "score": 0.85, + "content": "\\hat { f }", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 594, + 505, + 607 + ], + "score": 1.0, + "content": "can be a NODE, an HNN, or other models, and the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 88, + 605, + 449, + 618 + ], + "spans": [ + { + "bbox": [ + 88, + 608, + 100, + 617 + ], + "score": 1.0, + "content": "200", + "type": "text" + }, + { + "bbox": [ + 105, + 605, + 449, + 618 + ], + "score": 1.0, + "content": "numerical integrator can be a Runge–Kutta method, the leapfrog integrator, or others.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 88, + 621, + 505, + 723 + ], + "lines": [ + { + "bbox": [ + 86, + 621, + 504, + 633 + ], + "spans": [ + { + "bbox": [ + 86, + 624, + 99, + 633 + ], + "score": 1.0, + "content": "201", + "type": "text" + }, + { + "bbox": [ + 106, + 621, + 504, + 633 + ], + "score": 1.0, + "content": "Because dFINDE is an implicit method, it is computationally expensive for prediction. However, the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 86, + 630, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 86, + 635, + 99, + 644 + ], + "score": 1.0, + "content": "202", + "type": "text" + }, + { + "bbox": [ + 104, + 630, + 145, + 646 + ], + "score": 1.0, + "content": "next state", + "type": "text" + }, + { + "bbox": [ + 145, + 632, + 167, + 642 + ], + "score": 0.9, + "content": "\\boldsymbol { u } ^ { s + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 630, + 506, + 646 + ], + "score": 1.0, + "content": "is given for training, and the ADDA can explicitly obtain the discrete gradient w.r.t. the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 85, + 641, + 507, + 657 + ], + "spans": [ + { + "bbox": [ + 85, + 645, + 100, + 655 + ], + "score": 1.0, + "content": "203", + "type": "text" + }, + { + "bbox": [ + 104, + 641, + 125, + 657 + ], + "score": 1.0, + "content": "pair", + "type": "text" + }, + { + "bbox": [ + 126, + 643, + 170, + 655 + ], + "score": 0.92, + "content": "( { \\pmb u } ^ { s + 1 } , { \\pmb u } ^ { s } )", + "type": "inline_equation" + }, + { + "bbox": [ + 170, + 641, + 507, + 657 + ], + "score": 1.0, + "content": "as well as its computational graph. Thus, dFINDE can be computed explicitly and", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 85, + 654, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 85, + 656, + 100, + 667 + ], + "score": 1.0, + "content": "204", + "type": "text" + }, + { + "bbox": [ + 106, + 654, + 506, + 668 + ], + "score": 1.0, + "content": "optimized by standard backpropagation algorithms. Moreover, we suppose that dFINDE projects", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 85, + 665, + 506, + 681 + ], + "spans": [ + { + "bbox": [ + 85, + 669, + 100, + 679 + ], + "score": 1.0, + "content": "205", + "type": "text" + }, + { + "bbox": [ + 105, + 666, + 189, + 681 + ], + "score": 1.0, + "content": "the finite difference", + "type": "text" + }, + { + "bbox": [ + 189, + 665, + 198, + 679 + ], + "score": 0.87, + "content": "\\hat { \\psi }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 666, + 486, + 681 + ], + "score": 1.0, + "content": "only at every time step, whereas cFINDE projects the time-derivative", + "type": "text" + }, + { + "bbox": [ + 486, + 665, + 494, + 679 + ], + "score": 0.86, + "content": "\\hat { f }", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 666, + 506, + 681 + ], + "score": 1.0, + "content": "at", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 85, + 678, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 85, + 680, + 100, + 690 + ], + "score": 1.0, + "content": "206", + "type": "text" + }, + { + "bbox": [ + 105, + 678, + 505, + 691 + ], + "score": 1.0, + "content": "every substep inside a numerical integrator. Therefore, dFINDE is less computationally expensive", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 85, + 689, + 504, + 701 + ], + "spans": [ + { + "bbox": [ + 85, + 690, + 100, + 701 + ], + "score": 1.0, + "content": "207", + "type": "text" + }, + { + "bbox": [ + 106, + 689, + 504, + 701 + ], + "score": 1.0, + "content": "than cFINDE for training. 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See Appendix", + "type": "text" + }, + { + "bbox": [ + 304, + 291, + 312, + 300 + ], + "score": 0.31, + "content": "\\mathbf { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 290, + 501, + 302 + ], + "score": 1.0, + "content": "and the references [6, 23, 38] for more details.", + "type": "text" + } + ], + "index": 15, + "is_list_start_line": true + } + ], + "index": 11.5, + "bbox_fs": [ + 86, + 208, + 506, + 302 + ] + }, + { + "type": "text", + "bbox": [ + 92, + 306, + 505, + 330 + ], + "lines": [ + { + "bbox": [ + 88, + 306, + 506, + 319 + ], + "spans": [ + { + "bbox": [ + 88, + 306, + 401, + 319 + ], + "score": 1.0, + "content": "Following [11, 15], we introduce a discrete analogue to the tangent space 184", + "type": "text" + }, + { + "bbox": [ + 402, + 306, + 429, + 317 + ], + "score": 0.92, + "content": "\\mathcal { T } _ { \\mathbf { \\ b { u } } } \\mathcal { M } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 306, + 506, + 319 + ], + "score": 1.0, + "content": "called the discrete", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 87, + 316, + 456, + 330 + ], + "spans": [ + { + "bbox": [ + 87, + 316, + 163, + 330 + ], + "score": 1.0, + "content": "tangent space 185", + "type": "text" + }, + { + "bbox": [ + 163, + 317, + 204, + 330 + ], + "score": 0.94, + "content": "\\mathcal { T } _ { ( \\pmb { v } , \\pmb { u } ) } \\mathcal { M } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 316, + 301, + 330 + ], + "score": 1.0, + "content": ". 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We assume that the base model", + "type": "text" + }, + { + "bbox": [ + 311, + 418, + 320, + 432 + ], + "score": 0.84, + "content": "\\hat { \\psi }", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 420, + 505, + 432 + ], + "score": 1.0, + "content": "is composed of a continuous-time base model", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 86, + 428, + 471, + 449 + ], + "spans": [ + { + "bbox": [ + 86, + 434, + 101, + 446 + ], + "score": 1.0, + "content": "192", + "type": "text" + }, + { + "bbox": [ + 99, + 428, + 327, + 449 + ], + "score": 1.0, + "content": "ˆf and a numerical integrator. Then, the dFINDE us+1−∆ts", + "type": "text" + }, + { + "bbox": [ + 303, + 431, + 423, + 446 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\frac { { \\pmb u } ^ { s + 1 } - { \\pmb u } ^ { s } } { \\Delta t ^ { s } } = \\psi ( { \\pmb u } ^ { s + 1 } , { \\pmb u } ^ { s } ; \\Delta t ^ { s } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 430, + 471, + 449 + ], + "score": 1.0, + "content": "is given by", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24, + "bbox_fs": [ + 83, + 401, + 506, + 449 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 177, + 449, + 432, + 464 + ], + "lines": [ + { + "bbox": [ + 177, + 449, + 432, + 464 + ], + "spans": [ + { + "bbox": [ + 177, + 449, + 432, + 464 + ], + "score": 0.88, + "content": "\\psi ( \\pmb { u } ^ { s + 1 } , \\pmb { u } ^ { s } ; \\Delta t ^ { s } ) = \\hat { \\psi } ( \\pmb { u } ^ { s } ; \\Delta t ^ { s } ) - \\overline { { M } } ( \\pmb { u } ^ { s + 1 } , \\pmb { u } ^ { s } ) ^ { \\top } \\lambda ( \\pmb { u } ^ { s + 1 } , \\pmb { u } ^ { s } ) ,", + "type": "interline_equation", + "image_path": "d1a21837891a0816a8d90a934c2bd058b77755c9a71f9d7120f2e96416ef165f.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 177, + 449, + 432, + 464 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 89, + 467, + 506, + 491 + ], + "lines": [ + { + "bbox": [ + 84, + 465, + 507, + 482 + ], + "spans": [ + { + "bbox": [ + 84, + 465, + 134, + 482 + ], + "score": 1.0, + "content": "where 193", + "type": "text" + }, + { + "bbox": [ + 134, + 467, + 265, + 480 + ], + "score": 0.8, + "content": "\\overline { { M } } ( { \\pmb u } ^ { s + 1 } , { \\pmb u } ^ { s } ) = ( \\overline { { \\nabla } } V _ { 1 } ( { \\pmb u } ^ { s + 1 } , { \\pmb u } ^ { s } )", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 465, + 285, + 482 + ], + "score": 1.0, + "content": ". . .", + "type": "text" + }, + { + "bbox": [ + 285, + 466, + 359, + 480 + ], + "score": 0.87, + "content": "\\overline { { \\nabla } } V _ { K } ( \\pmb { u } ^ { s + 1 } , \\pmb { u } ^ { s } ) ) ^ { \\top }", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 465, + 507, + 482 + ], + "score": 1.0, + "content": ". As is the case in continuous time,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 85, + 478, + 300, + 492 + ], + "spans": [ + { + "bbox": [ + 85, + 478, + 253, + 492 + ], + "score": 1.0, + "content": "194 the preservation of the first integrals", + "type": "text" + }, + { + "bbox": [ + 253, + 480, + 264, + 490 + ], + "score": 0.86, + "content": "V _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 478, + 300, + 492 + ], + "score": 1.0, + "content": "leads to", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5, + "bbox_fs": [ + 84, + 465, + 507, + 492 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 126, + 493, + 457, + 511 + ], + "lines": [ + { + "bbox": [ + 126, + 493, + 457, + 511 + ], + "spans": [ + { + "bbox": [ + 126, + 493, + 457, + 511 + ], + "score": 0.87, + "content": "\\begin{array} { r } { \\mathbf { 0 } = \\frac { V ( u ^ { s + 1 } ) - V ( u ^ { s } ) } { \\Delta t ^ { s } } = \\overline { { M } } ( u ^ { s + 1 } , u ^ { s } ) \\frac { u ^ { s + 1 } - u ^ { s } } { \\Delta t ^ { s } } = \\overline { { M } } ( u ^ { s + 1 } , u ^ { s } ) \\psi ( u ^ { s + 1 } , u ^ { s } ; \\Delta t ^ { s } ) . } \\end{array}", + "type": "interline_equation", + "image_path": "c57172cf644c028312037ccb9029fe77af4e4c29c8dac75574ca1e6b7a921bd3.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 126, + 493, + 457, + 511 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 86, + 513, + 403, + 525 + ], + "lines": [ + { + "bbox": [ + 85, + 513, + 402, + 527 + ], + "spans": [ + { + "bbox": [ + 85, + 513, + 348, + 527 + ], + "score": 1.0, + "content": "195 Substituting Eq. (9) and eliminating the Lagrange multiplier", + "type": "text" + }, + { + "bbox": [ + 349, + 514, + 356, + 523 + ], + "score": 0.79, + "content": "\\boldsymbol { \\lambda }", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 513, + 402, + 527 + ], + "score": 1.0, + "content": ", we obtain", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30, + "bbox_fs": [ + 85, + 513, + 402, + 527 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 128, + 529, + 335, + 544 + ], + "lines": [ + { + "bbox": [ + 128, + 529, + 335, + 544 + ], + "spans": [ + { + "bbox": [ + 128, + 529, + 335, + 544 + ], + "score": 0.85, + "content": "\\psi ( \\pmb { u } ^ { s + 1 } , \\pmb { u } ^ { s } ; \\Delta t ^ { s } ) = ( I - \\overline { { Y } } ( \\pmb { u } ^ { s + 1 } , \\pmb { u } ^ { s } ) ) \\hat { \\psi } ( \\pmb { u } ^ { s } ; \\Delta t ^ { s } )", + "type": "interline_equation", + "image_path": "5cf3eab1461c710ba916e91303d63747b554722245d42fa5ae31f91a77890e74.jpg" + } + ] + } + ], + "index": 31, + "virtual_lines": [ + { + "bbox": [ + 128, + 529, + 335, + 544 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 97, + 547, + 507, + 575 + ], + "lines": [ + { + "bbox": [ + 103, + 544, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 103, + 544, + 425, + 566 + ], + "score": 1.0, + "content": "Remark 2 (discrete-time first integral preservation). The dFINDE us+1−us∆ts", + "type": "text" + }, + { + "bbox": [ + 103, + 547, + 215, + 577 + ], + "score": 1.0, + "content": "preserves all first integrals", + "type": "text" + }, + { + "bbox": [ + 215, + 563, + 227, + 574 + ], + "score": 0.84, + "content": "V _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 547, + 242, + 577 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 243, + 562, + 299, + 574 + ], + "score": 0.89, + "content": "k = 1 , \\ldots , K", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 547, + 383, + 577 + ], + "score": 1.0, + "content": "in discrete time, i.e.,", + "type": "text" + }, + { + "bbox": [ + 383, + 563, + 484, + 574 + ], + "score": 0.88, + "content": "V _ { k } ( { \\pmb u } ^ { s + 1 } ) - V _ { k } ( { \\pmb u } ^ { s } ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 547, + 505, + 562 + ], + "score": 0.89, + "content": "\\begin{array} { r } { \\frac { { \\pmb u } ^ { s + 1 } - { \\pmb u } ^ { s } } { \\Delta t ^ { s } } = \\psi ( { \\pmb u } ^ { s + 1 } , { \\pmb u } ^ { s } ; \\Delta t ^ { s } ) } \\end{array}", + "type": "inline_equation" + } + ], + "index": 32 + } + ], + "index": 32, + "bbox_fs": [ + 103, + 544, + 505, + 577 + ] + }, + { + "type": "index", + "bbox": [ + 90, + 581, + 505, + 617 + ], + "lines": [ + { + "bbox": [ + 89, + 582, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 89, + 584, + 100, + 593 + ], + "score": 1.0, + "content": "198", + "type": "text" + }, + { + "bbox": [ + 105, + 582, + 505, + 594 + ], + "score": 1.0, + "content": "Due to projection, dFINDE can be regarded as a projection method using discrete gradients. For the", + "type": "text" + } + ], + "index": 33, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 592, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 89, + 597, + 100, + 605 + ], + "score": 1.0, + "content": "199", + "type": "text" + }, + { + "bbox": [ + 105, + 594, + 153, + 607 + ], + "score": 1.0, + "content": "base model", + "type": "text" + }, + { + "bbox": [ + 154, + 592, + 162, + 606 + ], + "score": 0.86, + "content": "\\hat { \\psi }", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 594, + 293, + 607 + ], + "score": 1.0, + "content": ", the continuous-time base model", + "type": "text" + }, + { + "bbox": [ + 294, + 592, + 301, + 606 + ], + "score": 0.85, + "content": "\\hat { f }", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 594, + 505, + 607 + ], + "score": 1.0, + "content": "can be a NODE, an HNN, or other models, and the", + "type": "text" + } + ], + "index": 34, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 605, + 449, + 618 + ], + "spans": [ + { + "bbox": [ + 88, + 608, + 100, + 617 + ], + "score": 1.0, + "content": "200", + "type": "text" + }, + { + "bbox": [ + 105, + 605, + 449, + 618 + ], + "score": 1.0, + "content": "numerical integrator can be a Runge–Kutta method, the leapfrog integrator, or others.", + "type": "text" + } + ], + "index": 35, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 621, + 504, + 633 + ], + "spans": [ + { + "bbox": [ + 86, + 624, + 99, + 633 + ], + "score": 1.0, + "content": "201", + "type": "text" + }, + { + "bbox": [ + 106, + 621, + 504, + 633 + ], + "score": 1.0, + "content": "Because dFINDE is an implicit method, it is computationally expensive for prediction. However, the", + "type": "text" + } + ], + "index": 36, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 630, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 86, + 635, + 99, + 644 + ], + "score": 1.0, + "content": "202", + "type": "text" + }, + { + "bbox": [ + 104, + 630, + 145, + 646 + ], + "score": 1.0, + "content": "next state", + "type": "text" + }, + { + "bbox": [ + 145, + 632, + 167, + 642 + ], + "score": 0.9, + "content": "\\boldsymbol { u } ^ { s + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 630, + 506, + 646 + ], + "score": 1.0, + "content": "is given for training, and the ADDA can explicitly obtain the discrete gradient w.r.t. the", + "type": "text" + } + ], + "index": 37, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 641, + 507, + 657 + ], + "spans": [ + { + "bbox": [ + 85, + 645, + 100, + 655 + ], + "score": 1.0, + "content": "203", + "type": "text" + }, + { + "bbox": [ + 104, + 641, + 125, + 657 + ], + "score": 1.0, + "content": "pair", + "type": "text" + }, + { + "bbox": [ + 126, + 643, + 170, + 655 + ], + "score": 0.92, + "content": "( { \\pmb u } ^ { s + 1 } , { \\pmb u } ^ { s } )", + "type": "inline_equation" + }, + { + "bbox": [ + 170, + 641, + 507, + 657 + ], + "score": 1.0, + "content": "as well as its computational graph. Thus, dFINDE can be computed explicitly and", + "type": "text" + } + ], + "index": 38, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 654, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 85, + 656, + 100, + 667 + ], + "score": 1.0, + "content": "204", + "type": "text" + }, + { + "bbox": [ + 106, + 654, + 506, + 668 + ], + "score": 1.0, + "content": "optimized by standard backpropagation algorithms. Moreover, we suppose that dFINDE projects", + "type": "text" + } + ], + "index": 39, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 665, + 506, + 681 + ], + "spans": [ + { + "bbox": [ + 85, + 669, + 100, + 679 + ], + "score": 1.0, + "content": "205", + "type": "text" + }, + { + "bbox": [ + 105, + 666, + 189, + 681 + ], + "score": 1.0, + "content": "the finite difference", + "type": "text" + }, + { + "bbox": [ + 189, + 665, + 198, + 679 + ], + "score": 0.87, + "content": "\\hat { \\psi }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 666, + 486, + 681 + ], + "score": 1.0, + "content": "only at every time step, whereas cFINDE projects the time-derivative", + "type": "text" + }, + { + "bbox": [ + 486, + 665, + 494, + 679 + ], + "score": 0.86, + "content": "\\hat { f }", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 666, + 506, + 681 + ], + "score": 1.0, + "content": "at", + "type": "text" + } + ], + "index": 40, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 678, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 85, + 680, + 100, + 690 + ], + "score": 1.0, + "content": "206", + "type": "text" + }, + { + "bbox": [ + 105, + 678, + 505, + 691 + ], + "score": 1.0, + "content": "every substep inside a numerical integrator. Therefore, dFINDE is less computationally expensive", + "type": "text" + } + ], + "index": 41, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 689, + 504, + 701 + ], + "spans": [ + { + "bbox": [ + 85, + 690, + 100, + 701 + ], + "score": 1.0, + "content": "207", + "type": "text" + }, + { + "bbox": [ + 106, + 689, + 504, + 701 + ], + "score": 1.0, + "content": "than cFINDE for training. In contrast, a typical projection method requires much computational", + "type": "text" + } + ], + "index": 42, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 700, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 85, + 702, + 100, + 712 + ], + "score": 1.0, + "content": "208", + "type": "text" + }, + { + "bbox": [ + 105, + 700, + 506, + 713 + ], + "score": 1.0, + "content": "cost to solve an optimization problem for training, and standard backpropagation algorithms are not", + "type": "text" + } + ], + "index": 43, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 710, + 171, + 723 + ], + "spans": [ + { + "bbox": [ + 85, + 712, + 100, + 723 + ], + "score": 1.0, + "content": "209", + "type": "text" + }, + { + "bbox": [ + 104, + 710, + 171, + 723 + ], + "score": 1.0, + "content": "applicable to it.", + "type": "text" + } + ], + "index": 44, + "is_list_start_line": true + }, + { + "bbox": [ + 90, + 186, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 90, + 186, + 506, + 200 + ], + "score": 1.0, + "content": "10 Remark 3 (trainability). The dFINDE can be trained using the standard backpropagation algorithm,", + "type": "text", + "cross_page": true + } + ], + "index": 4, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 199, + 387, + 211 + ], + "spans": [ + { + "bbox": [ + 88, + 199, + 387, + 211 + ], + "score": 1.0, + "content": "211 whereas a straightforward application of a projection method cannot.", + "type": "text", + "cross_page": true + } + ], + "index": 5, + "is_list_start_line": true + } + ], + "index": 34, + "bbox_fs": [ + 88, + 582, + 505, + 618 + ] + }, + { + "type": "index", + "bbox": [ + 88, + 621, + 505, + 723 + ], + "lines": [], + "index": 40, + "bbox_fs": [ + 85, + 621, + 507, + 723 + ], + "lines_deleted": true + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 106, + 89, + 505, + 167 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 208, + 78, + 403, + 88 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 206, + 76, + 403, + 91 + ], + "spans": [ + { + "bbox": [ + 206, + 76, + 403, + 91 + ], + "score": 1.0, + "content": "Table 2: Datasets, Dynamics, and First Integrals.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 106, + 89, + 505, + 167 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 89, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 106, + 89, + 505, + 167 + ], + "score": 0.98, + "html": "
DatasetDynamicsNFirst Integrals
EnergyMomentumMassConstraint
Two-body problemCanonical Hamiltonian81
Discretized KdV equationNon-canonical Hamiltonian50广1
Double pendulumPoisson8√√
FitzHugh-Nagumo modelDirac4√
", + "type": "table", + "image_path": "9532b26093d205096da34a1d05959c144853a2f40d48bde0cd43c48af1534f6f.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 106, + 89, + 505, + 115.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 106, + 115.0, + 505, + 141.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 106, + 141.0, + 505, + 167.0 + ], + "spans": [], + "index": 3 + } + ] + } + ], + "index": 1.0 + }, + { + "type": "text", + "bbox": [ + 92, + 187, + 505, + 210 + ], + "lines": [ + { + "bbox": [ + 90, + 186, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 90, + 186, + 506, + 200 + ], + "score": 1.0, + "content": "10 Remark 3 (trainability). The dFINDE can be trained using the standard backpropagation algorithm,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 88, + 199, + 387, + 211 + ], + "spans": [ + { + "bbox": [ + 88, + 199, + 387, + 211 + ], + "score": 1.0, + "content": "211 whereas a straightforward application of a projection method cannot.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5 + }, + { + "type": "title", + "bbox": [ + 105, + 230, + 191, + 243 + ], + "lines": [ + { + "bbox": [ + 104, + 227, + 193, + 247 + ], + "spans": [ + { + "bbox": [ + 104, + 227, + 193, + 247 + ], + "score": 1.0, + "content": "4 Experiments", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "title", + "bbox": [ + 102, + 255, + 224, + 267 + ], + "lines": [ + { + "bbox": [ + 105, + 253, + 226, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 226, + 270 + ], + "score": 1.0, + "content": "4.1 Experimental Settings", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 105, + 275, + 505, + 439 + ], + "lines": [ + { + "bbox": [ + 107, + 276, + 506, + 287 + ], + "spans": [ + { + "bbox": [ + 107, + 276, + 506, + 287 + ], + "score": 1.0, + "content": "Target Systems We evaluated FINDE and base models using datasets associated with first integrals,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 286, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 505, + 299 + ], + "score": 1.0, + "content": "summarized in Table 2. A gravitational two-body problem (2-body) on a 2-dimensional configuration", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 297, + 506, + 310 + ], + "spans": [ + { + "bbox": [ + 104, + 297, + 506, + 310 + ], + "score": 1.0, + "content": "space is a typical Hamiltonian system in the canonical form. In addition to the total energy, it has first", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 308, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 106, + 308, + 505, + 321 + ], + "score": 1.0, + "content": "integrals related to symmetries in space, namely, the linear and angular momenta. The Korteweg–De", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 318, + 506, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 506, + 332 + ], + "score": 1.0, + "content": "Vries (KdV) equation is a PDE model of shallow water waves. This is a Hamiltonian system in a", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 330, + 506, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 506, + 343 + ], + "score": 1.0, + "content": "non-canonical form and has the Hamiltonian, total mass, and many other quantities as first integrals.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 342, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 505, + 353 + ], + "score": 1.0, + "content": "A double pendulum (2-pend) is a Hamiltonian system in polar coordinates. However, we transformed", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 352, + 505, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 505, + 364 + ], + "score": 1.0, + "content": "it to Cartesian coordinates; it was no longer a Hamiltonian system but a Poisson system. The lengths", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 362, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 505, + 375 + ], + "score": 1.0, + "content": "of two rods work as holonomic constraints and lead to four first integrals. The FitzHugh–Nagumo", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 373, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 505, + 387 + ], + "score": 1.0, + "content": "model is a biological neuron model as an electric circuit, which exhibits a rapid and transient change", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 384, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 106, + 384, + 505, + 397 + ], + "score": 1.0, + "content": "of voltage called a spike. 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DatasetDynamicsNFirst Integrals
EnergyMomentumMassConstraint
Two-body problemCanonical Hamiltonian81
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Double pendulumPoisson8√√
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Due to energy dissipation in the resistor, the model is not a Poisson system, but", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 428, + 385, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 385, + 440 + ], + "score": 1.0, + "content": "one can find a Dirac structure [53]. See Appendix C for more details.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 15, + "bbox_fs": [ + 104, + 276, + 506, + 440 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 452, + 505, + 518 + ], + "lines": [ + { + "bbox": [ + 105, + 450, + 506, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 506, + 466 + ], + "score": 1.0, + "content": "Implementation We implemented the proposed FINDE and evaluated it under the following", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 462, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 505, + 475 + ], + "score": 1.0, + "content": "settings. We implemented all codes by modifying the officially released codes of HNN [26] 2 and", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 473, + 507, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 136, + 487 + ], + "score": 1.0, + "content": "DGNet", + "type": "text" + }, + { + "bbox": [ + 137, + 473, + 158, + 485 + ], + "score": 0.27, + "content": "[ 3 8 ] ^ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 473, + 507, + 487 + ], + "score": 1.0, + "content": ". We used Python v3.8.12 with packages scipy v1.7.3, pytorch v1.10.2, torchdiffeq v0.1.1,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 484, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 506, + 498 + ], + "score": 1.0, + "content": "functorch v1.10 preview, and gplearn v0.4.2. We used the Dormand–Prince method (dopri5) [18]", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 496, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 505, + 509 + ], + "score": 1.0, + "content": "as the numerical integrator, unless otherwise stated. All experiments were performed on a single", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 506, + 347, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 347, + 519 + ], + "score": 1.0, + "content": "NVIDIA A100 provided by (ANONYMOUS PROVIDER).", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 450, + 507, + 519 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 523, + 505, + 627 + ], + "lines": [ + { + "bbox": [ + 106, + 523, + 504, + 536 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 402, + 536 + ], + "score": 1.0, + "content": "Following HNN [26] and DGNet [38], we represented the first integrals", + "type": "text" + }, + { + "bbox": [ + 403, + 523, + 413, + 533 + ], + "score": 0.51, + "content": "V", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 523, + 493, + 536 + ], + "score": 1.0, + "content": ", NODE, and HNN", + "type": "text" + }, + { + "bbox": [ + 493, + 523, + 504, + 533 + ], + "score": 0.62, + "content": "H", + "type": "inline_equation" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 534, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 505, + 546 + ], + "score": 1.0, + "content": "using fully-connected neural networks with two hidden layers. Each hidden layer had 200 units", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 545, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 505, + 557 + ], + "score": 1.0, + "content": "and preceded a hyperbolic tangent activation function. Each weight matrix was initialized as an", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 556, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 106, + 556, + 284, + 568 + ], + "score": 1.0, + "content": "orthogonal matrix. The input was the state", + "type": "text" + }, + { + "bbox": [ + 284, + 558, + 292, + 566 + ], + "score": 0.62, + "content": "\\textbf { \\em u }", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 556, + 479, + 568 + ], + "score": 1.0, + "content": ", and the output represented the first integrals", + "type": "text" + }, + { + "bbox": [ + 479, + 556, + 489, + 566 + ], + "score": 0.72, + "content": "V", + "type": "inline_equation" + }, + { + "bbox": [ + 490, + 556, + 505, + 568 + ], + "score": 1.0, + "content": "for", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 567, + 506, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 201, + 581 + ], + "score": 1.0, + "content": "FINDE, time-derivative", + "type": "text" + }, + { + "bbox": [ + 201, + 567, + 209, + 580 + ], + "score": 0.87, + "content": "\\hat { f }", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 568, + 336, + 581 + ], + "score": 1.0, + "content": "for NODE, and the Hamiltonian", + "type": "text" + }, + { + "bbox": [ + 337, + 569, + 347, + 578 + ], + "score": 0.75, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 568, + 506, + 581 + ], + "score": 1.0, + "content": "for HNN. For the KdV dataset, we used", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 579, + 507, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 507, + 592 + ], + "score": 1.0, + "content": "a 1-dimensional convolutional neural network (CNN), each of whose layers had a kernel size of 3.", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 589, + 507, + 606 + ], + "spans": [ + { + "bbox": [ + 104, + 589, + 428, + 605 + ], + "score": 1.0, + "content": "The double pendulum is a second–order system, implying that the time-derivative", + "type": "text" + }, + { + "bbox": [ + 424, + 589, + 507, + 606 + ], + "score": 1.0, + "content": "ddt q of the position", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 601, + 508, + 622 + ], + "spans": [ + { + "bbox": [ + 106, + 606, + 114, + 616 + ], + "score": 0.61, + "content": "\\pmb q", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 601, + 211, + 622 + ], + "score": 1.0, + "content": "is known as the velocity", + "type": "text" + }, + { + "bbox": [ + 212, + 606, + 218, + 614 + ], + "score": 0.7, + "content": "\\textbf { { v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 601, + 380, + 622 + ], + "score": 1.0, + "content": ". Hence, we treated only the acceleration", + "type": "text" + }, + { + "bbox": [ + 380, + 603, + 397, + 618 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\frac { \\mathrm { d } } { \\mathrm { d } t } \\pmb { v } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 601, + 508, + 622 + ], + "score": 1.0, + "content": "as the output to learn. This", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 616, + 486, + 627 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 486, + 627 + ], + "score": 1.0, + "content": "assumption slightly improved the absolute performances but did not change the relative trends.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 33, + "bbox_fs": [ + 104, + 523, + 508, + 627 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 632, + 505, + 691 + ], + "lines": [ + { + "bbox": [ + 105, + 631, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 158, + 644 + ], + "score": 1.0, + "content": "We used the", + "type": "text" + }, + { + "bbox": [ + 158, + 633, + 164, + 642 + ], + "score": 0.47, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 631, + 505, + 644 + ], + "score": 1.0, + "content": "-step error as the loss function to be minimized. In particular, it is the mean squared", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 642, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 104, + 642, + 280, + 658 + ], + "score": 1.0, + "content": "error (MSE) between the ground truth state", + "type": "text" + }, + { + "bbox": [ + 280, + 643, + 298, + 655 + ], + "score": 0.91, + "content": "\\pmb { u } _ { \\mathrm { G T } } ^ { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 642, + 351, + 658 + ], + "score": 1.0, + "content": "and the state", + "type": "text" + }, + { + "bbox": [ + 351, + 643, + 372, + 657 + ], + "score": 0.91, + "content": "\\pmb { u } _ { \\mathrm { p r e d . } } ^ { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 642, + 506, + 658 + ], + "score": 1.0, + "content": ". predicted from the previous step", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 651, + 509, + 674 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 129, + 669 + ], + "score": 0.91, + "content": "u _ { \\mathrm { G T } } ^ { s - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 651, + 509, + 674 + ], + "score": 1.0, + "content": ". The base model and FINDE were jointly trained using the Adam optimizer [33] with the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 104, + 668, + 504, + 681 + ], + "spans": [ + { + "bbox": [ + 104, + 668, + 152, + 681 + ], + "score": 1.0, + "content": "parameters", + "type": "text" + }, + { + "bbox": [ + 153, + 668, + 246, + 681 + ], + "score": 0.92, + "content": "( \\beta _ { 1 } , \\beta _ { 2 } ) = ( 0 . 9 , 0 . 9 9 9 )", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 668, + 482, + 681 + ], + "score": 1.0, + "content": "and a batch size of 200. The learning rate was initialized to", + "type": "text" + }, + { + "bbox": [ + 482, + 668, + 504, + 679 + ], + "score": 0.88, + "content": "1 0 ^ { - 3 }", + "type": "inline_equation" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 680, + 307, + 692 + ], + "spans": [ + { + "bbox": [ + 106, + 680, + 307, + 692 + ], + "score": 1.0, + "content": "and decayed to zero with a cosine annealing [36].", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 40, + "bbox_fs": [ + 104, + 631, + 509, + 692 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 85, + 72, + 505, + 172 + ], + "lines": [ + { + "bbox": [ + 85, + 73, + 506, + 84 + ], + "spans": [ + { + "bbox": [ + 85, + 74, + 100, + 84 + ], + "score": 1.0, + "content": "249", + "type": "text" + }, + { + "bbox": [ + 105, + 73, + 506, + 84 + ], + "score": 1.0, + "content": "Evaluation Metric As an evaluation metric, we used the 1-step error, which is identical to the loss", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 85, + 83, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 85, + 85, + 100, + 95 + ], + "score": 1.0, + "content": "250", + "type": "text" + }, + { + "bbox": [ + 105, + 83, + 268, + 96 + ], + "score": 1.0, + "content": "function. We displayed it at the scale of", + "type": "text" + }, + { + "bbox": [ + 268, + 83, + 297, + 94 + ], + "score": 0.91, + "content": "\\times 1 0 ^ { - 9 }", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 83, + 506, + 96 + ], + "score": 1.0, + "content": ". The lower this indicator, the better, as emphasized", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 85, + 94, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 85, + 96, + 100, + 106 + ], + "score": 1.0, + "content": "251", + "type": "text" + }, + { + "bbox": [ + 106, + 94, + 119, + 107 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 120, + 95, + 127, + 106 + ], + "score": 0.33, + "content": "\\downarrow", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 94, + 506, + 107 + ], + "score": 1.0, + "content": ". While several studies used the MSEs of the state or system energy over the whole time", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 85, + 106, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 85, + 107, + 100, + 118 + ], + "score": 1.0, + "content": "252", + "type": "text" + }, + { + "bbox": [ + 106, + 106, + 505, + 118 + ], + "score": 1.0, + "content": "series [26, 38], we consider these indicators are misleading, as pointed in several studies [4]. For", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 86, + 116, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 86, + 118, + 99, + 127 + ], + "score": 1.0, + "content": "253", + "type": "text" + }, + { + "bbox": [ + 105, + 116, + 506, + 129 + ], + "score": 1.0, + "content": "example, in the case of a periodic orbit, an orbit that is correctly learned except for a slight difference", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 85, + 127, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 85, + 129, + 100, + 139 + ], + "score": 1.0, + "content": "254", + "type": "text" + }, + { + "bbox": [ + 105, + 127, + 506, + 140 + ], + "score": 1.0, + "content": "in angular velocity will have the same MSE as an orbit that never moves from its initial position.", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 86, + 137, + 506, + 151 + ], + "spans": [ + { + "bbox": [ + 86, + 140, + 100, + 149 + ], + "score": 1.0, + "content": "255", + "type": "text" + }, + { + "bbox": [ + 105, + 137, + 272, + 151 + ], + "score": 1.0, + "content": "Instead, we used the valid prediction time", + "type": "text" + }, + { + "bbox": [ + 272, + 138, + 297, + 149 + ], + "score": 0.3, + "content": "( V P T )", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 137, + 455, + 151 + ], + "score": 1.0, + "content": "[4, 32, 54]. VPT denotes the time point", + "type": "text" + }, + { + "bbox": [ + 455, + 140, + 461, + 148 + ], + "score": 0.73, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 137, + 506, + 151 + ], + "score": 1.0, + "content": "divided by", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 85, + 147, + 504, + 165 + ], + "spans": [ + { + "bbox": [ + 85, + 151, + 100, + 161 + ], + "score": 1.0, + "content": "256", + "type": "text" + }, + { + "bbox": [ + 104, + 147, + 148, + 165 + ], + "score": 1.0, + "content": "the length", + "type": "text" + }, + { + "bbox": [ + 148, + 150, + 156, + 159 + ], + "score": 0.81, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 147, + 370, + 165 + ], + "score": 1.0, + "content": "of time series at which the MSE of the predicted state", + "type": "text" + }, + { + "bbox": [ + 371, + 150, + 392, + 162 + ], + "score": 0.9, + "content": "\\pmb { u } _ { \\mathrm { p r e d . } } ^ { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 147, + 497, + 165 + ], + "score": 1.0, + "content": ". exceeds a given threshold", + "type": "text" + }, + { + "bbox": [ + 498, + 150, + 504, + 159 + ], + "score": 0.81, + "content": "\\theta", + "type": "inline_equation" + } + ], + "index": 7 + }, + { + "bbox": [ + 85, + 160, + 313, + 173 + ], + "spans": [ + { + "bbox": [ + 85, + 163, + 100, + 173 + ], + "score": 1.0, + "content": "257", + "type": "text" + }, + { + "bbox": [ + 105, + 160, + 313, + 173 + ], + "score": 1.0, + "content": "for the first time in an initial value problem, that is,", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 4 + }, + { + "type": "interline_equation", + "bbox": [ + 148, + 176, + 462, + 191 + ], + "lines": [ + { + "bbox": [ + 148, + 176, + 462, + 191 + ], + "spans": [ + { + "bbox": [ + 148, + 176, + 462, + 191 + ], + "score": 0.88, + "content": "\\begin{array} { r } { V P T ( u _ { \\mathrm { p r e d . } } ; u _ { \\mathrm { G T } } ) = \\frac { 1 } { S } \\arg \\operatorname* { m a x } _ { s _ { f } } \\{ s _ { f } | \\mathrm { M S E } ( u _ { \\mathrm { p r e d . } } ^ { s } , u _ { \\mathrm { G T } } ^ { s } ) < \\theta \\mathrm { f o r } \\mathrm { a l l } s \\leq s _ { f } \\} . } \\end{array}", + "type": "interline_equation", + "image_path": "811dd5f21e7701a7467a228c4cae451c6d1973664d4fe839e7bf95ebdae7362c.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 148, + 176, + 462, + 191 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 86, + 196, + 505, + 251 + ], + "lines": [ + { + "bbox": [ + 86, + 196, + 505, + 207 + ], + "spans": [ + { + "bbox": [ + 86, + 198, + 100, + 207 + ], + "score": 1.0, + "content": "258", + "type": "text" + }, + { + "bbox": [ + 106, + 196, + 505, + 207 + ], + "score": 1.0, + "content": "To obtain VPTs, we normalized each element of state to have zero mean and unit variance in the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 86, + 206, + 506, + 219 + ], + "spans": [ + { + "bbox": [ + 86, + 209, + 99, + 218 + ], + "score": 1.0, + "content": "259", + "type": "text" + }, + { + "bbox": [ + 105, + 206, + 189, + 219 + ], + "score": 1.0, + "content": "training data and set", + "type": "text" + }, + { + "bbox": [ + 190, + 207, + 196, + 217 + ], + "score": 0.79, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 206, + 447, + 219 + ], + "score": 1.0, + "content": "to 0.01. The higher this indicator, the better, as emphasized by", + "type": "text" + }, + { + "bbox": [ + 448, + 207, + 454, + 218 + ], + "score": 0.76, + "content": "\\uparrow", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 206, + 506, + 219 + ], + "score": 1.0, + "content": ". Because of", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 86, + 217, + 505, + 230 + ], + "spans": [ + { + "bbox": [ + 86, + 219, + 100, + 228 + ], + "score": 1.0, + "content": "260", + "type": "text" + }, + { + "bbox": [ + 105, + 217, + 505, + 230 + ], + "score": 1.0, + "content": "the “spiking” behavior of the FitzHugh–Nagumo model, a small error in phase is regarded as a large", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 85, + 227, + 505, + 242 + ], + "spans": [ + { + "bbox": [ + 85, + 231, + 99, + 240 + ], + "score": 1.0, + "content": "261", + "type": "text" + }, + { + "bbox": [ + 104, + 227, + 505, + 242 + ], + "score": 1.0, + "content": "error in state. To measure the qualitative performance, we calculated VPTs by allowing for a delay", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 85, + 239, + 224, + 252 + ], + "spans": [ + { + "bbox": [ + 85, + 241, + 100, + 251 + ], + "score": 1.0, + "content": "262", + "type": "text" + }, + { + "bbox": [ + 105, + 239, + 224, + 252 + ], + "score": 1.0, + "content": "and advance of up to 5 steps.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12 + }, + { + "type": "title", + "bbox": [ + 105, + 263, + 348, + 275 + ], + "lines": [ + { + "bbox": [ + 106, + 262, + 349, + 277 + ], + "spans": [ + { + "bbox": [ + 106, + 262, + 349, + 277 + ], + "score": 1.0, + "content": "4.2 First Integral Preservation for Hamiltonian System", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 105, + 283, + 358, + 386 + ], + "lines": [ + { + "bbox": [ + 106, + 283, + 358, + 295 + ], + "spans": [ + { + "bbox": [ + 106, + 283, + 358, + 295 + ], + "score": 1.0, + "content": "Before learning first integrals from data, we first evaluated", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 293, + 358, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 358, + 307 + ], + "score": 1.0, + "content": "FINDE as a numerical integrator using a known mass-spring", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 304, + 359, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 247, + 318 + ], + "score": 1.0, + "content": "system. The system has the state", + "type": "text" + }, + { + "bbox": [ + 247, + 305, + 297, + 317 + ], + "score": 0.93, + "content": "\\mathbf { \\boldsymbol { \\mathscr { u } } } \\doteq ( q v ) ^ { \\top }", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 304, + 359, + 318 + ], + "score": 1.0, + "content": ", the dynamics", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 316, + 357, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 171, + 332 + ], + "score": 1.0, + "content": "ddt q = v and d", + "type": "text" + }, + { + "bbox": [ + 166, + 317, + 211, + 331 + ], + "score": 0.93, + "content": "\\begin{array} { r } { { \\frac { \\mathrm { d } } { \\mathrm { d } t } } v \\ = - q } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 317, + 313, + 331 + ], + "score": 1.0, + "content": ", and the system energy", + "type": "text" + }, + { + "bbox": [ + 313, + 317, + 357, + 330 + ], + "score": 0.87, + "content": "E ( q , v ) =", + "type": "inline_equation" + } + ], + "index": 19 + }, + { + "bbox": [ + 107, + 329, + 360, + 344 + ], + "spans": [ + { + "bbox": [ + 107, + 330, + 151, + 343 + ], + "score": 0.88, + "content": "{ \\scriptstyle { \\frac { 1 } { 2 } } } ( q ^ { 2 } + v ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 329, + 244, + 344 + ], + "score": 1.0, + "content": ". Using the initial value", + "type": "text" + }, + { + "bbox": [ + 244, + 329, + 288, + 343 + ], + "score": 0.9, + "content": "( 1 . 0 \\ 0 . 0 ) ^ { \\top }", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 329, + 360, + 344 + ], + "score": 1.0, + "content": "and the time step", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 342, + 358, + 354 + ], + "spans": [ + { + "bbox": [ + 106, + 342, + 125, + 354 + ], + "score": 1.0, + "content": "size", + "type": "text" + }, + { + "bbox": [ + 125, + 342, + 164, + 352 + ], + "score": 0.86, + "content": "\\Delta t = 0 . 2", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 342, + 358, + 354 + ], + "score": 1.0, + "content": ", we solved the initial value problem of the true", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 351, + 358, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 358, + 365 + ], + "score": 1.0, + "content": "ODE using the leapfrog integrator. We applied FINDE with the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 363, + 358, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 182, + 375 + ], + "score": 1.0, + "content": "true system energy", + "type": "text" + }, + { + "bbox": [ + 183, + 364, + 192, + 373 + ], + "score": 0.82, + "content": "E", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 363, + 267, + 375 + ], + "score": 1.0, + "content": "as the first integral", + "type": "text" + }, + { + "bbox": [ + 267, + 364, + 276, + 373 + ], + "score": 0.7, + "content": "V", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 363, + 358, + 375 + ], + "score": 1.0, + "content": ". Note that no neural", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 375, + 256, + 386 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 256, + 386 + ], + "score": 1.0, + "content": "networks nor training were involved.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 20 + }, + { + "type": "image", + "bbox": [ + 365, + 250, + 499, + 364 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 365, + 250, + 499, + 364 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 365, + 250, + 499, + 364 + ], + "spans": [ + { + "bbox": [ + 365, + 250, + 499, + 364 + ], + "score": 0.892, + "type": "image", + "image_path": "e62087da4d5eea6ebb9b01535f06401e481e011b62a41af256ef1f7e2091ccfd.jpg" + } + ] + } + ], + "index": 23.5, + "virtual_lines": [ + { + "bbox": [ + 365, + 250, + 499, + 307.0 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 365, + 307.0, + 499, + 364.0 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 367, + 366, + 504, + 388 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 365, + 365, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 365, + 365, + 505, + 376 + ], + "score": 1.0, + "content": "Figure 1: Integration of a known", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 365, + 376, + 449, + 388 + ], + "spans": [ + { + "bbox": [ + 365, + 376, + 449, + 388 + ], + "score": 1.0, + "content": "mass-spring system.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5 + } + ], + "index": 25.5 + }, + { + "type": "text", + "bbox": [ + 90, + 390, + 505, + 479 + ], + "lines": [ + { + "bbox": [ + 88, + 391, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 88, + 393, + 100, + 402 + ], + "score": 1.0, + "content": "273", + "type": "text" + }, + { + "bbox": [ + 106, + 391, + 505, + 403 + ], + "score": 1.0, + "content": "The results with the analytical solution are shown in Fig. 1. The upper panel shows that the time series", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 88, + 401, + 506, + 414 + ], + "spans": [ + { + "bbox": [ + 88, + 403, + 100, + 413 + ], + "score": 1.0, + "content": "274", + "type": "text" + }, + { + "bbox": [ + 106, + 401, + 506, + 414 + ], + "score": 1.0, + "content": "predicted by comparison methods overlap each other and are apparently almost identical. However,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 88, + 412, + 506, + 426 + ], + "spans": [ + { + "bbox": [ + 88, + 415, + 100, + 424 + ], + "score": 1.0, + "content": "275", + "type": "text" + }, + { + "bbox": [ + 106, + 412, + 506, + 426 + ], + "score": 1.0, + "content": "the lower panel shows that the energy obtained from the states predicted by the leapfrog integrator is", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 88, + 423, + 506, + 437 + ], + "spans": [ + { + "bbox": [ + 88, + 425, + 100, + 434 + ], + "score": 1.0, + "content": "276", + "type": "text" + }, + { + "bbox": [ + 105, + 423, + 506, + 437 + ], + "score": 1.0, + "content": "fluctuating. The same is true for the case with the cFINDE. This is because the symplectic integrator", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 87, + 434, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 87, + 435, + 100, + 446 + ], + "score": 1.0, + "content": "277", + "type": "text" + }, + { + "bbox": [ + 106, + 434, + 505, + 446 + ], + "score": 1.0, + "content": "and the cFINDE suffer from numerical errors caused by temporal discretization. In contrast, the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 88, + 445, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 88, + 448, + 100, + 457 + ], + "score": 1.0, + "content": "278", + "type": "text" + }, + { + "bbox": [ + 105, + 445, + 505, + 459 + ], + "score": 1.0, + "content": "dFINDE preserves the energy accurately. This is because, at every step, the dFINDE projects the state", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 87, + 455, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 87, + 457, + 100, + 469 + ], + "score": 1.0, + "content": "279", + "type": "text" + }, + { + "bbox": [ + 106, + 456, + 133, + 468 + ], + "score": 0.9, + "content": "( q v ) ^ { \\top }", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 455, + 263, + 470 + ], + "score": 1.0, + "content": "onto the discrete tangent space", + "type": "text" + }, + { + "bbox": [ + 264, + 456, + 304, + 469 + ], + "score": 0.93, + "content": "\\mathcal { T } _ { ( \\pmb { v } , \\pmb { u } ) } \\mathcal { M } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 455, + 505, + 470 + ], + "score": 1.0, + "content": ". Although a smaller step size reduces numerical", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 88, + 467, + 338, + 479 + ], + "spans": [ + { + "bbox": [ + 88, + 469, + 100, + 478 + ], + "score": 1.0, + "content": "280", + "type": "text" + }, + { + "bbox": [ + 106, + 467, + 338, + 479 + ], + "score": 1.0, + "content": "errors, this result demonstrates the advantage of dFINDE.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 32.5 + }, + { + "type": "title", + "bbox": [ + 93, + 491, + 379, + 503 + ], + "lines": [ + { + "bbox": [ + 88, + 490, + 381, + 506 + ], + "spans": [ + { + "bbox": [ + 88, + 490, + 381, + 506 + ], + "score": 1.0, + "content": "281 4.3 Learning First Integrals from Data of Hamiltonian System", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 95, + 513, + 505, + 623 + ], + "lines": [ + { + "bbox": [ + 92, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 92, + 516, + 99, + 524 + ], + "score": 1.0, + "content": "82", + "type": "text" + }, + { + "bbox": [ + 104, + 513, + 478, + 524 + ], + "score": 1.0, + "content": "We evaluated FINDE on learning from the 2-body dataset. We used HNN as the base model", + "type": "text" + }, + { + "bbox": [ + 478, + 511, + 485, + 524 + ], + "score": 0.85, + "content": "\\hat { f }", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 513, + 505, + 524 + ], + "score": 1.0, + "content": ". We", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 93, + 524, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 93, + 528, + 99, + 534 + ], + "score": 1.0, + "content": "83", + "type": "text" + }, + { + "bbox": [ + 104, + 524, + 441, + 535 + ], + "score": 1.0, + "content": "found that the FINDE got better performances if it did not treat the Hamiltonian", + "type": "text" + }, + { + "bbox": [ + 442, + 524, + 452, + 534 + ], + "score": 0.79, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 524, + 505, + 535 + ], + "score": 1.0, + "content": "of the HNN", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 93, + 534, + 506, + 547 + ], + "spans": [ + { + "bbox": [ + 93, + 538, + 99, + 545 + ], + "score": 1.0, + "content": "84", + "type": "text" + }, + { + "bbox": [ + 104, + 534, + 201, + 547 + ], + "score": 1.0, + "content": "as one of first integrals", + "type": "text" + }, + { + "bbox": [ + 201, + 535, + 213, + 546 + ], + "score": 0.88, + "content": "V _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 534, + 506, + 547 + ], + "score": 1.0, + "content": ". The medians and standard deviations of 5 trials are summarized in the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 93, + 545, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 93, + 550, + 99, + 556 + ], + "score": 1.0, + "content": "85", + "type": "text" + }, + { + "bbox": [ + 104, + 545, + 476, + 557 + ], + "score": 1.0, + "content": "leftmost column of Table 3. The cFINDE achieved better VPTs than the vanilla HNN with", + "type": "text" + }, + { + "bbox": [ + 476, + 546, + 505, + 556 + ], + "score": 0.89, + "content": "K = 1", + "type": "inline_equation" + } + ], + "index": 41 + }, + { + "bbox": [ + 92, + 556, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 92, + 560, + 99, + 567 + ], + "score": 1.0, + "content": "86", + "type": "text" + }, + { + "bbox": [ + 104, + 556, + 323, + 569 + ], + "score": 1.0, + "content": "to 2, and the performance was suddenly degraded for", + "type": "text" + }, + { + "bbox": [ + 323, + 557, + 351, + 567 + ], + "score": 0.91, + "content": "K = 3", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 556, + 506, + 569 + ], + "score": 1.0, + "content": ". The dFINDE showed a similar trend", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 94, + 568, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 94, + 572, + 98, + 576 + ], + "score": 1.0, + "content": "7", + "type": "text" + }, + { + "bbox": [ + 105, + 568, + 505, + 579 + ], + "score": 1.0, + "content": "with slightly better performances. The HNN with FINDE found two first integrals in addition to the", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 93, + 578, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 93, + 582, + 99, + 589 + ], + "score": 1.0, + "content": "88", + "type": "text" + }, + { + "bbox": [ + 105, + 578, + 159, + 591 + ], + "score": 1.0, + "content": "Hamiltonian", + "type": "text" + }, + { + "bbox": [ + 160, + 579, + 170, + 589 + ], + "score": 0.73, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 578, + 505, + 591 + ], + "score": 1.0, + "content": "of the HNN. Even though a two-body problem is a Hamiltonian system that HNN", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 93, + 589, + 506, + 602 + ], + "spans": [ + { + "bbox": [ + 93, + 593, + 99, + 599 + ], + "score": 1.0, + "content": "89", + "type": "text" + }, + { + "bbox": [ + 104, + 589, + 464, + 602 + ], + "score": 1.0, + "content": "can learn, the prior knowledge that there exist first integrals other than the Hamiltonian", + "type": "text" + }, + { + "bbox": [ + 464, + 590, + 475, + 599 + ], + "score": 0.76, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 589, + 506, + 602 + ], + "score": 1.0, + "content": "can be", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 93, + 600, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 93, + 604, + 99, + 610 + ], + "score": 1.0, + "content": "90", + "type": "text" + }, + { + "bbox": [ + 105, + 600, + 506, + 613 + ], + "score": 1.0, + "content": "a clue to better learning. The HNN with FINDE got worse 1-step errors, suggesting that without", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 104, + 610, + 480, + 624 + ], + "spans": [ + { + "bbox": [ + 104, + 610, + 480, + 624 + ], + "score": 1.0, + "content": "FINDE, HNN overfitted short-term change and had difficulty predicting long-term dynamics.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 42.5 + }, + { + "type": "text", + "bbox": [ + 100, + 627, + 505, + 661 + ], + "lines": [ + { + "bbox": [ + 106, + 627, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 321, + 640 + ], + "score": 1.0, + "content": "We performed a symbolic regression of first integrals", + "type": "text" + }, + { + "bbox": [ + 321, + 628, + 331, + 638 + ], + "score": 0.75, + "content": "V", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 627, + 474, + 640 + ], + "score": 1.0, + "content": "learned by the neural network. For", + "type": "text" + }, + { + "bbox": [ + 474, + 628, + 502, + 638 + ], + "score": 0.88, + "content": "K = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 627, + 506, + 640 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 638, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 210, + 651 + ], + "score": 1.0, + "content": "the learned first integrals", + "type": "text" + }, + { + "bbox": [ + 211, + 639, + 221, + 649 + ], + "score": 0.73, + "content": "V", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 638, + 401, + 651 + ], + "score": 1.0, + "content": "were identical to the linear momenta in the", + "type": "text" + }, + { + "bbox": [ + 401, + 640, + 408, + 648 + ], + "score": 0.56, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 638, + 430, + 651 + ], + "score": 1.0, + "content": "- and", + "type": "text" + }, + { + "bbox": [ + 430, + 640, + 437, + 650 + ], + "score": 0.8, + "content": "y", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 638, + 506, + 651 + ], + "score": 1.0, + "content": "-directions up to", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 649, + 396, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 396, + 662 + ], + "score": 1.0, + "content": "affine transformation in most cases. See Appendix D.1 for more details.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 49 + }, + { + "type": "text", + "bbox": [ + 97, + 666, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 99, + 664, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 99, + 664, + 436, + 679 + ], + "score": 1.0, + "content": "We depict example results in Fig. 2. In the absence of FINDE, the mass positions", + "type": "text" + }, + { + "bbox": [ + 436, + 666, + 505, + 677 + ], + "score": 0.91, + "content": "( x _ { 1 } , y _ { 1 } ) , ( x _ { 2 } , y _ { 2 } )", + "type": "inline_equation" + } + ], + "index": 51 + }, + { + "bbox": [ + 101, + 676, + 504, + 691 + ], + "spans": [ + { + "bbox": [ + 101, + 676, + 394, + 691 + ], + "score": 1.0, + "content": "became inaccurate in a short time and the center-of-gravity position", + "type": "text" + }, + { + "bbox": [ + 394, + 678, + 504, + 691 + ], + "score": 0.88, + "content": "\\textstyle ( x _ { c } , y _ { c } ) = ( { \\frac { x _ { 1 } + x _ { 2 } } { 2 } } , { \\frac { y _ { 1 } + y _ { 2 } } { 2 } } )", + "type": "inline_equation" + } + ], + "index": 52 + }, + { + "bbox": [ + 104, + 687, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 104, + 687, + 506, + 702 + ], + "score": 1.0, + "content": "deviated rapidly. The HNN with cFINDE accurately predicted the state for a longer period. Even after", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 104, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 104, + 700, + 505, + 712 + ], + "score": 1.0, + "content": "errors in the mass positions became non-negligible, errors in the center-of-gravity position were still", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 103, + 710, + 507, + 723 + ], + "spans": [ + { + "bbox": [ + 103, + 710, + 424, + 723 + ], + "score": 1.0, + "content": "small. We show the absolute errors averaged over all trials in Fig. 3. In each of", + "type": "text" + }, + { + "bbox": [ + 425, + 713, + 432, + 721 + ], + "score": 0.45, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 710, + 453, + 723 + ], + "score": 1.0, + "content": "- and", + "type": "text" + }, + { + "bbox": [ + 453, + 712, + 460, + 722 + ], + "score": 0.81, + "content": "y", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 710, + 507, + 723 + ], + "score": 1.0, + "content": "-directions,", + "type": "text" + } + ], + "index": 55 + } + ], + "index": 53 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "spans": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "index", + "bbox": [ + 85, + 72, + 505, + 172 + ], + "lines": [ + { + "bbox": [ + 85, + 73, + 506, + 84 + ], + "spans": [ + { + "bbox": [ + 85, + 74, + 100, + 84 + ], + "score": 1.0, + "content": "249", + "type": "text" + }, + { + "bbox": [ + 105, + 73, + 506, + 84 + ], + "score": 1.0, + "content": "Evaluation Metric As an evaluation metric, we used the 1-step error, which is identical to the loss", + "type": "text" + } + ], + "index": 0, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 83, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 85, + 85, + 100, + 95 + ], + "score": 1.0, + "content": "250", + "type": "text" + }, + { + "bbox": [ + 105, + 83, + 268, + 96 + ], + "score": 1.0, + "content": "function. We displayed it at the scale of", + "type": "text" + }, + { + "bbox": [ + 268, + 83, + 297, + 94 + ], + "score": 0.91, + "content": "\\times 1 0 ^ { - 9 }", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 83, + 506, + 96 + ], + "score": 1.0, + "content": ". The lower this indicator, the better, as emphasized", + "type": "text" + } + ], + "index": 1, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 94, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 85, + 96, + 100, + 106 + ], + "score": 1.0, + "content": "251", + "type": "text" + }, + { + "bbox": [ + 106, + 94, + 119, + 107 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 120, + 95, + 127, + 106 + ], + "score": 0.33, + "content": "\\downarrow", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 94, + 506, + 107 + ], + "score": 1.0, + "content": ". While several studies used the MSEs of the state or system energy over the whole time", + "type": "text" + } + ], + "index": 2, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 106, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 85, + 107, + 100, + 118 + ], + "score": 1.0, + "content": "252", + "type": "text" + }, + { + "bbox": [ + 106, + 106, + 505, + 118 + ], + "score": 1.0, + "content": "series [26, 38], we consider these indicators are misleading, as pointed in several studies [4]. For", + "type": "text" + } + ], + "index": 3, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 116, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 86, + 118, + 99, + 127 + ], + "score": 1.0, + "content": "253", + "type": "text" + }, + { + "bbox": [ + 105, + 116, + 506, + 129 + ], + "score": 1.0, + "content": "example, in the case of a periodic orbit, an orbit that is correctly learned except for a slight difference", + "type": "text" + } + ], + "index": 4, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 127, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 85, + 129, + 100, + 139 + ], + "score": 1.0, + "content": "254", + "type": "text" + }, + { + "bbox": [ + 105, + 127, + 506, + 140 + ], + "score": 1.0, + "content": "in angular velocity will have the same MSE as an orbit that never moves from its initial position.", + "type": "text" + } + ], + "index": 5, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 137, + 506, + 151 + ], + "spans": [ + { + "bbox": [ + 86, + 140, + 100, + 149 + ], + "score": 1.0, + "content": "255", + "type": "text" + }, + { + "bbox": [ + 105, + 137, + 272, + 151 + ], + "score": 1.0, + "content": "Instead, we used the valid prediction time", + "type": "text" + }, + { + "bbox": [ + 272, + 138, + 297, + 149 + ], + "score": 0.3, + "content": "( V P T )", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 137, + 455, + 151 + ], + "score": 1.0, + "content": "[4, 32, 54]. VPT denotes the time point", + "type": "text" + }, + { + "bbox": [ + 455, + 140, + 461, + 148 + ], + "score": 0.73, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 137, + 506, + 151 + ], + "score": 1.0, + "content": "divided by", + "type": "text" + } + ], + "index": 6, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 147, + 504, + 165 + ], + "spans": [ + { + "bbox": [ + 85, + 151, + 100, + 161 + ], + "score": 1.0, + "content": "256", + "type": "text" + }, + { + "bbox": [ + 104, + 147, + 148, + 165 + ], + "score": 1.0, + "content": "the length", + "type": "text" + }, + { + "bbox": [ + 148, + 150, + 156, + 159 + ], + "score": 0.81, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 147, + 370, + 165 + ], + "score": 1.0, + "content": "of time series at which the MSE of the predicted state", + "type": "text" + }, + { + "bbox": [ + 371, + 150, + 392, + 162 + ], + "score": 0.9, + "content": "\\pmb { u } _ { \\mathrm { p r e d . } } ^ { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 147, + 497, + 165 + ], + "score": 1.0, + "content": ". exceeds a given threshold", + "type": "text" + }, + { + "bbox": [ + 498, + 150, + 504, + 159 + ], + "score": 0.81, + "content": "\\theta", + "type": "inline_equation" + } + ], + "index": 7, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 160, + 313, + 173 + ], + "spans": [ + { + "bbox": [ + 85, + 163, + 100, + 173 + ], + "score": 1.0, + "content": "257", + "type": "text" + }, + { + "bbox": [ + 105, + 160, + 313, + 173 + ], + "score": 1.0, + "content": "for the first time in an initial value problem, that is,", + "type": "text" + } + ], + "index": 8, + "is_list_start_line": true + } + ], + "index": 4, + "bbox_fs": [ + 85, + 73, + 506, + 173 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 148, + 176, + 462, + 191 + ], + "lines": [ + { + "bbox": [ + 148, + 176, + 462, + 191 + ], + "spans": [ + { + "bbox": [ + 148, + 176, + 462, + 191 + ], + "score": 0.88, + "content": "\\begin{array} { r } { V P T ( u _ { \\mathrm { p r e d . } } ; u _ { \\mathrm { G T } } ) = \\frac { 1 } { S } \\arg \\operatorname* { m a x } _ { s _ { f } } \\{ s _ { f } | \\mathrm { M S E } ( u _ { \\mathrm { p r e d . } } ^ { s } , u _ { \\mathrm { G T } } ^ { s } ) < \\theta \\mathrm { f o r } \\mathrm { a l l } s \\leq s _ { f } \\} . } \\end{array}", + "type": "interline_equation", + "image_path": "811dd5f21e7701a7467a228c4cae451c6d1973664d4fe839e7bf95ebdae7362c.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 148, + 176, + 462, + 191 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "index", + "bbox": [ + 86, + 196, + 505, + 251 + ], + "lines": [ + { + "bbox": [ + 86, + 196, + 505, + 207 + ], + "spans": [ + { + "bbox": [ + 86, + 198, + 100, + 207 + ], + "score": 1.0, + "content": "258", + "type": "text" + }, + { + "bbox": [ + 106, + 196, + 505, + 207 + ], + "score": 1.0, + "content": "To obtain VPTs, we normalized each element of state to have zero mean and unit variance in the", + "type": "text" + } + ], + "index": 10, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 206, + 506, + 219 + ], + "spans": [ + { + "bbox": [ + 86, + 209, + 99, + 218 + ], + "score": 1.0, + "content": "259", + "type": "text" + }, + { + "bbox": [ + 105, + 206, + 189, + 219 + ], + "score": 1.0, + "content": "training data and set", + "type": "text" + }, + { + "bbox": [ + 190, + 207, + 196, + 217 + ], + "score": 0.79, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 206, + 447, + 219 + ], + "score": 1.0, + "content": "to 0.01. The higher this indicator, the better, as emphasized by", + "type": "text" + }, + { + "bbox": [ + 448, + 207, + 454, + 218 + ], + "score": 0.76, + "content": "\\uparrow", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 206, + 506, + 219 + ], + "score": 1.0, + "content": ". Because of", + "type": "text" + } + ], + "index": 11, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 217, + 505, + 230 + ], + "spans": [ + { + "bbox": [ + 86, + 219, + 100, + 228 + ], + "score": 1.0, + "content": "260", + "type": "text" + }, + { + "bbox": [ + 105, + 217, + 505, + 230 + ], + "score": 1.0, + "content": "the “spiking” behavior of the FitzHugh–Nagumo model, a small error in phase is regarded as a large", + "type": "text" + } + ], + "index": 12, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 227, + 505, + 242 + ], + "spans": [ + { + "bbox": [ + 85, + 231, + 99, + 240 + ], + "score": 1.0, + "content": "261", + "type": "text" + }, + { + "bbox": [ + 104, + 227, + 505, + 242 + ], + "score": 1.0, + "content": "error in state. 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The system has the state", + "type": "text" + }, + { + "bbox": [ + 247, + 305, + 297, + 317 + ], + "score": 0.93, + "content": "\\mathbf { \\boldsymbol { \\mathscr { u } } } \\doteq ( q v ) ^ { \\top }", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 304, + 359, + 318 + ], + "score": 1.0, + "content": ", the dynamics", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 316, + 357, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 171, + 332 + ], + "score": 1.0, + "content": "ddt q = v and d", + "type": "text" + }, + { + "bbox": [ + 166, + 317, + 211, + 331 + ], + "score": 0.93, + "content": "\\begin{array} { r } { { \\frac { \\mathrm { d } } { \\mathrm { d } t } } v \\ = - q } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 317, + 313, + 331 + ], + "score": 1.0, + "content": ", and the system energy", + "type": "text" + }, + { + "bbox": [ + 313, + 317, + 357, + 330 + ], + "score": 0.87, + "content": "E ( q , v ) =", + "type": "inline_equation" + } + ], + "index": 19 + }, + { + "bbox": [ + 107, + 329, + 360, + 344 + ], + "spans": [ + { + "bbox": [ + 107, + 330, + 151, + 343 + ], + "score": 0.88, + "content": "{ \\scriptstyle { \\frac { 1 } { 2 } } } ( q ^ { 2 } + v ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 329, + 244, + 344 + ], + "score": 1.0, + "content": ". Using the initial value", + "type": "text" + }, + { + "bbox": [ + 244, + 329, + 288, + 343 + ], + "score": 0.9, + "content": "( 1 . 0 \\ 0 . 0 ) ^ { \\top }", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 329, + 360, + 344 + ], + "score": 1.0, + "content": "and the time step", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 342, + 358, + 354 + ], + "spans": [ + { + "bbox": [ + 106, + 342, + 125, + 354 + ], + "score": 1.0, + "content": "size", + "type": "text" + }, + { + "bbox": [ + 125, + 342, + 164, + 352 + ], + "score": 0.86, + "content": "\\Delta t = 0 . 2", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 342, + 358, + 354 + ], + "score": 1.0, + "content": ", we solved the initial value problem of the true", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 351, + 358, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 358, + 365 + ], + "score": 1.0, + "content": "ODE using the leapfrog integrator. We applied FINDE with the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 363, + 358, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 182, + 375 + ], + "score": 1.0, + "content": "true system energy", + "type": "text" + }, + { + "bbox": [ + 183, + 364, + 192, + 373 + ], + "score": 0.82, + "content": "E", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 363, + 267, + 375 + ], + "score": 1.0, + "content": "as the first integral", + "type": "text" + }, + { + "bbox": [ + 267, + 364, + 276, + 373 + ], + "score": 0.7, + "content": "V", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 363, + 358, + 375 + ], + "score": 1.0, + "content": ". Note that no neural", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 375, + 256, + 386 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 256, + 386 + ], + "score": 1.0, + "content": "networks nor training were involved.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 283, + 360, + 386 + ] + }, + { + "type": "image", + "bbox": [ + 365, + 250, + 499, + 364 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 365, + 250, + 499, + 364 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 365, + 250, + 499, + 364 + ], + "spans": [ + { + "bbox": [ + 365, + 250, + 499, + 364 + ], + "score": 0.892, + "type": "image", + "image_path": "e62087da4d5eea6ebb9b01535f06401e481e011b62a41af256ef1f7e2091ccfd.jpg" + } + ] + } + ], + "index": 23.5, + "virtual_lines": [ + { + "bbox": [ + 365, + 250, + 499, + 307.0 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 365, + 307.0, + 499, + 364.0 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 367, + 366, + 504, + 388 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 365, + 365, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 365, + 365, + 505, + 376 + ], + "score": 1.0, + "content": "Figure 1: Integration of a known", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 365, + 376, + 449, + 388 + ], + "spans": [ + { + "bbox": [ + 365, + 376, + 449, + 388 + ], + "score": 1.0, + "content": "mass-spring system.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5 + } + ], + "index": 25.5 + }, + { + "type": "index", + "bbox": [ + 90, + 390, + 505, + 479 + ], + "lines": [ + { + "bbox": [ + 88, + 391, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 88, + 393, + 100, + 402 + ], + "score": 1.0, + "content": "273", + "type": "text" + }, + { + "bbox": [ + 106, + 391, + 505, + 403 + ], + "score": 1.0, + "content": "The results with the analytical solution are shown in Fig. 1. The upper panel shows that the time series", + "type": "text" + } + ], + "index": 29, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 401, + 506, + 414 + ], + "spans": [ + { + "bbox": [ + 88, + 403, + 100, + 413 + ], + "score": 1.0, + "content": "274", + "type": "text" + }, + { + "bbox": [ + 106, + 401, + 506, + 414 + ], + "score": 1.0, + "content": "predicted by comparison methods overlap each other and are apparently almost identical. However,", + "type": "text" + } + ], + "index": 30, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 412, + 506, + 426 + ], + "spans": [ + { + "bbox": [ + 88, + 415, + 100, + 424 + ], + "score": 1.0, + "content": "275", + "type": "text" + }, + { + "bbox": [ + 106, + 412, + 506, + 426 + ], + "score": 1.0, + "content": "the lower panel shows that the energy obtained from the states predicted by the leapfrog integrator is", + "type": "text" + } + ], + "index": 31, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 423, + 506, + 437 + ], + "spans": [ + { + "bbox": [ + 88, + 425, + 100, + 434 + ], + "score": 1.0, + "content": "276", + "type": "text" + }, + { + "bbox": [ + 105, + 423, + 506, + 437 + ], + "score": 1.0, + "content": "fluctuating. The same is true for the case with the cFINDE. This is because the symplectic integrator", + "type": "text" + } + ], + "index": 32, + "is_list_start_line": true + }, + { + "bbox": [ + 87, + 434, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 87, + 435, + 100, + 446 + ], + "score": 1.0, + "content": "277", + "type": "text" + }, + { + "bbox": [ + 106, + 434, + 505, + 446 + ], + "score": 1.0, + "content": "and the cFINDE suffer from numerical errors caused by temporal discretization. In contrast, the", + "type": "text" + } + ], + "index": 33, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 445, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 88, + 448, + 100, + 457 + ], + "score": 1.0, + "content": "278", + "type": "text" + }, + { + "bbox": [ + 105, + 445, + 505, + 459 + ], + "score": 1.0, + "content": "dFINDE preserves the energy accurately. This is because, at every step, the dFINDE projects the state", + "type": "text" + } + ], + "index": 34, + "is_list_start_line": true + }, + { + "bbox": [ + 87, + 455, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 87, + 457, + 100, + 469 + ], + "score": 1.0, + "content": "279", + "type": "text" + }, + { + "bbox": [ + 106, + 456, + 133, + 468 + ], + "score": 0.9, + "content": "( q v ) ^ { \\top }", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 455, + 263, + 470 + ], + "score": 1.0, + "content": "onto the discrete tangent space", + "type": "text" + }, + { + "bbox": [ + 264, + 456, + 304, + 469 + ], + "score": 0.93, + "content": "\\mathcal { T } _ { ( \\pmb { v } , \\pmb { u } ) } \\mathcal { M } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 455, + 505, + 470 + ], + "score": 1.0, + "content": ". Although a smaller step size reduces numerical", + "type": "text" + } + ], + "index": 35, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 467, + 338, + 479 + ], + "spans": [ + { + "bbox": [ + 88, + 469, + 100, + 478 + ], + "score": 1.0, + "content": "280", + "type": "text" + }, + { + "bbox": [ + 106, + 467, + 338, + 479 + ], + "score": 1.0, + "content": "errors, this result demonstrates the advantage of dFINDE.", + "type": "text" + } + ], + "index": 36, + "is_list_start_line": true + } + ], + "index": 32.5, + "bbox_fs": [ + 87, + 391, + 506, + 479 + ] + }, + { + "type": "title", + "bbox": [ + 93, + 491, + 379, + 503 + ], + "lines": [ + { + "bbox": [ + 88, + 490, + 381, + 506 + ], + "spans": [ + { + "bbox": [ + 88, + 490, + 381, + 506 + ], + "score": 1.0, + "content": "281 4.3 Learning First Integrals from Data of Hamiltonian System", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "index", + "bbox": [ + 95, + 513, + 505, + 623 + ], + "lines": [ + { + "bbox": [ + 92, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 92, + 516, + 99, + 524 + ], + "score": 1.0, + "content": "82", + "type": "text" + }, + { + "bbox": [ + 104, + 513, + 478, + 524 + ], + "score": 1.0, + "content": "We evaluated FINDE on learning from the 2-body dataset. We used HNN as the base model", + "type": "text" + }, + { + "bbox": [ + 478, + 511, + 485, + 524 + ], + "score": 0.85, + "content": "\\hat { f }", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 513, + 505, + 524 + ], + "score": 1.0, + "content": ". We", + "type": "text" + } + ], + "index": 38, + "is_list_start_line": true + }, + { + "bbox": [ + 93, + 524, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 93, + 528, + 99, + 534 + ], + "score": 1.0, + "content": "83", + "type": "text" + }, + { + "bbox": [ + 104, + 524, + 441, + 535 + ], + "score": 1.0, + "content": "found that the FINDE got better performances if it did not treat the Hamiltonian", + "type": "text" + }, + { + "bbox": [ + 442, + 524, + 452, + 534 + ], + "score": 0.79, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 524, + 505, + 535 + ], + "score": 1.0, + "content": "of the HNN", + "type": "text" + } + ], + "index": 39, + "is_list_start_line": true + }, + { + "bbox": [ + 93, + 534, + 506, + 547 + ], + "spans": [ + { + "bbox": [ + 93, + 538, + 99, + 545 + ], + "score": 1.0, + "content": "84", + "type": "text" + }, + { + "bbox": [ + 104, + 534, + 201, + 547 + ], + "score": 1.0, + "content": "as one of first integrals", + "type": "text" + }, + { + "bbox": [ + 201, + 535, + 213, + 546 + ], + "score": 0.88, + "content": "V _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 534, + 506, + 547 + ], + "score": 1.0, + "content": ". The medians and standard deviations of 5 trials are summarized in the", + "type": "text" + } + ], + "index": 40, + "is_list_start_line": true + }, + { + "bbox": [ + 93, + 545, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 93, + 550, + 99, + 556 + ], + "score": 1.0, + "content": "85", + "type": "text" + }, + { + "bbox": [ + 104, + 545, + 476, + 557 + ], + "score": 1.0, + "content": "leftmost column of Table 3. The cFINDE achieved better VPTs than the vanilla HNN with", + "type": "text" + }, + { + "bbox": [ + 476, + 546, + 505, + 556 + ], + "score": 0.89, + "content": "K = 1", + "type": "inline_equation" + } + ], + "index": 41, + "is_list_start_line": true + }, + { + "bbox": [ + 92, + 556, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 92, + 560, + 99, + 567 + ], + "score": 1.0, + "content": "86", + "type": "text" + }, + { + "bbox": [ + 104, + 556, + 323, + 569 + ], + "score": 1.0, + "content": "to 2, and the performance was suddenly degraded for", + "type": "text" + }, + { + "bbox": [ + 323, + 557, + 351, + 567 + ], + "score": 0.91, + "content": "K = 3", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 556, + 506, + 569 + ], + "score": 1.0, + "content": ". The dFINDE showed a similar trend", + "type": "text" + } + ], + "index": 42, + "is_list_start_line": true + }, + { + "bbox": [ + 94, + 568, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 94, + 572, + 98, + 576 + ], + "score": 1.0, + "content": "7", + "type": "text" + }, + { + "bbox": [ + 105, + 568, + 505, + 579 + ], + "score": 1.0, + "content": "with slightly better performances. The HNN with FINDE found two first integrals in addition to the", + "type": "text" + } + ], + "index": 43, + "is_list_start_line": true + }, + { + "bbox": [ + 93, + 578, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 93, + 582, + 99, + 589 + ], + "score": 1.0, + "content": "88", + "type": "text" + }, + { + "bbox": [ + 105, + 578, + 159, + 591 + ], + "score": 1.0, + "content": "Hamiltonian", + "type": "text" + }, + { + "bbox": [ + 160, + 579, + 170, + 589 + ], + "score": 0.73, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 578, + 505, + 591 + ], + "score": 1.0, + "content": "of the HNN. Even though a two-body problem is a Hamiltonian system that HNN", + "type": "text" + } + ], + "index": 44, + "is_list_start_line": true + }, + { + "bbox": [ + 93, + 589, + 506, + 602 + ], + "spans": [ + { + "bbox": [ + 93, + 593, + 99, + 599 + ], + "score": 1.0, + "content": "89", + "type": "text" + }, + { + "bbox": [ + 104, + 589, + 464, + 602 + ], + "score": 1.0, + "content": "can learn, the prior knowledge that there exist first integrals other than the Hamiltonian", + "type": "text" + }, + { + "bbox": [ + 464, + 590, + 475, + 599 + ], + "score": 0.76, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 589, + 506, + 602 + ], + "score": 1.0, + "content": "can be", + "type": "text" + } + ], + "index": 45, + "is_list_start_line": true + }, + { + "bbox": [ + 93, + 600, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 93, + 604, + 99, + 610 + ], + "score": 1.0, + "content": "90", + "type": "text" + }, + { + "bbox": [ + 105, + 600, + 506, + 613 + ], + "score": 1.0, + "content": "a clue to better learning. The HNN with FINDE got worse 1-step errors, suggesting that without", + "type": "text" + } + ], + "index": 46, + "is_list_start_line": true + }, + { + "bbox": [ + 104, + 610, + 480, + 624 + ], + "spans": [ + { + "bbox": [ + 104, + 610, + 480, + 624 + ], + "score": 1.0, + "content": "FINDE, HNN overfitted short-term change and had difficulty predicting long-term dynamics.", + "type": "text" + } + ], + "index": 47, + "is_list_start_line": true + } + ], + "index": 42.5, + "bbox_fs": [ + 92, + 511, + 506, + 624 + ] + }, + { + "type": "text", + "bbox": [ + 100, + 627, + 505, + 661 + ], + "lines": [ + { + "bbox": [ + 106, + 627, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 321, + 640 + ], + "score": 1.0, + "content": "We performed a symbolic regression of first integrals", + "type": "text" + }, + { + "bbox": [ + 321, + 628, + 331, + 638 + ], + "score": 0.75, + "content": "V", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 627, + 474, + 640 + ], + "score": 1.0, + "content": "learned by the neural network. For", + "type": "text" + }, + { + "bbox": [ + 474, + 628, + 502, + 638 + ], + "score": 0.88, + "content": "K = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 627, + 506, + 640 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 638, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 210, + 651 + ], + "score": 1.0, + "content": "the learned first integrals", + "type": "text" + }, + { + "bbox": [ + 211, + 639, + 221, + 649 + ], + "score": 0.73, + "content": "V", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 638, + 401, + 651 + ], + "score": 1.0, + "content": "were identical to the linear momenta in the", + "type": "text" + }, + { + "bbox": [ + 401, + 640, + 408, + 648 + ], + "score": 0.56, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 638, + 430, + 651 + ], + "score": 1.0, + "content": "- and", + "type": "text" + }, + { + "bbox": [ + 430, + 640, + 437, + 650 + ], + "score": 0.8, + "content": "y", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 638, + 506, + 651 + ], + "score": 1.0, + "content": "-directions up to", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 649, + 396, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 396, + 662 + ], + "score": 1.0, + "content": "affine transformation in most cases. See Appendix D.1 for more details.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 49, + "bbox_fs": [ + 105, + 627, + 506, + 662 + ] + }, + { + "type": "text", + "bbox": [ + 97, + 666, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 99, + 664, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 99, + 664, + 436, + 679 + ], + "score": 1.0, + "content": "We depict example results in Fig. 2. In the absence of FINDE, the mass positions", + "type": "text" + }, + { + "bbox": [ + 436, + 666, + 505, + 677 + ], + "score": 0.91, + "content": "( x _ { 1 } , y _ { 1 } ) , ( x _ { 2 } , y _ { 2 } )", + "type": "inline_equation" + } + ], + "index": 51 + }, + { + "bbox": [ + 101, + 676, + 504, + 691 + ], + "spans": [ + { + "bbox": [ + 101, + 676, + 394, + 691 + ], + "score": 1.0, + "content": "became inaccurate in a short time and the center-of-gravity position", + "type": "text" + }, + { + "bbox": [ + 394, + 678, + 504, + 691 + ], + "score": 0.88, + "content": "\\textstyle ( x _ { c } , y _ { c } ) = ( { \\frac { x _ { 1 } + x _ { 2 } } { 2 } } , { \\frac { y _ { 1 } + y _ { 2 } } { 2 } } )", + "type": "inline_equation" + } + ], + "index": 52 + }, + { + "bbox": [ + 104, + 687, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 104, + 687, + 506, + 702 + ], + "score": 1.0, + "content": "deviated rapidly. The HNN with cFINDE accurately predicted the state for a longer period. Even after", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 104, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 104, + 700, + 505, + 712 + ], + "score": 1.0, + "content": "errors in the mass positions became non-negligible, errors in the center-of-gravity position were still", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 103, + 710, + 507, + 723 + ], + "spans": [ + { + "bbox": [ + 103, + 710, + 424, + 723 + ], + "score": 1.0, + "content": "small. We show the absolute errors averaged over all trials in Fig. 3. In each of", + "type": "text" + }, + { + "bbox": [ + 425, + 713, + 432, + 721 + ], + "score": 0.45, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 710, + 453, + 723 + ], + "score": 1.0, + "content": "- and", + "type": "text" + }, + { + "bbox": [ + 453, + 712, + 460, + 722 + ], + "score": 0.81, + "content": "y", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 710, + 507, + 723 + ], + "score": 1.0, + "content": "-directions,", + "type": "text" + } + ], + "index": 55 + } + ], + "index": 53, + "bbox_fs": [ + 99, + 664, + 507, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 118, + 89, + 494, + 264 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 250, + 78, + 360, + 88 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 249, + 77, + 361, + 90 + ], + "spans": [ + { + "bbox": [ + 249, + 77, + 361, + 90 + ], + "score": 1.0, + "content": "Table 3: Results of FINDE.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 118, + 89, + 494, + 264 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 118, + 89, + 494, + 264 + ], + "spans": [ + { + "bbox": [ + 118, + 89, + 494, + 264 + ], + "score": 0.984, + "html": "
2-body + HNNKdV2-pendFitzHugh-Nagumo
ModelK1-step↓VPT个1-step↓VPT个1-step↓VPT个1-step↓VPT个
base model -5.17 ±0.57 0.362 ±0.0265.59 ±0.300.339 ±0.0380.82 ±0.020.110±0.03573.66 ±12.59 0.236 ±0.053
+ cFINDE17.10 ±1.250.374 ±0.0366.24 ±0.440.371 ±0.0880.75 ±0.040.156±0.04254.18 ±8.120.127 ±0.148
27.78 ±1.390.450 ±0.0522.59 ±0.110.608 ±0.0850.73±0.050.198 ±0.08837.03 ±3.810.437 ±0.084
3>1030.147 ±0.146*3.19 ±0.370.730 ±0.0910.69 ±0.030.411 ±0.093>1060.007 ±0.007*
4>1030.101 ±0.0053.65 ±0.300.641 ±0.0710.77 ±0.070.395 ±0.083一
5>1030.080±0.0144.68 ±0.430.601 ±0.0690.80±0.070.585 ±0.097
6>10³0.070 ±0.0197.79 ±0.510.425 ±0.06712.53±0.000.005 ±0.000*
+ dFINDE17.01 ±1.060.379 ±0.04011.61 ±6.600.288 ±0.0830.75 ±0.100.152 ±0.01747.07 ±8.030.117 ±0.122
27.03 ±1.000.475 ±0.0222.70 ±0.260.598 ±0.0590.74±0.050.271 ±0.11133.24 ±3.400.455 ±0.032
3 54.78 ±36.39 0.309 ±0.0243.78±0.270.636 ±0.0240.69±0.050.447 ±0.081319.70 ±91.11 0.049 ±0.007
4>100.102 ±0.0153.48±0.320.780 ±0.0590.71 ±0.030.454 ±0.060
5>1030.086±0.011*5.26 ±0.150.718±0.0380.86 ±0.090.591 ±0.087
6>1030.059 ±0.0179.60 ±3.610.573 ±0.121 58.88 ±22.98 0.037 ±0.039
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In contrast, when the cFINDE is present,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 85, + 470, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 85, + 474, + 100, + 485 + ], + "score": 1.0, + "content": "302", + "type": "text" + }, + { + "bbox": [ + 104, + 470, + 506, + 488 + ], + "score": 1.0, + "content": "errors in the center-of-gravity position were much smaller than those in the mass positions, implying", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 85, + 483, + 419, + 496 + ], + "spans": [ + { + "bbox": [ + 85, + 485, + 100, + 496 + ], + "score": 1.0, + "content": "303", + "type": "text" + }, + { + "bbox": [ + 105, + 483, + 419, + 496 + ], + "score": 1.0, + "content": "that errors in one mass position canceled out errors in the other mass position.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27.5 + }, + { + "type": "text", + "bbox": [ + 102, + 500, + 504, + 533 + ], + "lines": [ + { + "bbox": [ + 105, + 498, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 505, + 513 + ], + "score": 1.0, + "content": "Therefore, we conclude that FINDE not only had better prediction accuracy but also found and", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 511, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 506, + 523 + ], + "score": 1.0, + "content": "preserved linear momenta (which are related to symmetries in space) more accurately despite not", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 522, + 279, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 279, + 534 + ], + "score": 1.0, + "content": "having prior knowledge about symmetries.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31 + }, + { + "type": "title", + "bbox": [ + 92, + 551, + 371, + 564 + ], + "lines": [ + { + "bbox": [ + 88, + 550, + 373, + 566 + ], + "spans": [ + { + "bbox": [ + 88, + 550, + 373, + 566 + ], + "score": 1.0, + "content": "307 4.4 Learning First Integrals from Data of Unknown Systems", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 87, + 574, + 505, + 607 + ], + "lines": [ + { + "bbox": [ + 86, + 574, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 86, + 576, + 100, + 586 + ], + "score": 1.0, + "content": "308", + "type": "text" + }, + { + "bbox": [ + 104, + 574, + 505, + 586 + ], + "score": 1.0, + "content": "It is often unclear whether a target system is a Hamiltonian system or not, but one can expect that the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 86, + 585, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 86, + 588, + 99, + 596 + ], + "score": 1.0, + "content": "309", + "type": "text" + }, + { + "bbox": [ + 105, + 585, + 505, + 597 + ], + "score": 1.0, + "content": "target system has several first integrals. We evaluated FINDE using NODE as the base model. We", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 86, + 597, + 246, + 608 + ], + "spans": [ + { + "bbox": [ + 86, + 599, + 100, + 607 + ], + "score": 1.0, + "content": "310", + "type": "text" + }, + { + "bbox": [ + 105, + 597, + 246, + 608 + ], + "score": 1.0, + "content": "summarized the results in Table 3.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 88, + 612, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 85, + 613, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 85, + 614, + 100, + 624 + ], + "score": 1.0, + "content": "311", + "type": "text" + }, + { + "bbox": [ + 105, + 613, + 505, + 624 + ], + "score": 1.0, + "content": "For the KdV dataset, the NODE with FINDE got much better 1-step errors and VPTs for a wide", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 85, + 624, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 85, + 625, + 100, + 636 + ], + "score": 1.0, + "content": "312", + "type": "text" + }, + { + "bbox": [ + 105, + 624, + 142, + 636 + ], + "score": 1.0, + "content": "range of", + "type": "text" + }, + { + "bbox": [ + 142, + 624, + 152, + 633 + ], + "score": 0.76, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 624, + 505, + 636 + ], + "score": 1.0, + "content": ". Figure 4 shows an example result. The top panels show that the prediction results were", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 85, + 633, + 502, + 649 + ], + "spans": [ + { + "bbox": [ + 85, + 636, + 100, + 646 + ], + "score": 1.0, + "content": "313", + "type": "text" + }, + { + "bbox": [ + 104, + 633, + 420, + 649 + ], + "score": 1.0, + "content": "apparently similar. The bottom panels summarize mean absolute errors in states", + "type": "text" + }, + { + "bbox": [ + 420, + 636, + 428, + 645 + ], + "score": 0.71, + "content": "\\textbf { \\em u }", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 633, + 473, + 649 + ], + "score": 1.0, + "content": ", total mass", + "type": "text" + }, + { + "bbox": [ + 474, + 634, + 502, + 647 + ], + "score": 0.91, + "content": "\\textstyle \\sum _ { k } u _ { k }", + "type": "inline_equation" + } + ], + "index": 39 + }, + { + "bbox": [ + 85, + 645, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 85, + 647, + 100, + 657 + ], + "score": 1.0, + "content": "314", + "type": "text" + }, + { + "bbox": [ + 105, + 645, + 505, + 658 + ], + "score": 1.0, + "content": "and energy. In the absence of FINDE, the NODE increased all of its errors in proportion to time. With", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 85, + 656, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 85, + 658, + 100, + 668 + ], + "score": 1.0, + "content": "315", + "type": "text" + }, + { + "bbox": [ + 105, + 656, + 505, + 668 + ], + "score": 1.0, + "content": "the cFINDE, the error in total mass increased at the point where the two solitons collided but then", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 85, + 667, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 85, + 669, + 100, + 679 + ], + "score": 1.0, + "content": "316", + "type": "text" + }, + { + "bbox": [ + 105, + 667, + 505, + 680 + ], + "score": 1.0, + "content": "returned to the original level. Although the calculation is slightly inaccurate, the cFINDE learned", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 85, + 678, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 85, + 680, + 100, + 690 + ], + "score": 1.0, + "content": "317", + "type": "text" + }, + { + "bbox": [ + 105, + 678, + 506, + 691 + ], + "score": 1.0, + "content": "to preserve the total mass. 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2-body + HNNKdV2-pendFitzHugh-Nagumo
ModelK1-step↓VPT个1-step↓VPT个1-step↓VPT个1-step↓VPT个
base model -5.17 ±0.57 0.362 ±0.0265.59 ±0.300.339 ±0.0380.82 ±0.020.110±0.03573.66 ±12.59 0.236 ±0.053
+ cFINDE17.10 ±1.250.374 ±0.0366.24 ±0.440.371 ±0.0880.75 ±0.040.156±0.04254.18 ±8.120.127 ±0.148
27.78 ±1.390.450 ±0.0522.59 ±0.110.608 ±0.0850.73±0.050.198 ±0.08837.03 ±3.810.437 ±0.084
3>1030.147 ±0.146*3.19 ±0.370.730 ±0.0910.69 ±0.030.411 ±0.093>1060.007 ±0.007*
4>1030.101 ±0.0053.65 ±0.300.641 ±0.0710.77 ±0.070.395 ±0.083一
5>1030.080±0.0144.68 ±0.430.601 ±0.0690.80±0.070.585 ±0.097
6>10³0.070 ±0.0197.79 ±0.510.425 ±0.06712.53±0.000.005 ±0.000*
+ dFINDE17.01 ±1.060.379 ±0.04011.61 ±6.600.288 ±0.0830.75 ±0.100.152 ±0.01747.07 ±8.030.117 ±0.122
27.03 ±1.000.475 ±0.0222.70 ±0.260.598 ±0.0590.74±0.050.271 ±0.11133.24 ±3.400.455 ±0.032
3 54.78 ±36.39 0.309 ±0.0243.78±0.270.636 ±0.0240.69±0.050.447 ±0.081319.70 ±91.11 0.049 ±0.007
4>100.102 ±0.0153.48±0.320.780 ±0.0590.71 ±0.030.454 ±0.060
5>1030.086±0.011*5.26 ±0.150.718±0.0380.86 ±0.090.591 ±0.087
6>1030.059 ±0.0179.60 ±3.610.573 ±0.121 58.88 ±22.98 0.037 ±0.039
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"spans": [ + { + "bbox": [ + 324, + 415, + 501, + 428 + ], + "score": 1.0, + "content": "Figure 3: Mean absolute errors of states for", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 325, + 426, + 500, + 439 + ], + "spans": [ + { + "bbox": [ + 325, + 426, + 500, + 439 + ], + "score": 1.0, + "content": "the 2-body dataset with or without cFINDE.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5 + } + ], + "index": 22.0 + }, + { + "type": "index", + "bbox": [ + 87, + 451, + 505, + 495 + ], + "lines": [ + { + "bbox": [ + 86, + 450, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 86, + 453, + 99, + 462 + ], + "score": 1.0, + "content": "300", + "type": "text" + }, + { + "bbox": [ + 105, + 450, + 399, + 464 + ], + "score": 1.0, + "content": "the HNN without FINDE produced errors in the center-of-gravity position", + "type": "text" + }, + { + "bbox": [ + 400, + 452, + 411, + 462 + ], + "score": 0.86, + "content": "x _ { c }", + "type": "inline_equation" + }, 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In contrast, when the cFINDE is present,", + "type": "text" + } + ], + "index": 27, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 470, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 85, + 474, + 100, + 485 + ], + "score": 1.0, + "content": "302", + "type": "text" + }, + { + "bbox": [ + 104, + 470, + 506, + 488 + ], + "score": 1.0, + "content": "errors in the center-of-gravity position were much smaller than those in the mass positions, implying", + "type": "text" + } + ], + "index": 28, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 483, + 419, + 496 + ], + "spans": [ + { + "bbox": [ + 85, + 485, + 100, + 496 + ], + "score": 1.0, + "content": "303", + "type": "text" + }, + { + "bbox": [ + 105, + 483, + 419, + 496 + ], + "score": 1.0, + "content": "that errors in one mass position canceled out errors in the other mass position.", + "type": "text" + } + ], + "index": 29, + "is_list_start_line": true + } + ], + "index": 27.5, + "bbox_fs": [ + 85, + 450, + 506, + 496 + ] + }, + { + "type": "text", + "bbox": [ + 102, + 500, + 504, + 533 + ], + "lines": [ + { + "bbox": [ + 105, + 498, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 505, + 513 + ], + "score": 1.0, + "content": "Therefore, we conclude that FINDE not only had better prediction accuracy but also found and", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 511, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 506, + 523 + ], + "score": 1.0, + "content": "preserved linear momenta (which are related to symmetries in space) more accurately despite not", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 522, + 279, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 279, + 534 + ], + "score": 1.0, + "content": "having prior knowledge about symmetries.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 498, + 506, + 534 + ] + }, + { + "type": "title", + "bbox": [ + 92, + 551, + 371, + 564 + ], + "lines": [ + { + "bbox": [ + 88, + 550, + 373, + 566 + ], + "spans": [ + { + "bbox": [ + 88, + 550, + 373, + 566 + ], + "score": 1.0, + "content": "307 4.4 Learning First Integrals from Data of Unknown Systems", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "index", + "bbox": [ + 87, + 574, + 505, + 607 + ], + "lines": [ + { + "bbox": [ + 86, + 574, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 86, + 576, + 100, + 586 + ], + "score": 1.0, + "content": "308", + "type": "text" + }, + { + "bbox": [ + 104, + 574, + 505, + 586 + ], + "score": 1.0, + "content": "It is often unclear whether a target system is a Hamiltonian system or not, but one can expect that the", + "type": "text" + } + ], + "index": 34, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 585, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 86, + 588, + 99, + 596 + ], + "score": 1.0, + "content": "309", + "type": "text" + }, + { + "bbox": [ + 105, + 585, + 505, + 597 + ], + "score": 1.0, + "content": "target system has several first integrals. We evaluated FINDE using NODE as the base model. We", + "type": "text" + } + ], + "index": 35, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 597, + 246, + 608 + ], + "spans": [ + { + "bbox": [ + 86, + 599, + 100, + 607 + ], + "score": 1.0, + "content": "310", + "type": "text" + }, + { + "bbox": [ + 105, + 597, + 246, + 608 + ], + "score": 1.0, + "content": "summarized the results in Table 3.", + "type": "text" + } + ], + "index": 36, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 613, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 85, + 614, + 100, + 624 + ], + "score": 1.0, + "content": "311", + "type": "text" + }, + { + "bbox": [ + 105, + 613, + 505, + 624 + ], + "score": 1.0, + "content": "For the KdV dataset, the NODE with FINDE got much better 1-step errors and VPTs for a wide", + "type": "text" + } + ], + "index": 37, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 624, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 85, + 625, + 100, + 636 + ], + "score": 1.0, + "content": "312", + "type": "text" + }, + { + "bbox": [ + 105, + 624, + 142, + 636 + ], + "score": 1.0, + "content": "range of", + "type": "text" + }, + { + "bbox": [ + 142, + 624, + 152, + 633 + ], + "score": 0.76, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 624, + 505, + 636 + ], + "score": 1.0, + "content": ". Figure 4 shows an example result. The top panels show that the prediction results were", + "type": "text" + } + ], + "index": 38, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 633, + 502, + 649 + ], + "spans": [ + { + "bbox": [ + 85, + 636, + 100, + 646 + ], + "score": 1.0, + "content": "313", + "type": "text" + }, + { + "bbox": [ + 104, + 633, + 420, + 649 + ], + "score": 1.0, + "content": "apparently similar. The bottom panels summarize mean absolute errors in states", + "type": "text" + }, + { + "bbox": [ + 420, + 636, + 428, + 645 + ], + "score": 0.71, + "content": "\\textbf { \\em u }", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 633, + 473, + 649 + ], + "score": 1.0, + "content": ", total mass", + "type": "text" + }, + { + "bbox": [ + 474, + 634, + 502, + 647 + ], + "score": 0.91, + "content": "\\textstyle \\sum _ { k } u _ { k }", + "type": "inline_equation" + } + ], + "index": 39, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 645, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 85, + 647, + 100, + 657 + ], + "score": 1.0, + "content": "314", + "type": "text" + }, + { + "bbox": [ + 105, + 645, + 505, + 658 + ], + "score": 1.0, + "content": "and energy. In the absence of FINDE, the NODE increased all of its errors in proportion to time. With", + "type": "text" + } + ], + "index": 40, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 656, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 85, + 658, + 100, + 668 + ], + "score": 1.0, + "content": "315", + "type": "text" + }, + { + "bbox": [ + 105, + 656, + 505, + 668 + ], + "score": 1.0, + "content": "the cFINDE, the error in total mass increased at the point where the two solitons collided but then", + "type": "text" + } + ], + "index": 41, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 667, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 85, + 669, + 100, + 679 + ], + "score": 1.0, + "content": "316", + "type": "text" + }, + { + "bbox": [ + 105, + 667, + 505, + 680 + ], + "score": 1.0, + "content": "returned to the original level. Although the calculation is slightly inaccurate, the cFINDE learned", + "type": "text" + } + ], + "index": 42, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 678, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 85, + 680, + 100, + 690 + ], + "score": 1.0, + "content": "317", + "type": "text" + }, + { + "bbox": [ + 105, + 678, + 506, + 691 + ], + "score": 1.0, + "content": "to preserve the total mass. The rightmost panel shows that the error in energy continued to increase", + "type": "text" + } + ], + "index": 43, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 689, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 85, + 690, + 100, + 700 + ], + "score": 1.0, + "content": "318", + "type": "text" + }, + { + "bbox": [ + 106, + 689, + 121, + 700 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 122, + 689, + 153, + 699 + ], + "score": 0.89, + "content": "K = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 689, + 313, + 700 + ], + "score": 1.0, + "content": ", but it stayed within a small range for", + "type": "text" + }, + { + "bbox": [ + 314, + 691, + 344, + 699 + ], + "score": 0.88, + "content": "K = 3", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 689, + 505, + 700 + ], + "score": 1.0, + "content": ". These results suggest that the first or", + "type": "text" + } + ], + "index": 44, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 700, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 85, + 702, + 100, + 712 + ], + "score": 1.0, + "content": "319", + "type": "text" + }, + { + "bbox": [ + 105, + 700, + 505, + 713 + ], + "score": 1.0, + "content": "second quantity learned by the cFINDE was total mass, the third quantity was system energy, and the", + "type": "text" + } + ], + "index": 45, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 711, + 470, + 723 + ], + "spans": [ + { + "bbox": [ + 86, + 713, + 100, + 722 + ], + "score": 1.0, + "content": "320", + "type": "text" + }, + { + "bbox": [ + 104, + 711, + 470, + 723 + ], + "score": 1.0, + "content": "remaining quantity may correspond to one of the many first integrals of the KdV equation.", + "type": "text" + } + ], + "index": 46, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 367, + 506, + 379 + ], + "spans": [ + { + "bbox": [ + 86, + 369, + 99, + 378 + ], + "score": 1.0, + "content": "321", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 367, + 456, + 379 + ], + "score": 1.0, + "content": "For the 2-pend dataset, the NODE with FINDE got better 1-step errors and VPTs for", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 456, + 367, + 486, + 377 + ], + "score": 0.9, + "content": "K = 1", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 486, + 367, + 506, + 379 + ], + "score": 1.0, + "content": "to 5", + "type": "text", + "cross_page": true + } + ], + "index": 18, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 376, + 506, + 390 + ], + "spans": [ + { + "bbox": [ + 86, + 380, + 99, + 389 + ], + "score": 1.0, + "content": "322", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 376, + 295, + 390 + ], + "score": 1.0, + "content": "except for the 1-step error of the dFINDE with", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 295, + 378, + 324, + 388 + ], + "score": 0.9, + "content": "K = 5", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 324, + 376, + 506, + 390 + ], + "score": 1.0, + "content": ". In addition to the system energy, the double", + "type": "text", + "cross_page": true + } + ], + "index": 19, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 388, + 506, + 401 + ], + "spans": [ + { + "bbox": [ + 86, + 391, + 99, + 399 + ], + "score": 1.0, + "content": "323", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 388, + 506, + 401 + ], + "score": 1.0, + "content": "pendulum has two holonomic constraints on the position, which lead to two additional constraints", + "type": "text", + "cross_page": true + } + ], + "index": 20, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 399, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 85, + 401, + 100, + 411 + ], + "score": 1.0, + "content": "324", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 399, + 505, + 411 + ], + "score": 1.0, + "content": "involving the velocity (see Appendix C for details). Thus, it is reasonable that the NODE with FINDE", + "type": "text", + "cross_page": true + } + ], + "index": 21, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 409, + 506, + 423 + ], + "spans": [ + { + "bbox": [ + 86, + 412, + 99, + 422 + ], + "score": 1.0, + "content": "325", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 409, + 196, + 423 + ], + "score": 1.0, + "content": "got the best VPTs for", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 196, + 411, + 225, + 420 + ], + "score": 0.9, + "content": "K = 5", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 225, + 409, + 418, + 423 + ], + "score": 1.0, + "content": "first integrals and totally failed when assuming", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 419, + 410, + 447, + 421 + ], + "score": 0.9, + "content": "K > 5", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 448, + 409, + 506, + 423 + ], + "score": 1.0, + "content": "first integrals.", + "type": "text", + "cross_page": true + } + ], + "index": 22, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 420, + 506, + 435 + ], + "spans": [ + { + "bbox": [ + 86, + 423, + 99, + 432 + ], + "score": 1.0, + "content": "326", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 420, + 506, + 435 + ], + "score": 1.0, + "content": "As exemplified in Fig. 5, the NODE without FINDE did not preserve the lengths of rods, making", + "type": "text", + "cross_page": true + } + ], + "index": 23, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 432, + 506, + 446 + ], + "spans": [ + { + "bbox": [ + 86, + 435, + 99, + 443 + ], + "score": 1.0, + "content": "327", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 432, + 506, + 446 + ], + "score": 1.0, + "content": "the states deviate gradually. See Appendix D.2 for the case when actual constraints are known. For", + "type": "text", + "cross_page": true + } + ], + "index": 24, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 443, + 506, + 455 + ], + "spans": [ + { + "bbox": [ + 86, + 445, + 99, + 455 + ], + "score": 1.0, + "content": "328", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 443, + 506, + 455 + ], + "score": 1.0, + "content": "the FitzHugh–Nagumo dataset, the NODE with FINDE got much better 1-step errors and VPTs for", + "type": "text", + "cross_page": true + } + ], + "index": 25, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 454, + 506, + 466 + ], + "spans": [ + { + "bbox": [ + 85, + 456, + 99, + 465 + ], + "score": 1.0, + "content": "329", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 454, + 135, + 464 + ], + "score": 0.88, + "content": "K = 2", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 136, + 454, + 506, + 466 + ], + "score": 1.0, + "content": ". 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FINDE projects", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 574, + 505, + 587 + ], + "spans": [ + { + "bbox": [ + 106, + 574, + 505, + 587 + ], + "score": 1.0, + "content": "the time evolution onto the submanifold defined using the (discrete) gradients of first integrals", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 585, + 506, + 598 + ], + "spans": [ + { + "bbox": [ + 104, + 585, + 506, + 598 + ], + "score": 1.0, + "content": "represented by a neural network. With an appropriate number of assumed first integrals, FINDE", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 596, + 506, + 609 + ], + "spans": [ + { + "bbox": [ + 104, + 596, + 506, + 609 + ], + "score": 1.0, + "content": "predicted future states more accurately than base models. 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IEEE Transactions on Neural Networks, 9(2):294–307.", + "type": "text" + } + ], + "index": 29, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 475, + 507, + 490 + ], + "spans": [ + { + "bbox": [ + 88, + 478, + 100, + 488 + ], + "score": 1.0, + "content": "476", + "type": "text" + }, + { + "bbox": [ + 104, + 475, + 507, + 490 + ], + "score": 1.0, + "content": "[56] Zhong, G. and Marsden, J. E. (1988). Lie-Poisson Hamilton-Jacobi Theory and Lie-Poisson Integrators.", + "type": "text" + } + ], + "index": 30, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 486, + 229, + 499 + ], + "spans": [ + { + "bbox": [ + 88, + 487, + 100, + 498 + ], + "score": 1.0, + "content": "477", + "type": "text" + }, + { + "bbox": [ + 116, + 486, + 229, + 499 + ], + "score": 1.0, + "content": "Physics Letters A, 133(3):3–8.", + "type": "text" + } + ], + "index": 31, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 504, + 506, + 517 + ], + "spans": [ + { + "bbox": [ + 88, + 505, + 100, + 515 + ], + "score": 1.0, + "content": "478", + "type": "text" + }, + { + "bbox": [ + 105, + 504, + 506, + 517 + ], + "score": 1.0, + "content": "[57] Zhong, Y. D., Dey, B., and Chakraborty, A. (2020). Dissipative SymODEN: Encoding Hamiltonian", + "type": "text" + } + ], + "index": 32, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 514, + 403, + 527 + ], + "spans": [ + { + "bbox": [ + 88, + 515, + 100, + 527 + ], + "score": 1.0, + "content": "479", + "type": "text" + }, + { + "bbox": [ + 116, + 514, + 403, + 527 + ], + "score": 1.0, + "content": "Dynamics with Dissipation and Control into Deep Learning. arXiv, pages 1–6.", + "type": "text" + } + ], + "index": 33, + "is_list_start_line": true + } + ], + "index": 16.5, + "bbox_fs": [ + 87, + 73, + 508, + 527 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 131, + 92, + 208, + 103 + ], + "lines": [ + { + "bbox": [ + 129, + 90, + 210, + 105 + ], + "spans": [ + { + "bbox": [ + 129, + 90, + 210, + 105 + ], + "score": 1.0, + "content": "1. For all authors...", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 144, + 106, + 505, + 243 + ], + "lines": [ + { + "bbox": [ + 146, + 106, + 506, + 119 + ], + "spans": [ + { + "bbox": [ + 146, + 106, + 506, + 119 + ], + "score": 1.0, + "content": "(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 161, + 117, + 506, + 130 + ], + "spans": [ + { + "bbox": [ + 161, + 117, + 506, + 130 + ], + "score": 1.0, + "content": "contributions and scope? [Yes] The theoretical contributions are summarized in", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 161, + 128, + 505, + 141 + ], + "spans": [ + { + "bbox": [ + 161, + 128, + 505, + 141 + ], + "score": 1.0, + "content": "Remarks 1 and 2. The performance improvements were validated numerically in", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 162, + 139, + 297, + 151 + ], + "spans": [ + { + "bbox": [ + 162, + 139, + 297, + 151 + ], + "score": 1.0, + "content": "Table 3 and visually in Figs. 1–6.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 146, + 152, + 505, + 164 + ], + "spans": [ + { + "bbox": [ + 146, + 152, + 505, + 164 + ], + "score": 1.0, + "content": "(b) Did you describe the limitations of your work? [Yes] We have discussed an increase in", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 161, + 163, + 505, + 175 + ], + "spans": [ + { + "bbox": [ + 161, + 163, + 505, + 175 + ], + "score": 1.0, + "content": "computational complexity at the bottoms of Sections 3.1 and 3.2. We also presented", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 161, + 173, + 505, + 187 + ], + "spans": [ + { + "bbox": [ + 161, + 173, + 505, + 187 + ], + "score": 1.0, + "content": "the limitations in Appendix D.2 while their situations were originally outside the scope", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 161, + 184, + 261, + 197 + ], + "spans": [ + { + "bbox": [ + 161, + 184, + 261, + 197 + ], + "score": 1.0, + "content": "of the proposed method.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 146, + 197, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 146, + 198, + 160, + 209 + ], + "score": 1.0, + "content": "(c)", + "type": "text" + }, + { + "bbox": [ + 161, + 197, + 505, + 210 + ], + "score": 1.0, + "content": "Did you discuss any potential negative societal impacts of your work? [N/A] No", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 161, + 209, + 276, + 220 + ], + "spans": [ + { + "bbox": [ + 161, + 209, + 276, + 220 + ], + "score": 1.0, + "content": "societal impact is supposed.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 146, + 220, + 505, + 234 + ], + "spans": [ + { + "bbox": [ + 146, + 220, + 505, + 234 + ], + "score": 1.0, + "content": "(d) Have you read the ethics review guidelines and ensured that your paper conforms to", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 162, + 231, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 162, + 231, + 506, + 244 + ], + "score": 1.0, + "content": "them? [N/A] We have read the guidelines carefully, but no ethical impact is supposed.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 132, + 246, + 302, + 258 + ], + "lines": [ + { + "bbox": [ + 129, + 245, + 304, + 259 + ], + "spans": [ + { + "bbox": [ + 129, + 245, + 304, + 259 + ], + "score": 1.0, + "content": "2. If you are including theoretical results...", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 143, + 261, + 505, + 339 + ], + "lines": [ + { + "bbox": [ + 145, + 261, + 506, + 273 + ], + "spans": [ + { + "bbox": [ + 145, + 261, + 506, + 273 + ], + "score": 1.0, + "content": "(a) Did you state the full set of assumptions of all theoretical results? [Yes] We have", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 161, + 272, + 506, + 284 + ], + "spans": [ + { + "bbox": [ + 161, + 272, + 506, + 284 + ], + "score": 1.0, + "content": "introduced the background of the proposed method and provided the full set of as-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 162, + 283, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 162, + 283, + 505, + 294 + ], + "score": 1.0, + "content": "sumptions in Section 2. Even though a slight modification may make the proposed", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 161, + 294, + 505, + 305 + ], + "spans": [ + { + "bbox": [ + 161, + 294, + 505, + 305 + ], + "score": 1.0, + "content": "method available on a general manifold, we have clearly stated that our theoretical and", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 161, + 304, + 475, + 317 + ], + "spans": [ + { + "bbox": [ + 161, + 304, + 475, + 317 + ], + "score": 1.0, + "content": "experimental results were limited to the finite-dimensional Eucleadian spaces.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 146, + 317, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 146, + 318, + 160, + 327 + ], + "score": 1.0, + "content": "(b)", + "type": "text" + }, + { + "bbox": [ + 160, + 317, + 505, + 329 + ], + "score": 1.0, + "content": "Did you include complete proofs of all theoretical results? [Yes] We have provided a", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 161, + 329, + 335, + 339 + ], + "spans": [ + { + "bbox": [ + 161, + 329, + 335, + 339 + ], + "score": 1.0, + "content": "proof just before each of Remarks 1 and 2.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 131, + 343, + 241, + 354 + ], + "lines": [ + { + "bbox": [ + 128, + 340, + 243, + 357 + ], + "spans": [ + { + "bbox": [ + 128, + 340, + 243, + 357 + ], + "score": 1.0, + "content": "3. If you ran experiments...", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 145, + 357, + 506, + 515 + ], + "lines": [ + { + "bbox": [ + 145, + 356, + 507, + 370 + ], + "spans": [ + { + "bbox": [ + 145, + 356, + 507, + 370 + ], + "score": 1.0, + "content": "(a) Did you include the code, data, and instructions needed to reproduce the main exper-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 162, + 368, + 505, + 380 + ], + "spans": [ + { + "bbox": [ + 162, + 368, + 505, + 380 + ], + "score": 1.0, + "content": "imental results (either in the supplemental material or as a URL)? [Yes] We have", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 161, + 379, + 487, + 392 + ], + "spans": [ + { + "bbox": [ + 161, + 379, + 487, + 392 + ], + "score": 1.0, + "content": "enclosed the source code to reproduce all experiments in supplemental materials.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 146, + 391, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 146, + 391, + 505, + 405 + ], + "score": 1.0, + "content": "(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 161, + 402, + 506, + 414 + ], + "spans": [ + { + "bbox": [ + 161, + 402, + 253, + 414 + ], + "score": 1.0, + "content": "were chosen)? [Yes]", + "type": "text" + }, + { + "bbox": [ + 255, + 403, + 506, + 414 + ], + "score": 1.0, + "content": "We have provided the software and hardware environment,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 161, + 413, + 506, + 426 + ], + "spans": [ + { + "bbox": [ + 161, + 413, + 506, + 426 + ], + "score": 1.0, + "content": "network architectures, and hyperparameters in Section 4.1. We have also provided", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 162, + 424, + 409, + 437 + ], + "spans": [ + { + "bbox": [ + 162, + 424, + 409, + 437 + ], + "score": 1.0, + "content": "detailed hyperparameters to generate datasets in Appendix C.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 146, + 437, + 507, + 450 + ], + "spans": [ + { + "bbox": [ + 146, + 437, + 507, + 450 + ], + "score": 1.0, + "content": "(c) Did you report error bars (e.g., with respect to the random seed after running experi-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 161, + 448, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 161, + 448, + 505, + 459 + ], + "score": 1.0, + "content": "ments multiple times)? [Yes] We have summarized the standard deviations over five", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 162, + 459, + 229, + 469 + ], + "spans": [ + { + "bbox": [ + 162, + 459, + 229, + 469 + ], + "score": 1.0, + "content": "trials in Table 3.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 146, + 470, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 146, + 470, + 505, + 484 + ], + "score": 1.0, + "content": "(d) Did you include the total amount of compute and the type of resources used (e.g., type", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 161, + 482, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 161, + 482, + 505, + 495 + ], + "score": 1.0, + "content": "of GPUs, internal cluster, or cloud provider)? [Yes] We have provided the hardware", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 161, + 492, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 161, + 492, + 505, + 505 + ], + "score": 1.0, + "content": "environment in Section 4.1, but we have anonymized the cloud service providers to", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 161, + 504, + 376, + 517 + ], + "spans": [ + { + "bbox": [ + 161, + 504, + 376, + 517 + ], + "score": 1.0, + "content": "avoid a potential violation of the double-blind policy.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 28.5 + }, + { + "type": "text", + "bbox": [ + 132, + 519, + 504, + 530 + ], + "lines": [ + { + "bbox": [ + 129, + 517, + 507, + 532 + ], + "spans": [ + { + "bbox": [ + 129, + 517, + 507, + 532 + ], + "score": 1.0, + "content": "4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 146, + 533, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 145, + 532, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 145, + 532, + 505, + 545 + ], + "score": 1.0, + "content": "(a) If your work uses existing assets, did you cite the creators? [Yes] We have made", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 160, + 543, + 506, + 557 + ], + "spans": [ + { + "bbox": [ + 160, + 543, + 506, + 557 + ], + "score": 1.0, + "content": "the source code for all experiments by modifying the source codes of HNN [26] and", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 160, + 554, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 160, + 554, + 505, + 568 + ], + "score": 1.0, + "content": "DGNet [38]. We have cited these references and added links to respective repositories", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 161, + 565, + 229, + 578 + ], + "spans": [ + { + "bbox": [ + 161, + 565, + 229, + 578 + ], + "score": 1.0, + "content": "in the footnotes.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 146, + 578, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 146, + 578, + 505, + 590 + ], + "score": 1.0, + "content": "(b) Did you mention the license of the assets? [Yes] We have verified that the source codes", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 162, + 589, + 506, + 603 + ], + "spans": [ + { + "bbox": [ + 162, + 589, + 506, + 603 + ], + "score": 1.0, + "content": "of HNN [26] and DGNet [38] are provided in Apache-2.0 License and MIT License,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 161, + 600, + 418, + 613 + ], + "spans": [ + { + "bbox": [ + 161, + 600, + 418, + 613 + ], + "score": 1.0, + "content": "respectively. We have clearly stated these facts in the footnotes.", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 146, + 613, + 503, + 626 + ], + "spans": [ + { + "bbox": [ + 146, + 613, + 503, + 626 + ], + "score": 1.0, + "content": "(c) Did you include any new assets either in the supplemental material or as a URL? [Yes]", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 160, + 622, + 488, + 637 + ], + "spans": [ + { + "bbox": [ + 160, + 622, + 488, + 637 + ], + "score": 1.0, + "content": "We have enclosed the source code to generate datasets in supplemental materials.", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 146, + 636, + 505, + 648 + ], + "spans": [ + { + "bbox": [ + 146, + 637, + 160, + 648 + ], + "score": 1.0, + "content": "(d)", + "type": "text" + }, + { + "bbox": [ + 161, + 636, + 505, + 648 + ], + "score": 1.0, + "content": "Did you discuss whether and how consent was obtained from people whose data you’re", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 162, + 648, + 253, + 659 + ], + "spans": [ + { + "bbox": [ + 162, + 648, + 253, + 659 + ], + "score": 1.0, + "content": "using/curating? 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The performance improvements were validated numerically in", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 162, + 139, + 297, + 151 + ], + "spans": [ + { + "bbox": [ + 162, + 139, + 297, + 151 + ], + "score": 1.0, + "content": "Table 3 and visually in Figs. 1–6.", + "type": "text" + } + ], + "index": 4, + "is_list_end_line": true + }, + { + "bbox": [ + 146, + 152, + 505, + 164 + ], + "spans": [ + { + "bbox": [ + 146, + 152, + 505, + 164 + ], + "score": 1.0, + "content": "(b) Did you describe the limitations of your work? [Yes] We have discussed an increase in", + "type": "text" + } + ], + "index": 5, + "is_list_start_line": true + }, + { + "bbox": [ + 161, + 163, + 505, + 175 + ], + "spans": [ + { + "bbox": [ + 161, + 163, + 505, + 175 + ], + "score": 1.0, + "content": "computational complexity at the bottoms of Sections 3.1 and 3.2. We also presented", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 161, + 173, + 505, + 187 + ], + "spans": [ + { + "bbox": [ + 161, + 173, + 505, + 187 + ], + "score": 1.0, + "content": "the limitations in Appendix D.2 while their situations were originally outside the scope", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 161, + 184, + 261, + 197 + ], + "spans": [ + { + "bbox": [ + 161, + 184, + 261, + 197 + ], + "score": 1.0, + "content": "of the proposed method.", + "type": "text" + } + ], + "index": 8, + "is_list_end_line": true + }, + { + "bbox": [ + 146, + 197, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 146, + 198, + 160, + 209 + ], + "score": 1.0, + "content": "(c)", + "type": "text" + }, + { + "bbox": [ + 161, + 197, + 505, + 210 + ], + "score": 1.0, + "content": "Did you discuss any potential negative societal impacts of your work? [N/A] No", + "type": "text" + } + ], + "index": 9, + "is_list_start_line": true + }, + { + "bbox": [ + 161, + 209, + 276, + 220 + ], + "spans": [ + { + "bbox": [ + 161, + 209, + 276, + 220 + ], + "score": 1.0, + "content": "societal impact is supposed.", + "type": "text" + } + ], + "index": 10, + "is_list_end_line": true + }, + { + "bbox": [ + 146, + 220, + 505, + 234 + ], + "spans": [ + { + "bbox": [ + 146, + 220, + 505, + 234 + ], + "score": 1.0, + "content": "(d) Have you read the ethics review guidelines and ensured that your paper conforms to", + "type": "text" + } + ], + "index": 11, + "is_list_start_line": true + }, + { + "bbox": [ + 162, + 231, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 162, + 231, + 506, + 244 + ], + "score": 1.0, + "content": "them? [N/A] We have read the guidelines carefully, but no ethical impact is supposed.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 6.5, + "bbox_fs": [ + 146, + 106, + 506, + 244 + ] + }, + { + "type": "text", + "bbox": [ + 132, + 246, + 302, + 258 + ], + "lines": [ + { + "bbox": [ + 129, + 245, + 304, + 259 + ], + "spans": [ + { + "bbox": [ + 129, + 245, + 304, + 259 + ], + "score": 1.0, + "content": "2. If you are including theoretical results...", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13, + "bbox_fs": [ + 129, + 245, + 304, + 259 + ] + }, + { + "type": "list", + "bbox": [ + 143, + 261, + 505, + 339 + ], + "lines": [ + { + "bbox": [ + 145, + 261, + 506, + 273 + ], + "spans": [ + { + "bbox": [ + 145, + 261, + 506, + 273 + ], + "score": 1.0, + "content": "(a) Did you state the full set of assumptions of all theoretical results? [Yes] We have", + "type": "text" + } + ], + "index": 14, + "is_list_start_line": true + }, + { + "bbox": [ + 161, + 272, + 506, + 284 + ], + "spans": [ + { + "bbox": [ + 161, + 272, + 506, + 284 + ], + "score": 1.0, + "content": "introduced the background of the proposed method and provided the full set of as-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 162, + 283, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 162, + 283, + 505, + 294 + ], + "score": 1.0, + "content": "sumptions in Section 2. Even though a slight modification may make the proposed", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 161, + 294, + 505, + 305 + ], + "spans": [ + { + "bbox": [ + 161, + 294, + 505, + 305 + ], + "score": 1.0, + "content": "method available on a general manifold, we have clearly stated that our theoretical and", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 161, + 304, + 475, + 317 + ], + "spans": [ + { + "bbox": [ + 161, + 304, + 475, + 317 + ], + "score": 1.0, + "content": "experimental results were limited to the finite-dimensional Eucleadian spaces.", + "type": "text" + } + ], + "index": 18, + "is_list_end_line": true + }, + { + "bbox": [ + 146, + 317, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 146, + 318, + 160, + 327 + ], + "score": 1.0, + "content": "(b)", + "type": "text" + }, + { + "bbox": [ + 160, + 317, + 505, + 329 + ], + "score": 1.0, + "content": "Did you include complete proofs of all theoretical results? [Yes] We have provided a", + "type": "text" + } + ], + "index": 19, + "is_list_start_line": true + }, + { + "bbox": [ + 161, + 329, + 335, + 339 + ], + "spans": [ + { + "bbox": [ + 161, + 329, + 335, + 339 + ], + "score": 1.0, + "content": "proof just before each of Remarks 1 and 2.", + "type": "text" + } + ], + "index": 20, + "is_list_end_line": true + } + ], + "index": 17, + "bbox_fs": [ + 145, + 261, + 506, + 339 + ] + }, + { + "type": "text", + "bbox": [ + 131, + 343, + 241, + 354 + ], + "lines": [ + { + "bbox": [ + 128, + 340, + 243, + 357 + ], + "spans": [ + { + "bbox": [ + 128, + 340, + 243, + 357 + ], + "score": 1.0, + "content": "3. If you ran experiments...", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21, + "bbox_fs": [ + 128, + 340, + 243, + 357 + ] + }, + { + "type": "list", + "bbox": [ + 145, + 357, + 506, + 515 + ], + "lines": [ + { + "bbox": [ + 145, + 356, + 507, + 370 + ], + "spans": [ + { + "bbox": [ + 145, + 356, + 507, + 370 + ], + "score": 1.0, + "content": "(a) Did you include the code, data, and instructions needed to reproduce the main exper-", + "type": "text" + } + ], + "index": 22, + "is_list_start_line": true + }, + { + "bbox": [ + 162, + 368, + 505, + 380 + ], + "spans": [ + { + "bbox": [ + 162, + 368, + 505, + 380 + ], + "score": 1.0, + "content": "imental results (either in the supplemental material or as a URL)? [Yes] We have", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 161, + 379, + 487, + 392 + ], + "spans": [ + { + "bbox": [ + 161, + 379, + 487, + 392 + ], + "score": 1.0, + "content": "enclosed the source code to reproduce all experiments in supplemental materials.", + "type": "text" + } + ], + "index": 24, + "is_list_end_line": true + }, + { + "bbox": [ + 146, + 391, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 146, + 391, + 505, + 405 + ], + "score": 1.0, + "content": "(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they", + "type": "text" + } + ], + "index": 25, + "is_list_start_line": true + }, + { + "bbox": [ + 161, + 402, + 506, + 414 + ], + "spans": [ + { + "bbox": [ + 161, + 402, + 253, + 414 + ], + "score": 1.0, + "content": "were chosen)? 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energymonentum massconstraint learning invariants exact conservation
NODE [8]
HNN [26]√
LieConv [19]√厂
DGNet [38]√√
CHNN [20]√√
continuous FINDE (proposed)√√√√√
discrete FINDE (proposed)√厂厂丁
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DatasetDynamicsNFirst Integrals
EnergyMomentumMassConstraint
Two-body problemCanonical Hamiltonian81
Discretized KdV equationNon-canonical Hamiltonian50广1
Double pendulumPoisson8√√
FitzHugh-Nagumo modelDirac4√
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2-body + HNNKdV2-pendFitzHugh-Nagumo
ModelK1-step↓VPT个1-step↓VPT个1-step↓VPT个1-step↓VPT个
base model -5.17 ±0.57 0.362 ±0.0265.59 ±0.300.339 ±0.0380.82 ±0.020.110±0.03573.66 ±12.59 0.236 ±0.053
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27.78 ±1.390.450 ±0.0522.59 ±0.110.608 ±0.0850.73±0.050.198 ±0.08837.03 ±3.810.437 ±0.084
3>1030.147 ±0.146*3.19 ±0.370.730 ±0.0910.69 ±0.030.411 ±0.093>1060.007 ±0.007*
4>1030.101 ±0.0053.65 ±0.300.641 ±0.0710.77 ±0.070.395 ±0.083一
5>1030.080±0.0144.68 ±0.430.601 ±0.0690.80±0.070.585 ±0.097
6>10³0.070 ±0.0197.79 ±0.510.425 ±0.06712.53±0.000.005 ±0.000*
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27.03 ±1.000.475 ±0.0222.70 ±0.260.598 ±0.0590.74±0.050.271 ±0.11133.24 ±3.400.455 ±0.032
3 54.78 ±36.39 0.309 ±0.0243.78±0.270.636 ±0.0240.69±0.050.447 ±0.081319.70 ±91.11 0.049 ±0.007
4>100.102 ±0.0153.48±0.320.780 ±0.0590.71 ±0.030.454 ±0.060
5>1030.086±0.011*5.26 ±0.150.718±0.0380.86 ±0.090.591 ±0.087
6>1030.059 ±0.0179.60 ±3.610.573 ±0.121 58.88 ±22.98 0.037 ±0.039
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a/parse/dev/xMJWUKJnFSw/xMJWUKJnFSw.md b/parse/dev/xMJWUKJnFSw/xMJWUKJnFSw.md new file mode 100644 index 0000000000000000000000000000000000000000..9393be63ea722e244e763ff3e5c2c940d740e6d2 --- /dev/null +++ b/parse/dev/xMJWUKJnFSw/xMJWUKJnFSw.md @@ -0,0 +1,495 @@ +# NODEPIECE: COMPOSITIONAL AND PARAMETEREFFICIENT REPRESENTATIONS OF LARGE KNOWLEDGE GRAPHS + +Mikhail Galkin, Etienne Denis, Jiapeng Wu and William L. Hamilton +Mila, McGill University +Montreal, Canada +{mikhail.galkin,deniseti,jiapeng.wu,hamilton}@mila.quebec + +# ABSTRACT + +Conventional representation learning algorithms for knowledge graphs (KG) map each entity to a unique embedding vector. Such a shallow lookup results in a linear growth of memory consumption for storing the embedding matrix and incurs high computational costs when working with real-world KGs. Drawing parallels with subword tokenization commonly used in NLP, we explore the landscape of more parameter-efficient node embedding strategies. To this end, we propose NodePiece, an anchor-based approach to learn a fixed-size entity vocabulary. In NodePiece, a vocabulary of subword/sub-entity units is constructed from anchor nodes in a graph with known relation types. Given such a fixed-size vocabulary, it is possible to bootstrap an encoding and embedding for any entity, including those unseen during training. Experiments show that NodePiece performs competitively in node classification, link prediction, and relation prediction tasks while retaining less than $10 \%$ of explicit nodes in a graph as anchors and often having $1 0 \mathrm { x }$ fewer parameters. To this end, we show that a NodePiece-enabled model outperforms existing shallow models on a large OGB WikiKG 2 graph having $\mathrm { \sim } 7 0 \mathrm { x }$ fewer parameters1. + +# 1 INTRODUCTION + +Representation learning tasks on knowledge graphs (KGs) often require a parameterization of each unique atom in the graph with a vector or matrix. Traditionally, in multi-relational KGs such atoms constitute a set of all nodes $n \in N$ (entities) and relations (edge types) $r \in R$ (Nickel et al., 2016). Assuming parameterization with vectors, atoms are mapped to $d$ -dimensional vectors through shallow encoders $f _ { n } : n \to \mathbb { R } ^ { d }$ and $f _ { r } : r \to \mathbb { R } ^ { d }$ which scale linearly to the number of nodes and edge types2, i.e., having $O ( | N | )$ space complexity of the entity embedding matrix. Albeit efficient on small conventional benchmarking datasets based on Freebase (Toutanova & Chen, 2015) $( \sim 1 5 \mathrm { K }$ nodes) and WordNet (Dettmers et al., 2018) $- 4 0 \mathrm { K }$ nodes), training on larger graphs (e.g., YAGO 3-10 (Mahdisoltani et al., 2015) of 120K nodes) becomes computationally challenging. Scaling it further up to larger subsets (Hu et al., 2020; Wang et al., 2021; Safavi & Koutra, 2020) of Wikidata (Vrandecic & Krötzsch, 2014) requires a top-level GPU or a CPU cluster as done in, e.g., PyTorch-BigGraph (Lerer et al., 2019) that maintains a $7 8 \mathbf { M } \times 2 0 0 d$ embeddings matrix in memory (we list sizes of current best performing models in Table 1). + +Taking the perspective from NLP, shallow node encoding in KGs corresponds to shallow word embedding popularized with word2vec (Mikolov et al., 2013) and GloVe (Pennington et al., 2014) that learned a vocabulary of 400K-2M most frequent words, treating rarer ones as out-of-vocabulary (OOV). The OOV issue was resolved with the ability to build infinite combinations with a finite vocabulary enabled by subword units. Subword-powered algorithms such as fastText (Bojanowski et al., 2017), Byte-Pair Encoding (Sennrich et al., 2016), and WordPiece (Schuster & Nakajima, 2012) became a standard step in preprocessing pipelines of large language models and allowed to construct fixed-size token vocabularies, e.g., BERT (Devlin et al., 2019) contains $\mathord { \sim } 3 0 \mathrm { K }$ tokens and + +Table 1: Node embedding sizes of state-of-the-art KG embedding models compared to BERT Large. Parameters of type float32 take 4 bytes each. FB15k-237, WN18RR, and YAGO3-10 models as reported in Sun et al. (2019), OGB WikiKG2 as in Zhang et al. (2020c), Wikidata 5M as in Wang et al. (2021), PBG Wikidata as in Lerer et al. (2019), and BERT Large as in Devlin et al. (2019). + +
FB15k-237WN18RRYAGO3-10OGBWikiKG2Wikidata 5MPBG WikidataBERT Large
Vocabulary size15k40k120k2.5M5M78M30k
Embedding dim2000100010002005122001024
GPU RAM, GB0.120.150.461.879.6958.10.12
+ +GPT-2 (Radford et al., 2019) employs $\mathord { \sim } 5 0 \mathrm { K }$ tokens. Importantly, relatively small input embedding matrices enabled investing the parameters budget into more efficient encoders (Kaplan et al., 2020). + +Drawing inspiration from subword embeddings in NLP, we explore how similar strategies for tokenizing entities in large graphs can dramatically reduce parameter complexity, increase generalization, and naturally represent new unseen entities as using the same fixed vocabulary. To do so, tokenization has to rely on atoms akin to subword units and not the total set of nodes. + +To this end, we propose NodePiece, an anchor-based approach to learn a fixed-size vocabulary $V$ $( | V | \ll | N | )$ of any connected multi-relational graph. In NodePiece, the set of atoms consists of anchors and all relation types that, together, allow to construct a combinatorial number of sequences from a limited atoms vocabulary. In contrast to shallow approaches, each node $n$ is first tokenized into a unique $h a s h ( n )$ of $k$ closest anchors and $m$ immediate relations. A key element to build a node embedding is a proper encoder function $e n c ( n ) : h a s h ( n ) \mathbb { R } ^ { d }$ which can be designed leveraging inductive biases of an underlying graph or downstream tasks. Therefore, the overall parameter budget is now defined by a small fixed-size vocabulary of atoms and the complexity of the encoder function. + +Our experimental findings suggest that a fixed-size NodePiece vocabulary paired with a simple encoder still yields competitive results on a variety of tasks including link prediction, node classification, and relation prediction. Furthermore, anchor-based hashing enables conventional embedding models to work in the inductive and out-of-sample scenarios when unseen entities arrive at inference time, which otherwise required tailored learning mechanisms. + +# 2 RELATED WORK + +Conventional KG embedding approaches. To the best of our knowledge, all contemporary embedding algorithms (Ji et al., 2020; Ali et al., 2020) for link prediction on KGs employ shallow embedding lookups mapping each entity to a unique embedding vector thus being linear $O ( | N | )$ to the total number of nodes $| N |$ and size of an embedding matrix. This holds for different embedding families, e.g., translational (Sun et al., 2019), tensor factorization (Lacroix et al., 2018), convolutional (Dettmers et al., 2018), and hyperbolic (Chami et al., 2020; Balazevic et al., 2019). The same applies to relation-aware graph neural network (GNN) encoders (Schlichtkrull et al., 2018; Vashishth et al., 2020) who still initialize each node with a learned embedding or feature vector before message passing. Furthermore, shallow encoding is also used in higher-order KG structures such as hypergraphs (Fatemi et al., 2020) and hyper-relational graphs (Rosso et al., 2020; Galkin et al., 2020). NodePiece can be used as a drop-in replacement of the embedding lookup with any of those models. + +Distillation and compression. Several recent techniques for reducing memory footprint of embedding matrices follow successful applications of distilling large language models in NLP (Sanh et al., 2019), i.e., distillation (Wang et al., 2020; Zhu et al., 2020) into low-dimensional counterparts, and compression of trained matrices into discrete codes (Sachan, 2020). However, all of them require a full embedding matrix as input which we aim to avoid designing NodePiece. + +Vocabulary reduction in recommender systems. Commonly, recommender systems operate on thousands of categorical features combined in sparse high-dimensional vectors.Recent approaches (Medini et al., 2021; Liang et al., 2021) employ anchor-based hashing techniques to factorize sparse feature vectors into dense embeddings. Contrary to those setups, we do not expect availability of feature vectors for arbitrary KGs and rather learn vocabulary embeddings from scratch. + +![](images/b2dd2102f8724e1a2b533377fadb48c87c7d1681b42eb24887a2b59d1548267d.jpg) +Figure 1: NodePiece tokenization strategy. Given three anchors $a _ { 1 } , a _ { 2 } , a _ { 3 }$ , a target node can be tokenized into a hash of top- $k$ closest anchors, their distances to the target node, and the relational context of outgoing relations from the target node. This hash sequence is passed through an injective encoder to obtain a unique embedding. Inverse relations are added to ensure connectivity. + +Entity descriptions and language models. A recent line of work such as KG-BERT (Yao et al., 2019), MLMLM (Clouâtre et al., 2021), BLP (Daza et al., 2021) utilize entity descriptions passed through a language model (LM) encoder as entity embeddings suitable for link prediction. We would like to emphasize that such approaches are rather orthogonal to NodePiece. Textual features are mostly available in Wikipedia-derived KGs like Wikidata but are often missing in domain-specific graphs like social networks and product graphs. We therefore assume textual features are not available and rather learn node representations based on their spatial characteristics. Still, textual features can be easily added by concatenating NodePiece-encoded features with LM-produced features. + +Out-of-sample representation learning. This task focuses on predictions involving previously unseen, or out-of-sample, entities that attach to a known KG with a few edges. These new edges are then utilized as a context to compute its embedding. Previous work (Wang et al., 2019; Hamaguchi et al., 2017; Albooyeh et al., 2020) proposed different neighborhood aggregation functions for this process or resorted to meta-learning (Chen et al., 2019; Baek et al., 2020; Zhang et al., 2020a). However, all of them follow the shallow embedding paradigm. Instead, NodePiece uses the new edges as a basis for anchor-based tokenization of new nodes in terms of an existing vocabulary. + +# 3 NODEPIECE VOCABULARY CONSTRUCTION + +Given a directed KG $G = ( N , E , R )$ consisting of $| N |$ nodes, $| E |$ edges, and $| R |$ relation types, our task is to reduce the original vocabulary size of $| N |$ nodes to a smaller, fixed-size vocabulary of node pieces akin to subword units. In this work, we represent node pieces through anchor nodes $a \in A , A \subset N$ , a pre-selected set of nodes in a graph following a deterministic or stochastic strategy. A full NodePiece vocabulary is then constructed from anchor nodes and relation types, i.e, $V = A + R$ . Note that in order to maintain reachability of each node and balance in- and out-degrees we enrich $G$ with inverse edges with inverse relation types, such that $| R | _ { i n v e r s e } = | R | _ { d i r e c t }$ and $| R | = | R | _ { d i r e c t } + | R | _ { i n v e r s e }$ . Using elements of the constructed vocabulary each node $n$ can be tokenized into $h a s h ( n )$ as a sequence of $k$ closest anchors, discrete anchor distances, and a relational context of $m$ immediate relations. Then, any encoder function $e n c ( n ) : h a s h ( n ) \to \mathbb { R } ^ { d }$ can be applied to embed the hash into a $d$ -dimensional vector. An intuition of the approach is presented in Fig. 1 with each step explained in more detail below. + +# 3.1 ANCHOR SELECTION + +Subword tokenization algorithms such as BPE (Sennrich et al., 2016) employ deterministic strategies to create tokens and construct a vocabulary, e.g., based on frequencies of co-occurring n-grams, such that more frequent words are tokenized with fewer subword units. On graphs, such strategies might employ centrality measures like degree centrality or Personalized PageRank (Page et al., 1999). However, in our preliminary experiments, we found random anchor selection to be as effective as centrality-based strategies. A choice for deterministic strategies might be justified when optimizing for certain task-specific topological characteristics, e.g., degree and PPR strategies indeed skew the distribution of shortest anchor distances towards smaller values thus increasing chances to find anchors in 2- or 3-hop neighborhood of any node (we provide more evidence for that in Appendix C). + +# 3.2 NODE TOKENIZATION + +Once the vocabulary $V = A + R$ is constructed, each node $n$ can be hashed (or tokenized) into a $h a s h ( n )$ using 1) $k$ nearest anchors and their discrete distances; 2) $m$ immediate outgoing relations from the relational context of $n$ . Since anchor nodes are concrete nodes in $G$ , they get hashed in the same way as other non-anchor nodes. + +Anchors per node. Given $| A |$ anchor nodes, it is impractical to use all of them for encoding each node. Instead, we select $k$ anchors per node and describe two possible strategies for that, i.e., random and deterministic. The basic random strategy uniformly samples an unordered set of $k$ anchors from $A$ yielding $\binom { | A | } { k }$ possible combinations. To avoid collisions when hashing the nodes, $| A |$ and $k$ are to be chosen according to the lower bound on possible combinations that is defined by the total number of nodes, e.g., $\bar { \binom { | A | } { k } } \geq | N |$ . Note that running depth-first search (DFS) to random anchors at inference time is inefficient and, therefore, $h a s h ( n )$ of the random strategy has to be pre-computed. + +On the other hand, the deterministic strategy selects an ordered sequence of $k$ nearest anchors. Hence, the anchors can be obtained via breadth-first search (BFS) in the $l$ -hop neighborhood of $n$ at inference time (or pre-computed for speed reasons). However, the combinatorial bound is not applicable in this strategy and we need more discriminative signals to avoid hash collisions since nearby nodes will have similar anchors (we elaborate on the uniqueness issue in Appendix K). Such signals have to better ground anchors to the underlying graph structure, and we accomplish that using anchor distances3 and relational context described below. + +A node residing in a disconnected component is assigned with an auxiliary [DISCONNECTED] token or can be turned into an anchor. However, the majority of existing KGs are graphs with one large connected component with very few disconnected nodes, such that this effect is negligible. + +Anchor Distances. Given a target node $n$ and an anchor $a _ { i }$ , we define anchor distance $z _ { a _ { i } } \in$ [0; diameter $\cdot ( G ) ]$ as an integer denoting the shortest path distance between $a _ { i }$ and $n$ in the original graph $G$ . Note that when tokenizing an anchor $a _ { j }$ with the deterministic strategy, the nearest anchor among top- $k$ is always $a _ { j }$ itself with distance 0. We then map each integer to a learnable $d$ -dimensional vector $f _ { z } : z _ { a _ { i } } \to \mathbb { R } ^ { d }$ akin to relative distance encoding scheme. + +Relational Context. We also leverage the multi-relational nature of an underlying KG. Commonly4, the amount of unique edge types in $G$ is orders of magnitude smaller than the total number of nodes, i.e., $| R | \ll | N |$ . This fact allows to include the entire $| R |$ in the NodePiece vocabulary $V _ { N P }$ and further featurize each node with a unique relational context. We construct a relational context of a node $n$ by randomly sampling a set of $m$ immediate unique outgoing relations starting from $n$ , i.e., $r c o n _ { n } = \mathsf { \bar { \{ } } r _ { j } \mathsf { \} } ^ { m } \subseteq \mathsf { \bar { \mathcal { N } } } _ { r } ( n ) \mathsf { \bar { \Psi } }$ where $\mathcal { N } _ { r } ( n )$ denotes all outgoing relation types. Due to a non-uniform degree distribution, if $| { \mathcal { N } } _ { r } ( n ) | < m$ , we add auxiliary [PAD] tokens to complete $r c o n _ { n }$ to size $m$ . + +# 3.3 ENCODING + +At this step, a node $n$ is tokenized into a sequence of $k$ anchors, their $k$ respective distances, and relational context of size $m$ : + +$$ +h a s h ( n ) = \Bigl [ \{ a _ { i } \} ^ { k } , \{ z _ { a _ { i } } \} ^ { k } , \{ r _ { j } \} ^ { m } \Bigr ] +$$ + +Taking anchors vectors $\mathbf { a _ { n } }$ and relation vectors $\mathbf { r _ { n } }$ from the learnable NodePiece vocabulary $\textbf { V } \in$ $\mathbb { R } ^ { | V | \times \bar { d } }$ , and anchor distances $\mathbf { z _ { a _ { n } } }$ from $\mathbf { Z } \in \mathbb { R } ^ { ( d i a m e t e r ( G ) + 1 ) \times d }$ , we obtain a vectorized hash: + +$$ +h a s h ( n ) = \left[ \mathbf { a _ { n } } + \mathbf { z _ { a _ { n } } } , \mathbf { r _ { n } } \right] = \left[ \hat { \mathbf { a } } _ { \mathbf { n } } , \mathbf { r _ { n } } \right] \in \mathbb { R } ^ { ( k + m ) \times d } +$$ + +Although other operations are certainly possible, in this work, we use anchor distances as positional encodings of corresponding anchors and sum up their representations that helps to maintain the overall hash dimension of $( \bar { k } + m ) \times d$ . + +Finally, an encoder function $e n c : \mathbb { R } ^ { ( k + m ) \times d } \mathbb { R } ^ { d }$ is applied to the vectorized hash to bootstrap an embedding of $n$ . In our experiments, we probe two basic encoders: 1) MLP that takes as input a concatenated hash vector $\mathbb { R } ^ { 1 \times ( k + m ) d }$ projecting it down to $\mathbb { R } ^ { d }$ ; 2) Transformer encoder (Vaswani et al., 2017) with average pooling that takes as input an original sequence $\mathbb { R } ^ { ( k + m ) \times d }$ . While MLP is faster and better scales to graphs with more edges, Transformer is slower but requires less trainable parameters. As the two encoders were chosen to illustrate the general applicability of the whole approach, we leave a study of even more efficient and effective encoders for future work. + +While the nearest-neighbor hashing function has a greater number of collisions, its non-arbitrary mapping means that it is effectively permutation invariant. We show this in Proposition 1 through the framework of Janossy pooling and permutation sampling based SGD, $\pi$ -SGD (Murphy et al., 2019). A proof is provided in Appendix H. + +Proposition 1. The nearest-anchor encoder with $\textstyle { \binom { | A | } { k } }$ anchors and $| m |$ subsampled relations, can be considered a $\pi$ -SGD approximation of $( k + | m | )$ -ary Janossy pooling with a canonical ordering induced by the anchor distances. + +Janossy pooling with $\pi$ -SGD can be used to learn a permutation-invariant function from a broad class of permutation-sensitve functions such as MLPs (Murphy et al., 2019). The permutation-invariant nature of the nearest-neighbor encoding scheme combined with the lack of transductive features such as node-specific embeddings mean that NodePiece can be used for inductive learning tasks as well. + +With a fixed-size vocabulary $V _ { N P }$ , the overall complexity and parameter budget of downstream models are largely defined by the complexity of the encoder and its inductive biases. By design, the NodePiece smaller vocabulary - larger encoder framework is similar to various Transformer-based language models (Qiu et al., 2020) whose vocabulary size remains rather stable with the encoder being the most important part responsible for the final performance. + +# 4 EXPERIMENTS + +We design the experimental program not seeking to outperform the best existing approaches but to show the versatility of NodePiece on a variety of KG-related tasks: transductive, inductive, out-ofsample link prediction, and node classification (with relation prediction results in Appendix I). With this desiderata, we formulate the following research questions: RQ 1) Is it necessary to map each node to a unique vector for an acceptable performance on KG tasks?; RQ 2) What is the effect of hashing features?; RQ 3) Is there an optimal number of anchors per node, after which diminishing returns hit the performance? + +# 4.1 TRANSDUCTIVE LINK PREDICTION + +Setup. We run experiments on five KGs of different sizes (Appendix A.1) varying the total number of nodes from $\mathord { \sim } 1 5 \mathrm { K }$ to ${ \sim } 2 . 5 \mathbf { M }$ . As a baseline, we compare to RotatE (Sun et al., 2019) that remains one of state-of-the-art shallow embedding models for transductive link prediction tasks. To balance with NodePiece, RotatE operates on a graph with added inverse edges as well. We report MRR with Hits $@ 1 0$ in the filtered (Bordes et al., 2013) setting as evaluation metrics, and count parameters for all models. On larger KGs, we also compare to a smaller RotatE with a similar parameter budget. + +In this task, NodePiece is equipped with a 2-layer MLP encoder. For a fair comparison, we also adopt the RotatE scoring function as a link prediction decoder. As to the NodePiece configuration, we generally keep the number of anchors below $10 \%$ of total nodes in respective graphs. We select $1 \mathrm { k } / 2 0$ for FB15k-237 (i.e., total 1000 anchors and 20 anchors per tokenized node) with 15 unique outgoing relations in the relational context; 500/50 with 4 relations for WN18RR; $7 \mathrm { k } / 2 0$ with 6 relations for CoDEx-L; $1 0 \mathrm { k } / 2 0$ with 5 relations for YAGO 3-10. Other hyperparameters are listed in Appendix A. + +Discussion. Generally, the results suggest that a fixed-size NodePiece vocabulary of ${ < } 1 0 \%$ of nodes sustains $80 \%$ of Hits $@ 1 0$ compared to 10x larger best shallow models. Some performance loss is expected due to the compositional and compressive nature of entity tokenization. On smaller graphs (Table 2), parameter saving might not be well pronounced due to the overall small number of nodes to embed. Still, taking even as few as 500 nodes as anchors on WN18RR retains $90 \%$ of the best model performance. On bigger graphs (Table 3), parameter efficiency is more pronounced, i.e., on YAGO 3-10, a RotatE model of comparable size is $2 0 \mathrm { H i t s } @ 1 0$ points worse than a NodePiece-based one. This observation can be attributed to the fact the shrinking shallow models results in shrinking the embedding dimension of each node (20d for RotatE) which is inefficient on small parameter budgets. In contrast, a small fixed-size vocabulary allows for larger anchor embedding dimensions (100d for NodePiece with RotatE) since most of the parameter budget is defined by the encoder. + +Table 2: Transductive link prediction on smaller KGs. $\dagger$ results taken from (Sun et al., 2019). $| V |$ denotes vocabulary size (anchors $^ +$ relations), $\# \mathrm { P }$ is a total parameter count (millions). $\%$ denotes the Hits $@ 1 0$ ratio based on the strongest model. + +
FB15k-237WN18RR
IV#P (M)MRRH@10%|VI#P (M)MRRH@10%
RotatE15k + 0.5k290.338t0.533t10040k +22410.476t0.571t100
NodePiece + RotatE1k + 0.5k3.20.2560.42079500+224.40.4030.51590
- no rel. context1k + 0.5k20.2580.42580500+224.20.2660.46581
- no distances1k + 0.5k3.20.2540.42179500+224.40.3910.51089
- no anchors,rels only0+ 0.5k1.40.2040.355670+220.30.0110.0190.3
+ +Table 3: Transductive link prediction on bigger KGs. The same denotation as in Table 2. Second RotatE has a similar parameter budget as a NodePiece-based model. + +
CoDEx-LYAGO 3-10
IVI#P(M)MRRH@10%|VI#P (M)MRRH@10%
RotatE (500d)77k + 138770.2580.387100123k + 741230.495t0.670t100
RotatE (20d)77k + 1383.80.1960.32283123k + 744.80.1210.26239
NodePiece+RotatE7k +1383.60.1900.3138110k +744.10.2470.48873
- no rel. context7k +1383.10.2010.3328610k+743.70.2490.48272
- no distances7k +1383.60.1790.3027810k + 744.10.2500.49173
- no anchors,rels only0+1380.60.0630.121310+740.50.0250.0416
+ +We further study the effect of different anchor selection combinations (Fig. 2). On WN18RR, fewer anchors with fewer anchors per node $( | A | / k )$ yield relatively low accuracy but starting from 50/20 $( \sim 0 . 1 \%$ of $4 0 \mathrm { k }$ nodes in the graph) the Hits $@ 1 0$ performance starts to saturate. On FB15k-237, as few as 25 anchors already exhibit the signs of saturation where a further increase to 500 or 1000 anchors only marginally improves the performance. We hypothesize such a difference can be explained by graph density, e.g., WN18RR is a sparse graph with a diameter of 23 and average anchor distance of 6 hops; while FB15k-237 is a denser graph with an average anchor distance of 2-3. Hence, on a sparse graph with longer distances, it takes more anchors to properly encode a node. + +However, more precise predictions (e.g., Hits $@ 1$ ) reflected in the MRR metric (see Appendix E) still remain a challenging task for small vocabulary NodePiece setups, and bigger $| A | / k$ combinations alleviate this issue. We also observe that diminishing returns, which make further vocabulary increase less rewarding, start to appear from anchor set sizes of ${ \sim } 1 \%$ of total nodes. + +Ablations. In the ablation study, we measure the impact of relational context and anchor distances on link prediction (Table 2). Removing relational context and anchor distances does not tangibly affect the denser FB15k-237 data but does impair the accuracy on the sparser WN18RR. Pushing vocabulary sizes to the limit, we also investigate NodePiece behavior in the absence of anchors at all, i.e., when hashes are defined only by the relational context of size $m$ . Interestingly, this still yields fair performance on FB15k-237 with just 7 points $\mathrm { H i t s } @ 1 0$ drop, but drops to zero the WN18RR performance. The fact that node embeddings might not be at all necessary but relations are more important supports the recent findings of Teru et al. (2020) that relies only on relations seen in a small subgraph around a target node. However, at this point, it seems to be a virtue of graphs with a diverse set of unique relations. That is, FB15k-237 has $2 0 \mathrm { x }$ more unique relations than WN18RR and resulting hashes have more diverse combinations of relations which lead to more discriminative node representations. Additionally, we visualize anchor embedding projections in Appendix D. + +![](images/8880db12bcff05dfd8a2c907eaa983fcd983dd4d64311541239dc80378c547a9.jpg) +Figure 2: Combinations of total anchors $A$ and anchors per node. Denser FB15k-237 saturates faster on smaller $A$ while sparse WN18RR saturates at around 500 anchors. + +# 4.1.1 OGB WIKIKG 2 + +To measure the benefits of NodePiece on large-scale KGs, we run a link prediction experiment on OGB WikiKG 2 (Hu et al., 2020), a subset of Wikidata that consists of $2 . 5 \mathbf { M }$ nodes and 16M edges. NodePiece is configured to sample a vocabulary 20K anchor nodes $( < 1 \%$ of total nodes) where each node is represented with $k = 2 0$ nearest anchors and a relational context of size $m = 1 2$ , and we use a 2-layer MLP as a hash encoder (other hyperparameters as in Appendix A). Generally, such a NodePiece configuration can be paired with any link prediction decoder and we chose a non-parametric AutoSF (Zhang et al., 2020b) as one of the strongest decoders on this graph. Overall, the NodePiece $^ +$ AutoSF model has only 6.9M parameters, about $7 0 \times$ smaller than top shallow models. Compared to the best reported shallow approaches, the NodePiece-enabled model exhibits (cf. Table 4, averaged over 10 seeds) even better performance achieved with a orders of magnitude smaller parameter budget. We believe this result shows the effectiveness of NodePiece on large KGs with a dramatic parameter size reduction without significant performance losses. Ablations report the duality of a relational context, i.e., removing it from hashes leads to even higher MRR scores. On the other hand, the relational context alone with 0 learnable anchors still yields considerably better results than $1 0 0 0 \times$ larger shallow models RotatE and TransE. + +Table 4: Test MRR and parameter budget on OGB WikiKG 2. + +
Model#ParamsMRR
NP + AutoSF - rel. context6.9M 5.9M0.570 ±0.003 0.592 ±0.003
- anc.dists - no anchors6.9M 1.3M0.570 ±0.004 0.476 ±0.001
AutoSF500M
PairRE0.546 ±0.005
500M0.521 ±0.003
RotatE1250M0.433 ±0.002
TransE1250M0.426 ±0.003
+ +# 4.2 INDUCTIVE LINK PREDICTION + +We conduct a set of experiments on the inductive link prediction benchmark introduced by Teru et al. (2020) to measure the performance of NodePiece features in the extreme case when anchor nodes are not available and only relational context can be used to compose entity representations. + +Setup. The unique feature of this benchmark compared to other evaluated tasks is that training and inference graphs are disjoint, i.e., inference at validation and test time is performed on a completely new graph comprised of new entities, and link prediction involves only entities unseen during training. As inference graphs are disconnected from training ones, learning anchors from the training graph is useless, so node hashes are built only using the $m$ -sized relational context. On top of the obtained NodePiece features we then employ a relational message passing GNN, CompGCN (Vashishth et al., 2020), with RotatE (Sun et al., 2019) as a scoring function for triples. More details on the setup and best hyperparameters for NodePiece are presented in Appendix J. + +Baselines. We compare NodePiece $^ +$ CompGCN with two families of models applicable in the inductive setting, i.e., rule-based methods, Neural LP (Yang et al., 2017), DRUM (Sadeghian et al., 2019), RuleN (Meilicke et al., 2018), and GNNs: GraIL (Teru et al., 2020) and recently proposed Neural Bellman-Ford Nets (NBFNet) (Zhu et al., 2021). + +Discussion. Generally, the results confirm the trend identified previously: relation-only features are strong performers in dense relation-rich graphs. NodePiece features paired with CompGCN significantly improve over path-based methods where performance gap might reach 37 absolute Hits $@ 1 0$ points, e.g., in FB15k-237 V1 and NELL-995 V1. Comparing to GNNs, NodePiece $^ +$ + +Table 5: Inductive link prediction results, $\mathrm { H i t s } @ 1 0 .$ . Best results are in bold, second best are underlined. $\dagger$ results taken from Teru et al. (2020). NBFNet results taken from Zhu et al. (2021). + +
ClassMethodFB15k-237WN18RRNELL-995
V1V2V3V4v1V2V3V4V1V2V3V4
PathNeural LP †0.5290.5890.5290.5590.7440.6890.4620.6710.4080.7870.8270.806
DRUM+0.5290.5870.5290.5590.7440.6890.4620.6710.1940.7860.8270.806
RuleN†0.4980.7780.8770.8560.8090.7820.5340.7160.5350.8180.7730.614
GNNGraIL +0.6420.8180.8280.8930.8250.7870.5840.7340.5950.9330.9140.732
NBFNet0.8340.9490.9510.9600.9480.9050.8930.890-、--
NP + CompGCN0.8730.9390.9440.9490.8300.8860.7850.8070.8900.9010.9360.893
+ +Table 6: Node classification results. $| V |$ denotes vocabulary size (anchors $^ +$ relations), #P is a total parameter count (millions). +WD50K labeled) + + +
WD50K (5%labeled)WD50K(10%labeled)
|VI#P (M)ROC-AUCPRC-AUCHard AccROC-AUCPRC-AUCHard Acc
MLP46k + 1k4.10.5030.0160.0010.5100.0170.002
CompGCN46k + 1k4.40.8360.2800.1760.8340.2650.161
NodePiece +GNN50+1k0.750.9810.4430.5130.9810.4500.516
- no rel. context50+1k0.640.9820.4460.5340.9820.4490.530
- no distances50+1k0.740.9810.4480.5160.9810.4480.513
- no anchors,rels only0+ 1k0.540.9840.4530.5320.9840.4560.533
+ +CompGCN outperforms GraIL by a large margin in all (except one) experiments and is competitive to NBFNet on relation-rich FB15k-237 splits. As expected, NodePiece features are less efficient on sparse graphs (like WN18RR with few unique relations) but still outperform topology-based GraIL. + +# 4.3 NODE CLASSIFICATION + +Setup. Due to the lack of established node classification datasets on multi-relational KGs, we design a multi-class multi-label task based on a triple version of a recent WD50K (Galkin et al., 2020) extracted from Wikidata. The pre-processing steps are described in Appendix F, and the final graph consists of 46K nodes and 222K edges. The task belongs to the family of transductive (the whole graph is seen during training) semi-supervised (only a fraction of nodes are labeled) problems, where labels are 465 classes as seen in Wikidata. In a semi-supervised mode, we test the models on a graph with $5 \%$ and $10 \%$ of labeled nodes. Node features have to be learned as node embeddings. + +As baselines, we compare to a 2-layer MLP and CompGCN (Vashishth et al., 2020) in a full-batch mode which is one of the strongest GNN encoders for multi-relational KGs. Both baselines learn a full entity vocabulary. We report ROC-AUC, PRC-AUC, and Hard Accuracy metrics as commonly done in standard graph benchmarks like OGB (Hu et al., 2020). For PRC-AUC and Hard Accuracy, we binarize predicted logits using a threshold of 0.5. Hard Accuracy corresponds to the exact match of a predicted sparse 465-dimensional vector to a sparse 465-dimensional labels vector. + +NodePiece is configured to have only 50 anchors and use 10 nearest anchors per node with 5 unique relations in the relational context. The dimensionality of anchors and relations is the same as in the baseline CompGCN. Each epoch, we first materialize all entity embeddings through the NodePiece encoder and then send the materialized matrix to CompGCN with the class predictor. + +Discussion. Surprisingly, ${ \sim } 1 0 0 0 \mathrm { x }$ vocabulary reduction ratio (50 anchors against 46k for shallow models) greatly outperforms the baselines (Table 6). MLP, as expected, is not able to cope with the task producing random predictions. CompGCN, in turn, outperforms MLP demonstrating nonrandom outputs as seen by the ROC-AUC score of 0.836 and higher PRC-AUC and Hard Accuracy metrics. Still, a NodePiece-equipped CompGCN with 50 anchors reaches even higher ROC-AUC of 0.98 with considerable improvements along other metrics, i.e., $+ 1 6 - 1 9$ PRC-AUC points and $3 \mathbf { x }$ boost along the hardest accuracy metric. We attribute such a noticeable performance difference to better generalization capabilities of the NodePiece model. That is, a generalization gap between training and validation metrics of the NodePiece $^ +$ CompGCN is much smaller compared to the baselines who overfit rather heavily ( cf. the training curves in the Appendix G). The effect remains after increasing the number of labeled nodes to $10 \%$ . Even with 50 anchors, the overall performance is saturated as the further increase of the vocabulary size did not bring any improvements. + +Table 7: Out-of-sample link prediction. $\dagger$ results are taken from (Albooyeh et al., 2020). $| V |$ denotes vocabulary size (anchors $^ +$ relations), #P is a total parameter count (millions). + +
oFB15k-2370YAGO 3-10 (117k)
|VI#P (M)MRRH@10%|VI#P (M)MRRH@10%
oDistMult-ERAvg11k + 0.5k2.40.256t0.420t100117k + 7423.40OM0OM-
NodePiece +DistMult1k + 0.5k10.2060.3728810k+742.70.1330.261100
- no rel. context1k + 0.5k10.1730.3297810k +742.70.1250.24594
- no distances1k + 0.5k10.2080.3728810k + 742.70.1330.26099
- no anchors,rels only0+ 0.5k0.80.0690.127300+740.70.0150.0176
+ +Ablations. We probe setups where NodePiece hashes use only anchors or only relational context, and find they both deliver a similar performance. Following the previous experiments on dense graphs with lots of unique relations, it appears that node classification can be performed rather accurately based only on the node relational context which is captured by NodePiece hashes. + +# 4.4 OUT-OF-SAMPLE LINK PREDICTION + +Setup. In the out-of-sample setup, validation and test splits contain unseen entities that arrive with a few edges connected to the seen nodes. For this experiment, we use the out-of-sample FB15k237 split (oFB15k-237) as designed in Albooyeh et al. (2020). We do not employ their version of WN18RR as the split contains too many disconnected entities and components in the train graph. Instead, using the authors script, we sample a much bigger out-of-sample version of YAGO 3-10. + +As a baseline, we compare to oDistMult (Albooyeh et al., 2020) which aggregates embeddings of all seen neighboring nodes around the unseen one (akin to 1-layer message passing with mean aggregator). We adopt the same evaluation protocol - given an unseen node with its connecting edges, we mask one of the edges and predict its tail or head using the rest of the edges, repeating this procedure for each edge. We report filtered MRR and Hits $@ 1 0$ as main metrics. + +NodePiece enables traditional transductive-only models to perform inductive inference as both seen and unseen nodes are tokenized using the same vocabulary. For a smaller oFB15k-237 the NodePiece vocabulary has $1 \mathrm { k } / 2 0$ configuration with 15 relations, while in a bigger oYAGO 3-10 we use $1 0 \mathrm { k } / 2 0$ with 5 relations. For this task, we apply a transformer encoder instead of MLP. For a fair comparison, we use DistMult as a scoring function as well. + +Discussion. The results in Table 7 show that a simple NodePiece-based model retains ${ \sim } 9 0 \%$ of the baseline performance on oFB15k-237, but achieved faster and computationally inexpensive compared to oDistMult. Moreover, while oDistMult is tailored specifically for the out-of-sample task, we did not do any task-specific modifications to the NodePiece-enabled model as it is inductive by design. Furthermore, oDistMult is not able to scale to a bigger oYAGO 3-10 on a 256 GB RAM machine due to the out of memory crash. Conversely, a NodePiece-equipped model has the same computational requirements as in other tasks and converges rather quickly (40 epochs). Performed ablations underline the importance of having both anchors and relational context for tokenizing unseen entities. We elaborate more on possible inference strategies for transductive and inductive tasks in Appendix A.6. + +# 5 CONCLUSION + +In this paper, we have introduced NodePiece, a compositional approach for representing nodes in multi-relational graphs with a fixed-size vocabulary. Similar to subword units, NodePiece allows to tokenize every node as a combination of anchors and relations where the number of anchors can be $1 0 { - } 1 0 0 \times$ smaller than the total number of nodes. We show that in some tasks, node embeddings are not even necessary for getting an acceptable accuracy thanks to a rich set of relation types. Moreover, NodePiece is inductive by design and is able to tokenize unseen entities and perform downstream prediction tasks in the same fashion as on seen ones. + +Reproducibility Statement. The source code is openly available on GitHub. All hyperparameters and implementation details are presented in Appendix A. Information on the used datasets is presented in Table 8 and we provide more details on dataset construction for node classification and out-ofsample link prediction tasks in Appendix F. The proof for Proposition 1 is given in the Appendix H. + +Ethics Statement. As NodePiece is a general graph representation learning method, we do not foresee immediate ethical consequences pertaining to the method itself. + +Acknowledgements. The authors would like to thank Koustuv Sinha, Gaurav Maheshwari, and Priyansh Trivedi for insightful and valuable discussions at earlier stages of this work. We also thank anonymous reviewers for the helpful comments. This work is partially supported by the Canada CIFAR AI Chair Program and Samsung AI grant (held at Mila). We thank Mila and Compute Canada for access to computational resources. + +# REFERENCES + +Marjan Albooyeh, Rishab Goel, and Seyed Mehran Kazemi. Out-of-sample representation learning for knowledge graphs. 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In OOS-LP, Nodes also shows the amount of unseen nodes in validation/test. + +
DatasetTaskNodesRelationsEdgesTrainValidationTest
FB15k-237(Toutanova & Chen,2015)LP,RP14,505237310.079272,11517,52620.438
WN18RR (Dettmers et al.,2018)LP,RP40,5591192.58386.83528242924
CoDEx-Large (Safavi & Koutra,2020)LP77,95169612,437551,19330,62230,622
YAGO 3-10 (Mahdisoltani et al., 2015)LP,RP123,143371,089,0001,079,04049784982
OGB WikiKG 2 (Hu et al.,2020)LP2,500,60453517,137,18116,109,182429,456598,543
WD50KNC46,164526222.5634600 (N)4600 (N)4600 (N)
oFB15k-237 (Albooyeh et al.,2020)OOS-LP11k/1395/1395234292,173193,49044,60154,082
oYAGO 3-10OOS-LP117k/2960/2959371,086,416988,12447,11251,180
+ +Table 9: Inductive relation prediction dataset statistics. Facts denote the size of the input graph while queries denote the triples to be predicted. Training sets contain all queries as facts. Note that in validation and test we receive a new graph disjoint from the training one, and queries are sent against this new inference graph (hence the number of entities and facts for validation and test is the same). + +
DatasetRelationsTrainValidationTest
EntityQueryFactsEntityQueryFactEntityQueryFacts
FB15k-237v11832.0004,2454,2451,5002061,9931,5002051,993
v22033,0009,7399,7392,0004694,1452.0004784,145
v32184,00017,98617,9863.0008667,4063,0008657,406
Vv42225,00027,20327,2033,5001,41611,7143,5001,42411,714
WN18RRv192,7465,4105,4109221851,6189221881,618
v2106,95415,26215,2622,9234114,0112.9234414,011
v31112.07825,90125,9015,0845386,3275.0846056.327
V493,8617,9407,9407,2081,39412,3347,2081,42912.334
NELL-995v1143,1034,6874,687225101833225100833
v2882,5648,2198,2194,9374594,5864,9374764,586
v31424,64716,39316,3934,9218118.0484,9218098.048
v4772,0927,5467,5463,2947167,0733,2947317,073
+ +NodePiece is implemented in Python using igraph library (licensed under GNU GPL 2) for computing centrality measures and perform basic tokenization. Downstream tasks employ NodePiece in conjunction with PyTorch (Paszke et al., 2019) (BSD-style license), PyKEEN (Ali et al., 2021) (MIT License), and PyTorch-Geometric (Fey & Lenssen, 2019) (MIT License). We ran experiments on a machine with one RTX 8000 GPU and 64 GB RAM. The OGB WikiKG 2 experiments were executed on a single Tesla V100 16 GB VRAM and 64 GB RAM. All used datasets are available under open licenses. + +For all downstream tasks and datasets we employ the deterministic anchor selection strategy where $40 \%$ of the total number of anchors $| A |$ are nodes with top PPR scores, $40 \%$ are top degree nodes, and remaining $20 \%$ are selected randomly. All anchor sets are non-overlapping and disjoint, i.e., if some top degree nodes have already been selected with the PPR policy, they will be skipped in favor of next nodes in the sorted list. The choice for this strategy is motivated in Appendix C. + +# A.1 DATASETS + +Details on the datasets for transductive link prediction, out-of-sample link prediction, relation prediction and node classification are collected in Table 8. The inductive link prediction benchmark introduced by Teru et al. (2020) includes 3 graphs, FB15k-237, WN18RR, and NELL-995, each has 4 different splits that vary in the number of unique relations, number of nodes and triples at training and inference time. Full dataset statistics is provided in Table 9. FB15k-237 and most splits of NELL-995 can be considered as relation-rich graphs while WN18RR is a sparse graph with few relation types. + +# A.2 TRANSDUCTIVE LINK PREDICTION + +The optimizer is Adam for all experiments. As RotatE is a scoring function in the complex space, the reported embedding dimensions are a sum of real and imaginary dimensions, e.g., 1000d means that both real and imaginary vectors are 500d. + +Table 10: NodePiece hyperparameters for transductive link prediction experiments + +
ParameterFB15k-237WN18RRCoDEx-LYAGO 3-10OGB WikiKG 2
# Anchors, |A|100050070001000020000
# Anchors per node,k2050202020
Relational context, m1546512
Vocabulary dim, d200200200200200
Batch size512512256512512
Learning rate0.00050.00050.00050.000250.0001
Epochs400600120600300k (steps)
Encoder typeMLPMLPMLPMLPMLP
Encoder dim400400400400400
Encoder layers22222
Encoder dropout0.10.10.10.10.1
Loss functionBCENSSALBCENSSALNSSAL
Margin-15-5050
# Negative samples=20110128
Label smoothing0.410.311
Training time, hours75.5262311
+ +Table 11: RotatE hyperparameters for transductive link prediction experiments. CoDEx-L and YAGO 3-10 also list the hyperparameters (after the symbol / ) for smaller models (reported in Table 3) of the same parameter budget as NodePiece + +
ParameterFB15k-237WN18RRCoDEx-LYAGO 3-10
Embedding dim, d200010001000 /501000 /40
Batch size1024512512/5121024 /512
Loss functionNSSALNSSALNSSALNSSAL
Margin9625/924/15
# Negative samples2561024100 /100400/100
+ +# A.3 RELATION PREDICTION + +Configurations (Table 12) for the compared models are almost identical to those of the transductive link prediction experiment. We mostly reduce the number of epochs and negative samples as models converge faster on this task. + +# A.4 NODE CLASSIFICATION + +In this experiment (Table 13), NodePiece is used at the initial step to bootstrap a node embeddings matrix which is then sent to the CompGCN graph encoder. In contrast, CompGCN and MLP baselines use directly a trained node embedding matrix as their initial input. + +# A.5 OUT-OF-SAMPLE LINK PREDICTION + +The set of NodePiece hyperparameters (Table 14) is similar to the set of the transductive experiments except the scoring function (DistMult), encoder function (Transformer), and number of epochs as the model converges faster. We do not provide a setup for the baseline oDistMult on oYAGO 3-10 as the model was not able to pre-process the dataset on a machine with 256 GB RAM. Reported training times for NodePiece models exclude evaluation. Training times of the baseline oDistMult were not reported by its authors. + +Table 12: Hyperparameters for relation prediction experiments. The content is largely identical to Table 10, only changed parameters are listed + +
NodePiece+RotatERotatE
ParameterFB15k-237WN18RRYAGO 3-10FB15k-237WN18RRYAGO 3-10
Batch size512512512512512512
Epochs201507150150150
Loss functionNSSALNSSALNSSALNSSALNSSALNSSAL
Margin151225935
# Negative samples202020202020
Training time,min253025281057
+ +Table 13: Hyperparameters for node classification experiments + +
ParameterNodePiece + CompGCNCompGCNMLP
# Anchors, |Al50=
# Anchors per node, k10=
Relational context,m5=1
Vocabulary dim, d100100100
Batch size512512512
Learning rate0.0010.0010.001
Epochs400040004000
NodePiece encoderMLP--
NodePiece encoder dim2001
NodePiece encoder layers2
NodePiece encoder dropout0.1==
GNN (MLP) layers333
GNN (MLP) dropout0.50.50.5
Loss functionBCEBCEBCE
Label smoothing0.10.10.1
Training time, hours14226
+ +A.6 DEPLOYMENT IN REAL-WORLD DYNAMIC KNOWLEDGE GRAPHS + +NodePiece, on account of its compositional representation, can be applied to dynamic real-world knowledge graphs where nodes are added and removed over time. That is, training on a graph snapshot we can obtain the embeddings of new nodes without re-computing and updating representations of every other node in the graph. This is valuable in settings where there are latency requirements such as many online services. For example, if a user creates an account on a social media service and begins liking content (represented as a “like” edge in the social network graph between the user and the content), it would be desirable to make future content recommendations rapidly reflect this new data without waiting for the next batched retraining to update the users embedding. In order to reduce latency the entire embedding matrix can be materialized and cached ahead of time, updating embeddings as new nodes and edges are added. The parameter efficiency and compositionality of NodePeice means that for large real-world graphs NodePiece subsumes what would before have been a complex system of a large-scale embedding framework like Pytorch-BigGraph (Lerer et al., 2019), an OOV embedding method (e.g,. ERAvg) and a shallow embedding method (e.g., RotatE). + +Table 14: Hyperparameters for out-of-sample link prediction experiments. The content is largely identical to Table 10, only changed parameters are listed + +
ParameterNodePiece+DistMultoDistMult
oFB15k-237oYAGO 3-10
# Anchors, |A|100010000
# Anchors per node,k2020=
Relational context, m155=
Vocabulary dim,d200200200
Batch size2562561000
Learning rate0.00050.00050.01
Epochs40401000
NodePiece encoderTransformerTransformer-
NodePiece encoder dim512512■
NodePiece encoder layers22
NodePiece encoder dropout0.10.1
Loss functionSoftplusSoftplusSoftplus
# Negative samples551
Training time,hours28
+ +# B LIMITATIONS AND FUTURE WORK + +Our experimental results demonstrate the promise of using NodePeice to significantly reduce the parameter complexity of node embeddings. While it is difficult to prove, we also hypothesize that the parameters required by NodePeice to maintain the same level of performance (as the graph scales) increase sublinearly according to the size of the graph. The intuition for this is twofold. First, the number of unique anchor combinations of size $k$ that can be encoded increases according to $\textstyle { \binom { | A | } { k } }$ (i.e. $\mathcal { O } ( | A | ^ { k } ) )$ if randomly sampled — if the sampling is done via nearest neighbor anchor selection then the number of unique permutations is expected to increase polynomially. Second, increasing the size of the graph will only require sublinear increase in the number of anchors in order to maintain the same average node-anchor distance. Although proving causality is difficult, we believe that maintaining hashing uniqueness and node-anchor distances stable will be sufficient to maintain equivalent performance. + +# C ANCHOR SELECTION STRATEGIES + +Here, we provide more details as to anchor configurations ( $k$ nearest from total $A$ anchors) and anchor distances. Recall that there exist several ways to select the total set of anchors $A$ as stated in Section 3.1, i.e., random or centrality-measure based. Then, $k$ anchors per node can be chosen either as $k$ nearest (default NodePiece mode) or $k$ random anchors. Figure 3 depicts the effect of those strategies on the distribution of anchor distances (number of hops between a target node and its anchors). We use the configurations used in the main experiments, i.e., 1000 anchors and 20 anchors per node for FB15k-237, and 500 anchors with 50 anchors per node on WN18RR. + +First, we observe that PPR, degree, and mixed ( $40 \%$ PPR, $40 \%$ degree, $20 \%$ random) strategies generally skew the distribution towards smaller anchor distances compared to random strategies. This fact supports the hypothesis that deterministic strategies improve the chances to find an anchor in a closer $l$ -hop neighborhood of a target node. Second, varying the way of selecting $k$ anchors per node between nearest (left column) and random (right column), we also observe the skew of a distribution of anchor distances. + +Next, we fix the anchor selection strategy to the mix, fix the number of anchors per node (50 for WN18RR and 20 for FB15k-237), and vary a total number of anchors $A$ (50 to 1000 for WN18RR and 20 to 1000 for FB15k-237) along with the method of sampling $k$ anchors per node, i.e., nearest and random. Figure 4 shows that increasing the total number of anchors together with $k$ nearest anchors again skews the distribution of anchor distances towards smaller values and, hence, to higher probabilities of finding anchors in a closer neighborhood of a target node. + +![](images/057e081baea2ec49fd398141c91fb1aaf275530faf7f731a530bcf6c764d7cbc.jpg) +Figure 3: Distribution of anchor distances under various anchor selection strategies. Top-bottom: mixed, degree-based, PPR-based, random. For each dataset, left: selecting $k$ nearest anchors, right: $k$ random anchors. (a) Selecting a fixed 500/50 configuration on WN18RR; (b) Selecting a fixed 1000/20 configuration on FB15k237. Generally, all strategies except random ones skew the distributions towards nearest anchors. + +![](images/4d1a1400fc473a8e8f824414e521480429db385e107068b82d278aee5c8aac3c.jpg) +Figure 4: Distribution of anchor distances under the fixed mix anchor selection strategy when varying the total number of anchors $A$ (50–1000 for WN18RR, 20–1000 for $\mathbf { F b 1 5 k - 2 3 7 }$ ). For each dataset, left column - $k$ nearest anchors, right - $k$ random anchors. On both graphs, increase in $A$ with the nearest anchors always leads to shorter anchor distances. + +We would recommend using centrality-based strategies to select $A$ with $k$ nearest anchors per node if anchor distances and probability of finding anchors in a closer neighborhood are of higher importance. + +Finally, we fix the anchor selection strategy as mix, obtain nearest anchors per node, and under this setup study average anchor distances varying $k$ - the number of anchors per node in various combinations of total anchors $A$ . The results presented on Figure 5 suggest that sparser graphs (like WN18RR) benefit more from increasing the number of anchors $A$ , i.e., the delta between distances is much larger than that of dense FB15k-237. The difference in distances on Figure 5 might also explain the performance on Figure 8, i.e., generally, smaller $A / k$ configurations like 25/5 are inferior on sparser graphs but perform competitively on denser ones. + +![](images/5c83762ad70aceab22b28829e29e8a4217f2f51bb030fca7fc3d9b6ffed58232.jpg) +Figure 5: Average node-anchor distances when varying the total number of anchors $A$ from 25 to 1000 and $k$ nearest anchors per node from 5 to 50. Note that on a sparser WN18RR the gap between min and max values is much wider than of denser FB15k-237. Signs of saturation suggest that further increasing $A$ is not beneficial. + +# D EMBEDDING VISUALIZATIONS + +To further study learned representations of anchors and capabilities of the encoder, we build tSNE and UMAP projections from subsamples of FB15k-237 and WN18RR based on trained models from the transductive link prediction experiments (hyperparameters listed in Table A). + +For FB15k-237, we randomly sample 1000 entities (out of total 15K) and find their top-100 most common anchors. The anchor embeddings are extracted from the learned tensor while 1000 entity embeddings are obtained through the NodePiece encoder. Similarly for WN18RR, we sample 4000 entities (out of total 40K) keeping their top-100 most common anchors. As we use the RotatE decoder that assumes entities and anchors are modeled in a complex space, we visualize their real parts (e.g., first 100 dimensions out of 200). + +Recalling that link prediction performance of NodePiece $^ +$ RotatE retains $80 \%$ of the state of the art models performance, the results on Fig. 6 and Fig. 7 demonstrate that (1) NodePiece encoder is able to reconstruct clusters of similar entities; (2) anchors are well-scattered among communities. Albeit entity embeddings are built as a composition of $k$ anchors, it can be seen that all communities have "specialized" nearby anchors. On a higher level, common anchors tend to be well-scattered in the space. Less common anchors, as seen on FB15k-237 and Fig. 6, tend to group together. However, thanks to the non-linear nature of the NodePiece encoder, resulting entity embeddings still form different clusters and communities not concentrated around one point. We believe this is the effect of a compositional encoder and plan to investigate this phenomenon further. + +# E TRANSDUCTIVE LINK PREDICTION RESULTS: MRR + +In addition to Figure 2 that presents Hits $@ 1 0$ , we report variations of mean reciprocal rank (MRR) depending on combinations of $A$ and $k$ on Figure 8 from the same set of experiments. On sparser WN18RR, smaller $A / k$ combinations like 25/5 or 50/10 struggle with more precise predictions like + +![](images/1af61e1e10b58335056878a25e13f8e5248e69fd3b55b7b56b177dfd7f48b627.jpg) +Figure 6: tSNE (left) and UMAP (right) projections of 1000 encoded entities sampled randomly from FB $1 5 \mathrm { k }$ -237 and their top 100 most common anchors. + +![](images/8957842fac60ab0a93c114967577de387a5575380933e8686ecfe472945834f4.jpg) +Figure 7: tSNE (left) and UMAP (right) projections of 4000 encoded entities sampled randomly from WN18RR and their top 100 most common anchors. + +Hits $@ 1$ which is captured by low values of MRR. Starting from 500/10, the WN18RR performance starts to saturate. On the other hand, on denser FB15k-237, the difference between minimum and maximum MRR is less than 4 points, and performance exhibits signs of saturation already at 50/10. + +# F DATASETS CONSTRUCTION + +# F.1 NODE CLASSIFICATION: WD50K NC + +The original WD50K (Galkin et al., 2020) contains a triple-only KG version on which we base a new dataset for semi-supervised multi-class multi-label node classification. First, we remove all triples containing Wikidata properties P31 (instance of ) and P279 (subclass of ) as they already contain class information. We then remove nodes that became disconnected after removing those edges. Third, using SPARQL queries, for each remaining node in a graph, we extract a 3-hop class hierarchy of Wikidata classes and their superclasses. We only keep class labels that occur at least 50 times in the training set. Then, we sample $10 \%$ of nodes with labels for validation and $10 \%$ for test, and of remaining $80 \%$ we sample a set of nodes for the semi-supervised setup, i.e., we keep only $5 \%$ and + +![](images/5a8194a73210c9ecf515ecad0695a36c6cc94619e6c07cb6c8a88372a7ca6b56.jpg) +Figure 8: Combinations of total anchors $A$ and anchors per node. Denser FB15k-237 saturates faster on smaller $A$ while sparse WN18RR saturates at around 500 anchors. MRR metric captures all ranks. Note that performance gap on FB15k-237 is very small indicating that saturation has occurred already with small anchor configurations. + +$10 \%$ of those nodes. The resulting graph has 46k nodes, 526 distinct relation types, and 465 class labels. + +# F.2 OUF-OF-SAMPLE LINK PREDICTION: OYAGO 3-10 + +For sampling the out-of-sample version of a bigger YAGO 3-10 we largely follow the same original procedure described in Section 4 of (Albooyeh et al., 2020). We first merge the train, validation and test triples from the original dataset for transductive link prediction. Then, from all entities appearing in at least two triples, we randomly sample $5 \%$ of nodes to be the out-of-sample entities for validation and $5 \%$ for test. All triples containing the out-of-sample entities on subject or object positions are put into validation or test, respectively, as edges that connect an unseen entity with the seen graph. + +# G NODE CLASSIFICATION: TRAINING CURVES + +Figure 9 depicts train and validation values of Hard Accuracy and PRC-AUC metrics for all the compared models on WD50K NC with $5 \%$ of labeled nodes. The NodePiece model has only 50 total anchors with 10 nearest anchors per node, and 5 unique relation types in the relational context. The performance on the dataset with $10 \%$ of nodes is almost the same, so we report the charts only on $5 \%$ dataset. By the generalization gap we understand the delta between training and validation values. + +The MLP baseline quickly overfits but fails to generalize on the validation. The generalization gap of CompGCN is smaller compared to MLP but is still significant, i.e., validation performance is $2 { - } 3 \times$ smaller than train. Finally, the NodePiece-enabled model has the smallest generalization gaps, especially along the Hard Accuracy metric where the validation performance is very close to that of train. Similarly, the gap on PRC-AUC is smaller than 10 points. + +As shown in the ablation study in Table 6, it appears that explicit node embeddings do not contribute to the classification performance. Hence, the baseline models tend to be overparameterized where learnable node embeddings add noise, while the NodePiece model has only a few anchors (or no anchors at all when using only the relational context), much fewer parameters, and therefore generalizes better. This hypothesis also explains the observation that the node classification performance does not improve when increasing $A / k$ anchor configurations. + +# H PROOFS + +Proposition 2. The nearest-anchor encoder with $\binom { | A | } { k }$ anchors and $| m |$ subsampled relations, can be considered a $\pi$ -SGD approximation of $( k + | m | )$ -ary Janossy pooling with a canonical ordering induced by the anchor distances. + +![](images/0e115ed23e92c50662171a6d4f54a45b264834f88d611131451b294b96f29048.jpg) +Figure 9: Generalization gap on WD50K $5 \%$ labeled nodes). NodePiece-based model has observably smaller generalization gaps compared to the baselines. + +# Proof. + +We begin by providing the definition of Janossy pooling as it was presented in the original paper Murphy et al. (2019). + +Definition 1 (Janossy pooling). Let $\mathbb { H } ^ { \cup }$ be the union of all anchors and relations. Consider a function $\vec { f } : \mathbb { N } \times \mathbb { H } ^ { \cup } \times \mathbb { R } ^ { d } \stackrel { \left. } { \right. } \mathbb { F }$ on variable-length but finite sequences h, parameterized by $\pmb { \theta } ^ { ( f ) } \in \mathbb { R } ^ { d }$ , $d > 0$ . A permutation-invariant function $\overline { { \overline { { f } } } } : \mathbb { N } \times \mathbb { H } ^ { \cup } \times \mathbb { R } ^ { d } \mathbb { F }$ is the Janossy function associated with $\vec { \boldsymbol { f } }$ if + +$$ +\overline { { \overline { { f } } } } ( | { \bf h } | , { \bf h } ; \theta ^ { ( f ) } ) = \frac { 1 } { | { \bf h } | ! } \sum _ { \pi \in \Pi _ { | { \bf h } | } } \vec { f } ( | { \bf h } | , { \bf h } _ { \pi } ; \theta ^ { ( f ) } ) , +$$ + +where $\Pi _ { | \mathbf { h } | }$ is the set of all permutations of the integers 1 to $| \mathbf { h } |$ , and $\mathbf { h } _ { \pi }$ represents a particular reordering of the elements of sequence h according to $\pi \in \Pi _ { | \mathbf { h } | }$ . We refer the operation used to construct $\overline { { \overline { { f } } } }$ from $\vec { \boldsymbol { f } }$ as Janossy pooling. + +While Janossy pooling provides a simple approach to construct permutation-invariant functions from arbitrary permutation sensitive functions, it is computationally intractable due to the need to sum over all computations. Three general strategies proposed under this framework to overcome this combinatorial challenge: canonical orderings, $k$ -ary Janossy pooling, and $\pi$ -SGD approximations. + +A very effective way of reducing the complexity is to constrain the permutations to a canonical ordering that is independent of a specific adjacency matrix ordering over a given graph. More precisely, one defines as a function CANONICAL : $\mathbb { H } ^ { \cup } \mathbb { H } ^ { \cup }$ such that CANONICAL $( { \bf h } ) =$ $\mathrm { C A N O N I C A L } ( \mathbf { h } _ { \pi } ) \forall \pi \in \Pi _ { | \mathbf { h } | }$ and only considers functions $\vec { \boldsymbol { f } }$ based on the composition ${ \vec { f } } =$ $\mathrm { C A N O N I C A L } \circ \vec { f } ^ { \prime }$ (Murphy et al., 2019). In the case of NodePiece we are able to define this ordering for the anchors according to their distance to the target node. Assuming that the number of relations is fixed or grows at slow rate throughout the life-cycle of a graph we can define an arbitrary ordering for relations as a canonical ordering for the relational context. However, since anchors can be equidistant such a canonical ordering does fully satisfy permutation invariance. We propose a trivial relaxation of the original definition of canonical orderings simply requiring that an ordering greatly reduce the number of unique permutations since in practice an exact canonical ordering is rarely feasible. Specifically, $| \{ \mathrm { C A } \mathrm { \hat { N } O N I C A L } ( \mathbf { h } _ { \pi } ) \forall \pi \in \Pi _ { | \mathbf { h } | } \} | \ll | \{ ( \mathbf { h } _ { \pi } ) \forall \pi \in \Pi _ { | \mathbf { h } | } \} |$ . + +To further reduce the number of permutations we can truncate our ordered sequence $\mathbf { h }$ . This is known as $k$ -ary Janossy pooling pooling (Definition 2) and is implicitly performed by the NodePeice algorithm by varying the anchor per node parameter, $k$ , and the size of the relational context, $| m |$ . + +Definition 2 ( $k$ -ary Janossy pooling). Fix $k \in \mathbb N$ . For any sequence h, define $\downarrow _ { k }$ (h) as its projection to a length $k$ sequence; in particular, if $| { \bf h } | \ge k$ , we keep the first $k$ elements. Then, a $k$ -ary permutation-invariant Janossy function $\overline { { \overline { { f } } } }$ is given by + +Table 15: Relation prediction results. $| V |$ denotes vocabulary size (anchors $^ +$ relations). + +
FB15k-237WN18RRYAGO 3-10
|V|MRRH@10|V|MRRH@10|VIMRRH@10
RotatE15k +0.5k0.9050.97940k +220.7740.897123k +740.9090.992
NodePiece +RotatE1k + 0.5k0.8740.971500 +220.7610.98510k + 740.9510.997
- no rel. context1k + 0.5k0.8760.968500+220.5410.95810k + 740.8980.993
- no distances1k + 0.5k0.8770.970500+220.7460.97510k + 740.9430.997
- no anchors,rels only0+ 0.5k0.8730.9710+220.5450.9470+740.9510.998
+ +$$ +\overline { { \overline { { f } } } } ( | { \bf h } | , { \bf h } ; \pmb { \theta } ^ { ( f ) } ) = \frac { 1 } { | { \bf h } | ! } \sum _ { \pi \in \Pi _ { | { \bf h } | } } \vec { f } ( | { \bf h } | , \downarrow _ { k } ( { \bf h } _ { \pi } ) ; \pmb { \theta } ^ { ( f ) } ) . +$$ + +Since an imperfect truncated canonical ordering may still result in a potentially intractable number of permutations, we use permutation sampling also known as $\pi$ -SGD to learn arbitrary functions that approximate $( k + | m | )$ -ary Janossy pooling. This is done by randomly ordering anchors that are equidistant resulting in a uniform sampling of possible permutations during training and evaluation. For more details on the formal definition of $\pi$ -SGD we point the reader to the original paper (Murphy et al., 2019). + +# I RELATION PREDICTION + +Setup. We conduct the relation prediction experiment on the same FB15k-237, WN18RR, and YAGO 3-10 datasets. While link prediction deals with entities, the relation prediction model has to rank a correct relation given a (head, ?, tail) query. We report MRR and Hits $@ 1 0$ in the filtered setting as evaluation metrics. Similar to the link prediction configuration, we use NodePiece $^ { + 2 }$ -layer MLP and compare against RotatE of the same total parameter count. + +Discussion. The reported results (Table 15) demonstrate a competitive performance of NodePiecebased models with reduced vocabulary sizes bringing more than $9 7 \%$ Hits $@ 1 0$ across graphs of different sizes. In the case of WN18RR and YAGO 3-10, NodePiece models with fewer anchors even slightly improve the accuracy upon the shallow embedding baseline. The ablation study suggests that on dense graphs with a reasonable amount of unique relations having explicit learnable node embeddings might not be needed at all for this task. That is, we see that on $\mathrm { F B } 1 5 \mathrm { k } { - } 2 3 7$ and YAGO 3-10 the NodePiece hashes comprised only of the relational context deliver the same performance without any performance drop confirming the findings from the previous experiment. + +# J INDUCTIVE LINK PREDICTION + +Setup. Nodes in the inference graphs do not have any associated feature vectors which makes this benchmark very relevant for graph representation learning. Importantly, the set of relation types in the inference graphs is a subset of those seen in the training set. Since the relation embedding matrix can be learned on the training graph, we therefore have a uniform method for constructing node representations both on seen and unseen graphs. As an encoder we try MLP and Transformer. + +Evaluation Protocol. Following the original work (Teru et al., 2020), we employ a filtered setting and rank each triple against 50 random negative triples reporting the Hits $@ 1 0$ metric. This setup is motivated by computational complexity of GraIL at inference time, while NodePiece $^ +$ CompGCN is as fast in the inductive inference as in the transductive regime reported in other experiments. + +Discussion. For each KG, there are 4 splits of increasing size of train and inference nodes and edges. The empirical results on this spectrum of various sizes demonstrates interesting scalability properties of NodePiece in inductive settings. Without anchor nodes, the NodePiece vocabulary size is independent of the number of nodes and edges, depending only on the number of relation types. On dense relation-rich graphs such a vocabulary is enough to yield very competitive performance. + +Table 16: Hyperparameters for inductive link prediction experiments. Entries are shared among 4 splits of each graph if not particularly specified. V1 | V2 | V3 | V4 otherwise. + +
ParameterFB15k-237WN18RRNELL-995
# Anchors, |A|
# Anchors per node,k=
Relational context, m1241414134161416
Vocabulary dim, d100100100
Batch size512512512
Learning rate0.00010.00010.0001
Num negatives323232
Epochs2500 V1|2000 rest590|2000|210丨20002000
NodePiece encoderMLPMLPMLPIMLPIMLP|Trf
NodePiece encoder dim200200200
NodePiece encoder layers222
NodePiece encoder dropout0.10.10.1
CompGCNlayers3316161103141313
CompGCN attentionyesyesyes
CompGCN dropout0.10.10.210.110.110.1
Loss functionNSSALNSSALNSSAL
Margin25115115125151151512015|20|30120
Training time, hours2141161191118|41106151818
+ +# K UNIQUENESS OF NODE HASHES + +NodePiece represents nodes as a sequence of tokens and a natural question in this context is how unique such sequences can be in light of different anchor selection and tokenization strategies. + +Assuming the input graph is a single connected component, when sampling $| A |$ total anchors and selecting randomly $k$ anchors per node, the number of possible hash combinations is bounded by $\textstyle { \binom { | A | } { k } }$ . In this scenario, uniqueness of hashes is achieved by having this number bigger than the number $N$ of nodes in a graph, $\binom { | A | } { k } > N$ , and this happens with high probability for any reasonably large $| A |$ , eg, $\binom { 5 0 } { 2 0 }$ encodes about $4 . 7 \cdot 1 0 ^ { 1 3 }$ combinations which covers all existing public KGs combined. + +In the deterministic selection of nearest anchors per node we do not have such guarantees. Nevertheless, additional sequences of relational contexts and anchor distances help to obtain more unique hashes. Collisions are possible in highly regular graphs like Wordnet, but real-world KGs like Wikidata and DBpedia do not exhibit such a regular structure. Similar to homonyms whose meaning depends on the surrounding context in a sentence, we hypothesize that adding message passing layers (that can be seen as encoding of a neighboring context for a given node) on top of NodePiece hashes might further improve the diversity of node representations. That said, the future work research agenda might include proving tighter theoretical bounds on hashes uniqueness and developing new anchor sampling and tokenization strategies. \ No newline at end of file diff --git a/parse/dev/xMJWUKJnFSw/xMJWUKJnFSw_content_list.json b/parse/dev/xMJWUKJnFSw/xMJWUKJnFSw_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..e5ab38f9d3d4c05ad9625ad7e3dae509e0afcb8c --- /dev/null +++ b/parse/dev/xMJWUKJnFSw/xMJWUKJnFSw_content_list.json @@ -0,0 +1,2671 @@ +[ + { + "type": "text", + "text": "NODEPIECE: COMPOSITIONAL AND PARAMETEREFFICIENT REPRESENTATIONS OF LARGE KNOWLEDGE GRAPHS ", + "text_level": 1, + "bbox": [ + 174, + 98, + 828, + 171 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Mikhail Galkin, Etienne Denis, Jiapeng Wu and William L. Hamilton \nMila, McGill University \nMontreal, Canada \n{mikhail.galkin,deniseti,jiapeng.wu,hamilton}@mila.quebec ", + "bbox": [ + 184, + 194, + 743, + 252 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 287, + 544, + 303 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Conventional representation learning algorithms for knowledge graphs (KG) map each entity to a unique embedding vector. Such a shallow lookup results in a linear growth of memory consumption for storing the embedding matrix and incurs high computational costs when working with real-world KGs. Drawing parallels with subword tokenization commonly used in NLP, we explore the landscape of more parameter-efficient node embedding strategies. To this end, we propose NodePiece, an anchor-based approach to learn a fixed-size entity vocabulary. In NodePiece, a vocabulary of subword/sub-entity units is constructed from anchor nodes in a graph with known relation types. Given such a fixed-size vocabulary, it is possible to bootstrap an encoding and embedding for any entity, including those unseen during training. Experiments show that NodePiece performs competitively in node classification, link prediction, and relation prediction tasks while retaining less than $10 \\%$ of explicit nodes in a graph as anchors and often having $1 0 \\mathrm { x }$ fewer parameters. To this end, we show that a NodePiece-enabled model outperforms existing shallow models on a large OGB WikiKG 2 graph having $\\mathrm { \\sim } 7 0 \\mathrm { x }$ fewer parameters1. ", + "bbox": [ + 233, + 319, + 766, + 527 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 555, + 334, + 570 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Representation learning tasks on knowledge graphs (KGs) often require a parameterization of each unique atom in the graph with a vector or matrix. Traditionally, in multi-relational KGs such atoms constitute a set of all nodes $n \\in N$ (entities) and relations (edge types) $r \\in R$ (Nickel et al., 2016). Assuming parameterization with vectors, atoms are mapped to $d$ -dimensional vectors through shallow encoders $f _ { n } : n \\to \\mathbb { R } ^ { d }$ and $f _ { r } : r \\to \\mathbb { R } ^ { d }$ which scale linearly to the number of nodes and edge types2, i.e., having $O ( | N | )$ space complexity of the entity embedding matrix. Albeit efficient on small conventional benchmarking datasets based on Freebase (Toutanova & Chen, 2015) $( \\sim 1 5 \\mathrm { K }$ nodes) and WordNet (Dettmers et al., 2018) $- 4 0 \\mathrm { K }$ nodes), training on larger graphs (e.g., YAGO 3-10 (Mahdisoltani et al., 2015) of 120K nodes) becomes computationally challenging. Scaling it further up to larger subsets (Hu et al., 2020; Wang et al., 2021; Safavi & Koutra, 2020) of Wikidata (Vrandecic & Krötzsch, 2014) requires a top-level GPU or a CPU cluster as done in, e.g., PyTorch-BigGraph (Lerer et al., 2019) that maintains a $7 8 \\mathbf { M } \\times 2 0 0 d$ embeddings matrix in memory (we list sizes of current best performing models in Table 1). ", + "bbox": [ + 173, + 587, + 825, + 767 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Taking the perspective from NLP, shallow node encoding in KGs corresponds to shallow word embedding popularized with word2vec (Mikolov et al., 2013) and GloVe (Pennington et al., 2014) that learned a vocabulary of 400K-2M most frequent words, treating rarer ones as out-of-vocabulary (OOV). The OOV issue was resolved with the ability to build infinite combinations with a finite vocabulary enabled by subword units. Subword-powered algorithms such as fastText (Bojanowski et al., 2017), Byte-Pair Encoding (Sennrich et al., 2016), and WordPiece (Schuster & Nakajima, 2012) became a standard step in preprocessing pipelines of large language models and allowed to construct fixed-size token vocabularies, e.g., BERT (Devlin et al., 2019) contains $\\mathord { \\sim } 3 0 \\mathrm { K }$ tokens and ", + "bbox": [ + 174, + 773, + 825, + 886 + ], + "page_idx": 0 + }, + { + "type": "table", + "img_path": "images/0db7686f97f75745e0756661a3105252a040a4b1996f7f2a6b81624760cf2ef1.jpg", + "table_caption": [ + "Table 1: Node embedding sizes of state-of-the-art KG embedding models compared to BERT Large. Parameters of type float32 take 4 bytes each. FB15k-237, WN18RR, and YAGO3-10 models as reported in Sun et al. (2019), OGB WikiKG2 as in Zhang et al. (2020c), Wikidata 5M as in Wang et al. (2021), PBG Wikidata as in Lerer et al. (2019), and BERT Large as in Devlin et al. (2019). " + ], + "table_footnote": [], + "table_body": "
FB15k-237WN18RRYAGO3-10OGBWikiKG2Wikidata 5MPBG WikidataBERT Large
Vocabulary size15k40k120k2.5M5M78M30k
Embedding dim2000100010002005122001024
GPU RAM, GB0.120.150.461.879.6958.10.12
", + "bbox": [ + 176, + 167, + 825, + 227 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "GPT-2 (Radford et al., 2019) employs $\\mathord { \\sim } 5 0 \\mathrm { K }$ tokens. Importantly, relatively small input embedding matrices enabled investing the parameters budget into more efficient encoders (Kaplan et al., 2020). ", + "bbox": [ + 176, + 262, + 823, + 291 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Drawing inspiration from subword embeddings in NLP, we explore how similar strategies for tokenizing entities in large graphs can dramatically reduce parameter complexity, increase generalization, and naturally represent new unseen entities as using the same fixed vocabulary. To do so, tokenization has to rely on atoms akin to subword units and not the total set of nodes. ", + "bbox": [ + 174, + 297, + 825, + 353 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "To this end, we propose NodePiece, an anchor-based approach to learn a fixed-size vocabulary $V$ $( | V | \\ll | N | )$ of any connected multi-relational graph. In NodePiece, the set of atoms consists of anchors and all relation types that, together, allow to construct a combinatorial number of sequences from a limited atoms vocabulary. In contrast to shallow approaches, each node $n$ is first tokenized into a unique $h a s h ( n )$ of $k$ closest anchors and $m$ immediate relations. A key element to build a node embedding is a proper encoder function $e n c ( n ) : h a s h ( n ) \\mathbb { R } ^ { d }$ which can be designed leveraging inductive biases of an underlying graph or downstream tasks. Therefore, the overall parameter budget is now defined by a small fixed-size vocabulary of atoms and the complexity of the encoder function. ", + "bbox": [ + 173, + 359, + 825, + 472 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Our experimental findings suggest that a fixed-size NodePiece vocabulary paired with a simple encoder still yields competitive results on a variety of tasks including link prediction, node classification, and relation prediction. Furthermore, anchor-based hashing enables conventional embedding models to work in the inductive and out-of-sample scenarios when unseen entities arrive at inference time, which otherwise required tailored learning mechanisms. ", + "bbox": [ + 174, + 478, + 826, + 549 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 579, + 344, + 595 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Conventional KG embedding approaches. To the best of our knowledge, all contemporary embedding algorithms (Ji et al., 2020; Ali et al., 2020) for link prediction on KGs employ shallow embedding lookups mapping each entity to a unique embedding vector thus being linear $O ( | N | )$ to the total number of nodes $| N |$ and size of an embedding matrix. This holds for different embedding families, e.g., translational (Sun et al., 2019), tensor factorization (Lacroix et al., 2018), convolutional (Dettmers et al., 2018), and hyperbolic (Chami et al., 2020; Balazevic et al., 2019). The same applies to relation-aware graph neural network (GNN) encoders (Schlichtkrull et al., 2018; Vashishth et al., 2020) who still initialize each node with a learned embedding or feature vector before message passing. Furthermore, shallow encoding is also used in higher-order KG structures such as hypergraphs (Fatemi et al., 2020) and hyper-relational graphs (Rosso et al., 2020; Galkin et al., 2020). NodePiece can be used as a drop-in replacement of the embedding lookup with any of those models. ", + "bbox": [ + 173, + 617, + 826, + 770 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Distillation and compression. Several recent techniques for reducing memory footprint of embedding matrices follow successful applications of distilling large language models in NLP (Sanh et al., 2019), i.e., distillation (Wang et al., 2020; Zhu et al., 2020) into low-dimensional counterparts, and compression of trained matrices into discrete codes (Sachan, 2020). However, all of them require a full embedding matrix as input which we aim to avoid designing NodePiece. ", + "bbox": [ + 174, + 776, + 825, + 847 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Vocabulary reduction in recommender systems. Commonly, recommender systems operate on thousands of categorical features combined in sparse high-dimensional vectors.Recent approaches (Medini et al., 2021; Liang et al., 2021) employ anchor-based hashing techniques to factorize sparse feature vectors into dense embeddings. Contrary to those setups, we do not expect availability of feature vectors for arbitrary KGs and rather learn vocabulary embeddings from scratch. ", + "bbox": [ + 174, + 854, + 825, + 924 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/b2dd2102f8724e1a2b533377fadb48c87c7d1681b42eb24887a2b59d1548267d.jpg", + "image_caption": [ + "Figure 1: NodePiece tokenization strategy. Given three anchors $a _ { 1 } , a _ { 2 } , a _ { 3 }$ , a target node can be tokenized into a hash of top- $k$ closest anchors, their distances to the target node, and the relational context of outgoing relations from the target node. This hash sequence is passed through an injective encoder to obtain a unique embedding. Inverse relations are added to ensure connectivity. " + ], + "image_footnote": [], + "bbox": [ + 176, + 106, + 794, + 276 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Entity descriptions and language models. A recent line of work such as KG-BERT (Yao et al., 2019), MLMLM (Clouâtre et al., 2021), BLP (Daza et al., 2021) utilize entity descriptions passed through a language model (LM) encoder as entity embeddings suitable for link prediction. We would like to emphasize that such approaches are rather orthogonal to NodePiece. Textual features are mostly available in Wikipedia-derived KGs like Wikidata but are often missing in domain-specific graphs like social networks and product graphs. We therefore assume textual features are not available and rather learn node representations based on their spatial characteristics. Still, textual features can be easily added by concatenating NodePiece-encoded features with LM-produced features. ", + "bbox": [ + 174, + 377, + 825, + 488 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Out-of-sample representation learning. This task focuses on predictions involving previously unseen, or out-of-sample, entities that attach to a known KG with a few edges. These new edges are then utilized as a context to compute its embedding. Previous work (Wang et al., 2019; Hamaguchi et al., 2017; Albooyeh et al., 2020) proposed different neighborhood aggregation functions for this process or resorted to meta-learning (Chen et al., 2019; Baek et al., 2020; Zhang et al., 2020a). However, all of them follow the shallow embedding paradigm. Instead, NodePiece uses the new edges as a basis for anchor-based tokenization of new nodes in terms of an existing vocabulary. ", + "bbox": [ + 174, + 496, + 825, + 593 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 NODEPIECE VOCABULARY CONSTRUCTION ", + "text_level": 1, + "bbox": [ + 174, + 613, + 570, + 630 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Given a directed KG $G = ( N , E , R )$ consisting of $| N |$ nodes, $| E |$ edges, and $| R |$ relation types, our task is to reduce the original vocabulary size of $| N |$ nodes to a smaller, fixed-size vocabulary of node pieces akin to subword units. In this work, we represent node pieces through anchor nodes $a \\in A , A \\subset N$ , a pre-selected set of nodes in a graph following a deterministic or stochastic strategy. A full NodePiece vocabulary is then constructed from anchor nodes and relation types, i.e, $V = A + R$ . Note that in order to maintain reachability of each node and balance in- and out-degrees we enrich $G$ with inverse edges with inverse relation types, such that $| R | _ { i n v e r s e } = | R | _ { d i r e c t }$ and $| R | = | R | _ { d i r e c t } + | R | _ { i n v e r s e }$ . Using elements of the constructed vocabulary each node $n$ can be tokenized into $h a s h ( n )$ as a sequence of $k$ closest anchors, discrete anchor distances, and a relational context of $m$ immediate relations. Then, any encoder function $e n c ( n ) : h a s h ( n ) \\to \\mathbb { R } ^ { d }$ can be applied to embed the hash into a $d$ -dimensional vector. An intuition of the approach is presented in Fig. 1 with each step explained in more detail below. ", + "bbox": [ + 173, + 643, + 825, + 810 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.1 ANCHOR SELECTION ", + "text_level": 1, + "bbox": [ + 176, + 828, + 361, + 842 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Subword tokenization algorithms such as BPE (Sennrich et al., 2016) employ deterministic strategies to create tokens and construct a vocabulary, e.g., based on frequencies of co-occurring n-grams, such that more frequent words are tokenized with fewer subword units. On graphs, such strategies might employ centrality measures like degree centrality or Personalized PageRank (Page et al., 1999). However, in our preliminary experiments, we found random anchor selection to be as effective as centrality-based strategies. A choice for deterministic strategies might be justified when optimizing for certain task-specific topological characteristics, e.g., degree and PPR strategies indeed skew the distribution of shortest anchor distances towards smaller values thus increasing chances to find anchors in 2- or 3-hop neighborhood of any node (we provide more evidence for that in Appendix C). ", + "bbox": [ + 174, + 853, + 825, + 924 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 160 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.2 NODE TOKENIZATION ", + "text_level": 1, + "bbox": [ + 176, + 176, + 367, + 190 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Once the vocabulary $V = A + R$ is constructed, each node $n$ can be hashed (or tokenized) into a $h a s h ( n )$ using 1) $k$ nearest anchors and their discrete distances; 2) $m$ immediate outgoing relations from the relational context of $n$ . Since anchor nodes are concrete nodes in $G$ , they get hashed in the same way as other non-anchor nodes. ", + "bbox": [ + 174, + 202, + 825, + 257 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Anchors per node. Given $| A |$ anchor nodes, it is impractical to use all of them for encoding each node. Instead, we select $k$ anchors per node and describe two possible strategies for that, i.e., random and deterministic. The basic random strategy uniformly samples an unordered set of $k$ anchors from $A$ yielding $\\binom { | A | } { k }$ possible combinations. To avoid collisions when hashing the nodes, $| A |$ and $k$ are to be chosen according to the lower bound on possible combinations that is defined by the total number of nodes, e.g., $\\bar { \\binom { | A | } { k } } \\geq | N |$ . Note that running depth-first search (DFS) to random anchors at inference time is inefficient and, therefore, $h a s h ( n )$ of the random strategy has to be pre-computed. ", + "bbox": [ + 173, + 265, + 825, + 369 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "On the other hand, the deterministic strategy selects an ordered sequence of $k$ nearest anchors. Hence, the anchors can be obtained via breadth-first search (BFS) in the $l$ -hop neighborhood of $n$ at inference time (or pre-computed for speed reasons). However, the combinatorial bound is not applicable in this strategy and we need more discriminative signals to avoid hash collisions since nearby nodes will have similar anchors (we elaborate on the uniqueness issue in Appendix K). Such signals have to better ground anchors to the underlying graph structure, and we accomplish that using anchor distances3 and relational context described below. ", + "bbox": [ + 174, + 375, + 825, + 472 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "A node residing in a disconnected component is assigned with an auxiliary [DISCONNECTED] token or can be turned into an anchor. However, the majority of existing KGs are graphs with one large connected component with very few disconnected nodes, such that this effect is negligible. ", + "bbox": [ + 176, + 479, + 821, + 522 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Anchor Distances. Given a target node $n$ and an anchor $a _ { i }$ , we define anchor distance $z _ { a _ { i } } \\in$ [0; diameter $\\cdot ( G ) ]$ as an integer denoting the shortest path distance between $a _ { i }$ and $n$ in the original graph $G$ . Note that when tokenizing an anchor $a _ { j }$ with the deterministic strategy, the nearest anchor among top- $k$ is always $a _ { j }$ itself with distance 0. We then map each integer to a learnable $d$ -dimensional vector $f _ { z } : z _ { a _ { i } } \\to \\mathbb { R } ^ { d }$ akin to relative distance encoding scheme. ", + "bbox": [ + 174, + 529, + 825, + 601 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Relational Context. We also leverage the multi-relational nature of an underlying KG. Commonly4, the amount of unique edge types in $G$ is orders of magnitude smaller than the total number of nodes, i.e., $| R | \\ll | N |$ . This fact allows to include the entire $| R |$ in the NodePiece vocabulary $V _ { N P }$ and further featurize each node with a unique relational context. We construct a relational context of a node $n$ by randomly sampling a set of $m$ immediate unique outgoing relations starting from $n$ , i.e., $r c o n _ { n } = \\mathsf { \\bar { \\{ } } r _ { j } \\mathsf { \\} } ^ { m } \\subseteq \\mathsf { \\bar { \\mathcal { N } } } _ { r } ( n ) \\mathsf { \\bar { \\Psi } }$ where $\\mathcal { N } _ { r } ( n )$ denotes all outgoing relation types. Due to a non-uniform degree distribution, if $| { \\mathcal { N } } _ { r } ( n ) | < m$ , we add auxiliary [PAD] tokens to complete $r c o n _ { n }$ to size $m$ . ", + "bbox": [ + 173, + 607, + 826, + 705 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.3 ENCODING ", + "text_level": 1, + "bbox": [ + 174, + 720, + 290, + 736 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "At this step, a node $n$ is tokenized into a sequence of $k$ anchors, their $k$ respective distances, and relational context of size $m$ : ", + "bbox": [ + 173, + 747, + 825, + 775 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/53e1b6ffaf17e42668d4d3004092532643a326cd6dc8c91215856528f2315b00.jpg", + "text": "$$\nh a s h ( n ) = \\Bigl [ \\{ a _ { i } \\} ^ { k } , \\{ z _ { a _ { i } } \\} ^ { k } , \\{ r _ { j } \\} ^ { m } \\Bigr ]\n$$", + "text_format": "latex", + "bbox": [ + 380, + 776, + 617, + 804 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Taking anchors vectors $\\mathbf { a _ { n } }$ and relation vectors $\\mathbf { r _ { n } }$ from the learnable NodePiece vocabulary $\\textbf { V } \\in$ $\\mathbb { R } ^ { | V | \\times \\bar { d } }$ , and anchor distances $\\mathbf { z _ { a _ { n } } }$ from $\\mathbf { Z } \\in \\mathbb { R } ^ { ( d i a m e t e r ( G ) + 1 ) \\times d }$ , we obtain a vectorized hash: ", + "bbox": [ + 173, + 814, + 823, + 844 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/25acec3b121509d8ee1c5be12ca102e20b95262988485e761adbac1b0945af17.jpg", + "text": "$$\nh a s h ( n ) = \\left[ \\mathbf { a _ { n } } + \\mathbf { z _ { a _ { n } } } , \\mathbf { r _ { n } } \\right] = \\left[ \\hat { \\mathbf { a } } _ { \\mathbf { n } } , \\mathbf { r _ { n } } \\right] \\in \\mathbb { R } ^ { ( k + m ) \\times d }\n$$", + "text_format": "latex", + "bbox": [ + 325, + 848, + 673, + 876 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Although other operations are certainly possible, in this work, we use anchor distances as positional encodings of corresponding anchors and sum up their representations that helps to maintain the overall hash dimension of $( \\bar { k } + m ) \\times d$ . ", + "bbox": [ + 176, + 103, + 823, + 146 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Finally, an encoder function $e n c : \\mathbb { R } ^ { ( k + m ) \\times d } \\mathbb { R } ^ { d }$ is applied to the vectorized hash to bootstrap an embedding of $n$ . In our experiments, we probe two basic encoders: 1) MLP that takes as input a concatenated hash vector $\\mathbb { R } ^ { 1 \\times ( k + m ) d }$ projecting it down to $\\mathbb { R } ^ { d }$ ; 2) Transformer encoder (Vaswani et al., 2017) with average pooling that takes as input an original sequence $\\mathbb { R } ^ { ( k + m ) \\times d }$ . While MLP is faster and better scales to graphs with more edges, Transformer is slower but requires less trainable parameters. As the two encoders were chosen to illustrate the general applicability of the whole approach, we leave a study of even more efficient and effective encoders for future work. ", + "bbox": [ + 174, + 152, + 825, + 255 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "While the nearest-neighbor hashing function has a greater number of collisions, its non-arbitrary mapping means that it is effectively permutation invariant. We show this in Proposition 1 through the framework of Janossy pooling and permutation sampling based SGD, $\\pi$ -SGD (Murphy et al., 2019). A proof is provided in Appendix H. ", + "bbox": [ + 176, + 261, + 825, + 318 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Proposition 1. The nearest-anchor encoder with $\\textstyle { \\binom { | A | } { k } }$ anchors and $| m |$ subsampled relations, can be considered a $\\pi$ -SGD approximation of $( k + | m | )$ -ary Janossy pooling with a canonical ordering induced by the anchor distances. ", + "bbox": [ + 174, + 320, + 825, + 366 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Janossy pooling with $\\pi$ -SGD can be used to learn a permutation-invariant function from a broad class of permutation-sensitve functions such as MLPs (Murphy et al., 2019). The permutation-invariant nature of the nearest-neighbor encoding scheme combined with the lack of transductive features such as node-specific embeddings mean that NodePiece can be used for inductive learning tasks as well. ", + "bbox": [ + 174, + 376, + 825, + 433 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "With a fixed-size vocabulary $V _ { N P }$ , the overall complexity and parameter budget of downstream models are largely defined by the complexity of the encoder and its inductive biases. By design, the NodePiece smaller vocabulary - larger encoder framework is similar to various Transformer-based language models (Qiu et al., 2020) whose vocabulary size remains rather stable with the encoder being the most important part responsible for the final performance. ", + "bbox": [ + 174, + 439, + 825, + 510 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 530, + 326, + 545 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We design the experimental program not seeking to outperform the best existing approaches but to show the versatility of NodePiece on a variety of KG-related tasks: transductive, inductive, out-ofsample link prediction, and node classification (with relation prediction results in Appendix I). With this desiderata, we formulate the following research questions: RQ 1) Is it necessary to map each node to a unique vector for an acceptable performance on KG tasks?; RQ 2) What is the effect of hashing features?; RQ 3) Is there an optimal number of anchors per node, after which diminishing returns hit the performance? ", + "bbox": [ + 174, + 560, + 825, + 659 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4.1 TRANSDUCTIVE LINK PREDICTION", + "text_level": 1, + "bbox": [ + 176, + 674, + 457, + 688 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Setup. We run experiments on five KGs of different sizes (Appendix A.1) varying the total number of nodes from $\\mathord { \\sim } 1 5 \\mathrm { K }$ to ${ \\sim } 2 . 5 \\mathbf { M }$ . As a baseline, we compare to RotatE (Sun et al., 2019) that remains one of state-of-the-art shallow embedding models for transductive link prediction tasks. To balance with NodePiece, RotatE operates on a graph with added inverse edges as well. We report MRR with Hits $@ 1 0$ in the filtered (Bordes et al., 2013) setting as evaluation metrics, and count parameters for all models. On larger KGs, we also compare to a smaller RotatE with a similar parameter budget. ", + "bbox": [ + 173, + 700, + 825, + 784 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In this task, NodePiece is equipped with a 2-layer MLP encoder. For a fair comparison, we also adopt the RotatE scoring function as a link prediction decoder. As to the NodePiece configuration, we generally keep the number of anchors below $10 \\%$ of total nodes in respective graphs. We select $1 \\mathrm { k } / 2 0$ for FB15k-237 (i.e., total 1000 anchors and 20 anchors per tokenized node) with 15 unique outgoing relations in the relational context; 500/50 with 4 relations for WN18RR; $7 \\mathrm { k } / 2 0$ with 6 relations for CoDEx-L; $1 0 \\mathrm { k } / 2 0$ with 5 relations for YAGO 3-10. Other hyperparameters are listed in Appendix A. ", + "bbox": [ + 174, + 790, + 825, + 875 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Discussion. Generally, the results suggest that a fixed-size NodePiece vocabulary of ${ < } 1 0 \\%$ of nodes sustains $80 \\%$ of Hits $@ 1 0$ compared to 10x larger best shallow models. Some performance loss is expected due to the compositional and compressive nature of entity tokenization. On smaller graphs (Table 2), parameter saving might not be well pronounced due to the overall small number of nodes to embed. Still, taking even as few as 500 nodes as anchors on WN18RR retains $90 \\%$ of the best model performance. On bigger graphs (Table 3), parameter efficiency is more pronounced, i.e., on YAGO 3-10, a RotatE model of comparable size is $2 0 \\mathrm { H i t s } @ 1 0$ points worse than a NodePiece-based one. This observation can be attributed to the fact the shrinking shallow models results in shrinking the embedding dimension of each node (20d for RotatE) which is inefficient on small parameter budgets. In contrast, a small fixed-size vocabulary allows for larger anchor embedding dimensions (100d for NodePiece with RotatE) since most of the parameter budget is defined by the encoder. ", + "bbox": [ + 176, + 882, + 823, + 924 + ], + "page_idx": 4 + }, + { + "type": "table", + "img_path": "images/66b4734cb79ba1f9abda07094b916b95ba912ecc19fe62733588476f0e008b89.jpg", + "table_caption": [ + "Table 2: Transductive link prediction on smaller KGs. $\\dagger$ results taken from (Sun et al., 2019). $| V |$ denotes vocabulary size (anchors $^ +$ relations), $\\# \\mathrm { P }$ is a total parameter count (millions). $\\%$ denotes the Hits $@ 1 0$ ratio based on the strongest model. " + ], + "table_footnote": [], + "table_body": "
FB15k-237WN18RR
IV#P (M)MRRH@10%|VI#P (M)MRRH@10%
RotatE15k + 0.5k290.338t0.533t10040k +22410.476t0.571t100
NodePiece + RotatE1k + 0.5k3.20.2560.42079500+224.40.4030.51590
- no rel. context1k + 0.5k20.2580.42580500+224.20.2660.46581
- no distances1k + 0.5k3.20.2540.42179500+224.40.3910.51089
- no anchors,rels only0+ 0.5k1.40.2040.355670+220.30.0110.0190.3
", + "bbox": [ + 176, + 154, + 823, + 256 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/f90816be4d022610ef598e181c863797306c91412226cf8126b7b575de0a482e.jpg", + "table_caption": [ + "Table 3: Transductive link prediction on bigger KGs. The same denotation as in Table 2. Second RotatE has a similar parameter budget as a NodePiece-based model. " + ], + "table_footnote": [], + "table_body": "
CoDEx-LYAGO 3-10
IVI#P(M)MRRH@10%|VI#P (M)MRRH@10%
RotatE (500d)77k + 138770.2580.387100123k + 741230.495t0.670t100
RotatE (20d)77k + 1383.80.1960.32283123k + 744.80.1210.26239
NodePiece+RotatE7k +1383.60.1900.3138110k +744.10.2470.48873
- no rel. context7k +1383.10.2010.3328610k+743.70.2490.48272
- no distances7k +1383.60.1790.3027810k + 744.10.2500.49173
- no anchors,rels only0+1380.60.0630.121310+740.50.0250.0416
", + "bbox": [ + 176, + 313, + 823, + 426 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 457, + 825, + 569 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We further study the effect of different anchor selection combinations (Fig. 2). On WN18RR, fewer anchors with fewer anchors per node $( | A | / k )$ yield relatively low accuracy but starting from 50/20 $( \\sim 0 . 1 \\%$ of $4 0 \\mathrm { k }$ nodes in the graph) the Hits $@ 1 0$ performance starts to saturate. On FB15k-237, as few as 25 anchors already exhibit the signs of saturation where a further increase to 500 or 1000 anchors only marginally improves the performance. We hypothesize such a difference can be explained by graph density, e.g., WN18RR is a sparse graph with a diameter of 23 and average anchor distance of 6 hops; while FB15k-237 is a denser graph with an average anchor distance of 2-3. Hence, on a sparse graph with longer distances, it takes more anchors to properly encode a node. ", + "bbox": [ + 173, + 575, + 825, + 688 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "However, more precise predictions (e.g., Hits $@ 1$ ) reflected in the MRR metric (see Appendix E) still remain a challenging task for small vocabulary NodePiece setups, and bigger $| A | / k$ combinations alleviate this issue. We also observe that diminishing returns, which make further vocabulary increase less rewarding, start to appear from anchor set sizes of ${ \\sim } 1 \\%$ of total nodes. ", + "bbox": [ + 174, + 694, + 825, + 750 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Ablations. In the ablation study, we measure the impact of relational context and anchor distances on link prediction (Table 2). Removing relational context and anchor distances does not tangibly affect the denser FB15k-237 data but does impair the accuracy on the sparser WN18RR. Pushing vocabulary sizes to the limit, we also investigate NodePiece behavior in the absence of anchors at all, i.e., when hashes are defined only by the relational context of size $m$ . Interestingly, this still yields fair performance on FB15k-237 with just 7 points $\\mathrm { H i t s } @ 1 0$ drop, but drops to zero the WN18RR performance. The fact that node embeddings might not be at all necessary but relations are more important supports the recent findings of Teru et al. (2020) that relies only on relations seen in a small subgraph around a target node. However, at this point, it seems to be a virtue of graphs with a diverse set of unique relations. That is, FB15k-237 has $2 0 \\mathrm { x }$ more unique relations than WN18RR and resulting hashes have more diverse combinations of relations which lead to more discriminative node representations. Additionally, we visualize anchor embedding projections in Appendix D. ", + "bbox": [ + 174, + 757, + 825, + 924 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/8880db12bcff05dfd8a2c907eaa983fcd983dd4d64311541239dc80378c547a9.jpg", + "image_caption": [ + "Figure 2: Combinations of total anchors $A$ and anchors per node. Denser FB15k-237 saturates faster on smaller $A$ while sparse WN18RR saturates at around 500 anchors. " + ], + "image_footnote": [], + "bbox": [ + 289, + 99, + 707, + 224 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.1.1 OGB WIKIKG 2 ", + "text_level": 1, + "bbox": [ + 176, + 292, + 344, + 306 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "To measure the benefits of NodePiece on large-scale KGs, we run a link prediction experiment on OGB WikiKG 2 (Hu et al., 2020), a subset of Wikidata that consists of $2 . 5 \\mathbf { M }$ nodes and 16M edges. NodePiece is configured to sample a vocabulary 20K anchor nodes $( < 1 \\%$ of total nodes) where each node is represented with $k = 2 0$ nearest anchors and a relational context of size $m = 1 2$ , and we use a 2-layer MLP as a hash encoder (other hyperparameters as in Appendix A). Generally, such a NodePiece configuration can be paired with any link prediction decoder and we chose a non-parametric AutoSF (Zhang et al., 2020b) as one of the strongest decoders on this graph. Overall, the NodePiece $^ +$ AutoSF model has only 6.9M parameters, about $7 0 \\times$ smaller than top shallow models. Compared to the best reported shallow approaches, the NodePiece-enabled model exhibits (cf. Table 4, averaged over 10 seeds) even better performance achieved with a orders of magnitude smaller parameter budget. We believe this result shows the effectiveness of NodePiece on large KGs with a dramatic parameter size reduction without significant performance losses. Ablations report the duality of a relational context, i.e., removing it from hashes leads to even higher MRR scores. On the other hand, the relational context alone with 0 learnable anchors still yields considerably better results than $1 0 0 0 \\times$ larger shallow models RotatE and TransE. ", + "bbox": [ + 176, + 316, + 576, + 511 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/05715b3a672e5eb72b8f70a039245ddd1a562a1a41eadecf8cec11e5c6e4a596.jpg", + "table_caption": [ + "Table 4: Test MRR and parameter budget on OGB WikiKG 2. " + ], + "table_footnote": [], + "table_body": "
Model#ParamsMRR
NP + AutoSF - rel. context6.9M 5.9M0.570 ±0.003 0.592 ±0.003
- anc.dists - no anchors6.9M 1.3M0.570 ±0.004 0.476 ±0.001
AutoSF500M
PairRE0.546 ±0.005
500M0.521 ±0.003
RotatE1250M0.433 ±0.002
TransE1250M0.426 ±0.003
", + "bbox": [ + 591, + 373, + 820, + 491 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 511, + 825, + 593 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.2 INDUCTIVE LINK PREDICTION ", + "text_level": 1, + "bbox": [ + 176, + 611, + 424, + 626 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We conduct a set of experiments on the inductive link prediction benchmark introduced by Teru et al. (2020) to measure the performance of NodePiece features in the extreme case when anchor nodes are not available and only relational context can be used to compose entity representations. ", + "bbox": [ + 174, + 637, + 825, + 679 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Setup. The unique feature of this benchmark compared to other evaluated tasks is that training and inference graphs are disjoint, i.e., inference at validation and test time is performed on a completely new graph comprised of new entities, and link prediction involves only entities unseen during training. As inference graphs are disconnected from training ones, learning anchors from the training graph is useless, so node hashes are built only using the $m$ -sized relational context. On top of the obtained NodePiece features we then employ a relational message passing GNN, CompGCN (Vashishth et al., 2020), with RotatE (Sun et al., 2019) as a scoring function for triples. More details on the setup and best hyperparameters for NodePiece are presented in Appendix J. ", + "bbox": [ + 173, + 686, + 825, + 799 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Baselines. We compare NodePiece $^ +$ CompGCN with two families of models applicable in the inductive setting, i.e., rule-based methods, Neural LP (Yang et al., 2017), DRUM (Sadeghian et al., 2019), RuleN (Meilicke et al., 2018), and GNNs: GraIL (Teru et al., 2020) and recently proposed Neural Bellman-Ford Nets (NBFNet) (Zhu et al., 2021). ", + "bbox": [ + 174, + 804, + 825, + 861 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Discussion. Generally, the results confirm the trend identified previously: relation-only features are strong performers in dense relation-rich graphs. NodePiece features paired with CompGCN significantly improve over path-based methods where performance gap might reach 37 absolute Hits $@ 1 0$ points, e.g., in FB15k-237 V1 and NELL-995 V1. Comparing to GNNs, NodePiece $^ +$ ", + "bbox": [ + 174, + 867, + 825, + 924 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/f48ec65dc1b122e598bade987edd3f0babf946e3daa4bfe57adda5b94d6d061e.jpg", + "table_caption": [ + "Table 5: Inductive link prediction results, $\\mathrm { H i t s } @ 1 0 .$ . Best results are in bold, second best are underlined. $\\dagger$ results taken from Teru et al. (2020). NBFNet results taken from Zhu et al. (2021). " + ], + "table_footnote": [], + "table_body": "
ClassMethodFB15k-237WN18RRNELL-995
V1V2V3V4v1V2V3V4V1V2V3V4
PathNeural LP †0.5290.5890.5290.5590.7440.6890.4620.6710.4080.7870.8270.806
DRUM+0.5290.5870.5290.5590.7440.6890.4620.6710.1940.7860.8270.806
RuleN†0.4980.7780.8770.8560.8090.7820.5340.7160.5350.8180.7730.614
GNNGraIL +0.6420.8180.8280.8930.8250.7870.5840.7340.5950.9330.9140.732
NBFNet0.8340.9490.9510.9600.9480.9050.8930.890-、--
NP + CompGCN0.8730.9390.9440.9490.8300.8860.7850.8070.8900.9010.9360.893
", + "bbox": [ + 176, + 140, + 821, + 247 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/dbbe4b623335aa6f2ceb8d498783c88373c9358c87ce5cb140612386961ccd3b.jpg", + "table_caption": [ + "Table 6: Node classification results. $| V |$ denotes vocabulary size (anchors $^ +$ relations), #P is a total parameter count (millions). ", + "WD50K labeled)", + "" + ], + "table_footnote": [], + "table_body": "
WD50K (5%labeled)WD50K(10%labeled)
|VI#P (M)ROC-AUCPRC-AUCHard AccROC-AUCPRC-AUCHard Acc
MLP46k + 1k4.10.5030.0160.0010.5100.0170.002
CompGCN46k + 1k4.40.8360.2800.1760.8340.2650.161
NodePiece +GNN50+1k0.750.9810.4430.5130.9810.4500.516
- no rel. context50+1k0.640.9820.4460.5340.9820.4490.530
- no distances50+1k0.740.9810.4480.5160.9810.4480.513
- no anchors,rels only0+ 1k0.540.9840.4530.5320.9840.4560.533
", + "bbox": [ + 176, + 306, + 823, + 410 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "CompGCN outperforms GraIL by a large margin in all (except one) experiments and is competitive to NBFNet on relation-rich FB15k-237 splits. As expected, NodePiece features are less efficient on sparse graphs (like WN18RR with few unique relations) but still outperform topology-based GraIL. ", + "bbox": [ + 174, + 438, + 825, + 481 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.3 NODE CLASSIFICATION ", + "text_level": 1, + "bbox": [ + 176, + 500, + 377, + 513 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Setup. Due to the lack of established node classification datasets on multi-relational KGs, we design a multi-class multi-label task based on a triple version of a recent WD50K (Galkin et al., 2020) extracted from Wikidata. The pre-processing steps are described in Appendix F, and the final graph consists of 46K nodes and 222K edges. The task belongs to the family of transductive (the whole graph is seen during training) semi-supervised (only a fraction of nodes are labeled) problems, where labels are 465 classes as seen in Wikidata. In a semi-supervised mode, we test the models on a graph with $5 \\%$ and $10 \\%$ of labeled nodes. Node features have to be learned as node embeddings. ", + "bbox": [ + 173, + 526, + 825, + 625 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "As baselines, we compare to a 2-layer MLP and CompGCN (Vashishth et al., 2020) in a full-batch mode which is one of the strongest GNN encoders for multi-relational KGs. Both baselines learn a full entity vocabulary. We report ROC-AUC, PRC-AUC, and Hard Accuracy metrics as commonly done in standard graph benchmarks like OGB (Hu et al., 2020). For PRC-AUC and Hard Accuracy, we binarize predicted logits using a threshold of 0.5. Hard Accuracy corresponds to the exact match of a predicted sparse 465-dimensional vector to a sparse 465-dimensional labels vector. ", + "bbox": [ + 174, + 631, + 825, + 715 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "NodePiece is configured to have only 50 anchors and use 10 nearest anchors per node with 5 unique relations in the relational context. The dimensionality of anchors and relations is the same as in the baseline CompGCN. Each epoch, we first materialize all entity embeddings through the NodePiece encoder and then send the materialized matrix to CompGCN with the class predictor. ", + "bbox": [ + 174, + 722, + 825, + 777 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Discussion. Surprisingly, ${ \\sim } 1 0 0 0 \\mathrm { x }$ vocabulary reduction ratio (50 anchors against 46k for shallow models) greatly outperforms the baselines (Table 6). MLP, as expected, is not able to cope with the task producing random predictions. CompGCN, in turn, outperforms MLP demonstrating nonrandom outputs as seen by the ROC-AUC score of 0.836 and higher PRC-AUC and Hard Accuracy metrics. Still, a NodePiece-equipped CompGCN with 50 anchors reaches even higher ROC-AUC of 0.98 with considerable improvements along other metrics, i.e., $+ 1 6 - 1 9$ PRC-AUC points and $3 \\mathbf { x }$ boost along the hardest accuracy metric. We attribute such a noticeable performance difference to better generalization capabilities of the NodePiece model. That is, a generalization gap between training and validation metrics of the NodePiece $^ +$ CompGCN is much smaller compared to the baselines who overfit rather heavily ( cf. the training curves in the Appendix G). The effect remains after increasing the number of labeled nodes to $10 \\%$ . Even with 50 anchors, the overall performance is saturated as the further increase of the vocabulary size did not bring any improvements. ", + "bbox": [ + 173, + 785, + 825, + 924 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/07c20997c24e66437e8a429e50adb5f396b8a72b678b575f26f49a5e4438c66e.jpg", + "table_caption": [ + "Table 7: Out-of-sample link prediction. $\\dagger$ results are taken from (Albooyeh et al., 2020). $| V |$ denotes vocabulary size (anchors $^ +$ relations), #P is a total parameter count (millions). " + ], + "table_footnote": [], + "table_body": "
oFB15k-2370YAGO 3-10 (117k)
|VI#P (M)MRRH@10%|VI#P (M)MRRH@10%
oDistMult-ERAvg11k + 0.5k2.40.256t0.420t100117k + 7423.40OM0OM-
NodePiece +DistMult1k + 0.5k10.2060.3728810k+742.70.1330.261100
- no rel. context1k + 0.5k10.1730.3297810k +742.70.1250.24594
- no distances1k + 0.5k10.2080.3728810k + 742.70.1330.26099
- no anchors,rels only0+ 0.5k0.80.0690.127300+740.70.0150.0176
", + "bbox": [ + 176, + 140, + 823, + 242 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 268, + 823, + 296 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Ablations. We probe setups where NodePiece hashes use only anchors or only relational context, and find they both deliver a similar performance. Following the previous experiments on dense graphs with lots of unique relations, it appears that node classification can be performed rather accurately based only on the node relational context which is captured by NodePiece hashes. ", + "bbox": [ + 174, + 303, + 825, + 359 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "4.4 OUT-OF-SAMPLE LINK PREDICTION", + "text_level": 1, + "bbox": [ + 176, + 377, + 464, + 391 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Setup. In the out-of-sample setup, validation and test splits contain unseen entities that arrive with a few edges connected to the seen nodes. For this experiment, we use the out-of-sample FB15k237 split (oFB15k-237) as designed in Albooyeh et al. (2020). We do not employ their version of WN18RR as the split contains too many disconnected entities and components in the train graph. Instead, using the authors script, we sample a much bigger out-of-sample version of YAGO 3-10. ", + "bbox": [ + 174, + 404, + 825, + 473 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "As a baseline, we compare to oDistMult (Albooyeh et al., 2020) which aggregates embeddings of all seen neighboring nodes around the unseen one (akin to 1-layer message passing with mean aggregator). We adopt the same evaluation protocol - given an unseen node with its connecting edges, we mask one of the edges and predict its tail or head using the rest of the edges, repeating this procedure for each edge. We report filtered MRR and Hits $@ 1 0$ as main metrics. ", + "bbox": [ + 174, + 479, + 825, + 550 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "NodePiece enables traditional transductive-only models to perform inductive inference as both seen and unseen nodes are tokenized using the same vocabulary. For a smaller oFB15k-237 the NodePiece vocabulary has $1 \\mathrm { k } / 2 0$ configuration with 15 relations, while in a bigger oYAGO 3-10 we use $1 0 \\mathrm { k } / 2 0$ with 5 relations. For this task, we apply a transformer encoder instead of MLP. For a fair comparison, we use DistMult as a scoring function as well. ", + "bbox": [ + 174, + 558, + 825, + 626 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Discussion. The results in Table 7 show that a simple NodePiece-based model retains ${ \\sim } 9 0 \\%$ of the baseline performance on oFB15k-237, but achieved faster and computationally inexpensive compared to oDistMult. Moreover, while oDistMult is tailored specifically for the out-of-sample task, we did not do any task-specific modifications to the NodePiece-enabled model as it is inductive by design. Furthermore, oDistMult is not able to scale to a bigger oYAGO 3-10 on a 256 GB RAM machine due to the out of memory crash. Conversely, a NodePiece-equipped model has the same computational requirements as in other tasks and converges rather quickly (40 epochs). Performed ablations underline the importance of having both anchors and relational context for tokenizing unseen entities. We elaborate more on possible inference strategies for transductive and inductive tasks in Appendix A.6. ", + "bbox": [ + 174, + 633, + 825, + 772 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "5 CONCLUSION ", + "text_level": 1, + "bbox": [ + 176, + 794, + 318, + 810 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In this paper, we have introduced NodePiece, a compositional approach for representing nodes in multi-relational graphs with a fixed-size vocabulary. Similar to subword units, NodePiece allows to tokenize every node as a combination of anchors and relations where the number of anchors can be $1 0 { - } 1 0 0 \\times$ smaller than the total number of nodes. We show that in some tasks, node embeddings are not even necessary for getting an acceptable accuracy thanks to a rich set of relation types. Moreover, NodePiece is inductive by design and is able to tokenize unseen entities and perform downstream prediction tasks in the same fashion as on seen ones. ", + "bbox": [ + 174, + 825, + 825, + 922 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Reproducibility Statement. The source code is openly available on GitHub. All hyperparameters and implementation details are presented in Appendix A. Information on the used datasets is presented in Table 8 and we provide more details on dataset construction for node classification and out-ofsample link prediction tasks in Appendix F. The proof for Proposition 1 is given in the Appendix H. ", + "bbox": [ + 174, + 103, + 825, + 160 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Ethics Statement. As NodePiece is a general graph representation learning method, we do not foresee immediate ethical consequences pertaining to the method itself. ", + "bbox": [ + 169, + 166, + 823, + 195 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Acknowledgements. The authors would like to thank Koustuv Sinha, Gaurav Maheshwari, and Priyansh Trivedi for insightful and valuable discussions at earlier stages of this work. We also thank anonymous reviewers for the helpful comments. This work is partially supported by the Canada CIFAR AI Chair Program and Samsung AI grant (held at Mila). We thank Mila and Compute Canada for access to computational resources. ", + "bbox": [ + 174, + 202, + 825, + 271 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 174, + 292, + 285, + 308 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Marjan Albooyeh, Rishab Goel, and Seyed Mehran Kazemi. Out-of-sample representation learning for knowledge graphs. In Findings of the Association for Computational Linguistics: EMNLP 2020, pp. 2657–2666, Online, November 2020. Association for Computational Linguistics. doi: 10.18653/v1/2020.findings-emnlp.241. 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", + "bbox": [ + 174, + 502, + 823, + 545 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Zhaocheng Zhu, Zuobai Zhang, Louis-Pascal Xhonneux, and Jian Tang. Neural bellman-ford networks: A general graph neural network framework for link prediction. In Neural Information Processing Systems, NeurIPS, 2021. ", + "bbox": [ + 174, + 554, + 823, + 597 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "A IMPLEMENTATION & HYPERPARAMETERS ", + "text_level": 1, + "bbox": [ + 176, + 102, + 558, + 118 + ], + "page_idx": 14 + }, + { + "type": "table", + "img_path": "images/f43f8d7bee8219b96632870d8f2619cdb8613a5f2996686a000c2b723d302399.jpg", + "table_caption": [ + "Table 8: Dataset statistics. LP - link prediction, RP - relation prediction, NC - node classification, OOS - out-of-sample. In OOS-LP, Nodes also shows the amount of unseen nodes in validation/test. " + ], + "table_footnote": [], + "table_body": "
DatasetTaskNodesRelationsEdgesTrainValidationTest
FB15k-237(Toutanova & Chen,2015)LP,RP14,505237310.079272,11517,52620.438
WN18RR (Dettmers et al.,2018)LP,RP40,5591192.58386.83528242924
CoDEx-Large (Safavi & Koutra,2020)LP77,95169612,437551,19330,62230,622
YAGO 3-10 (Mahdisoltani et al., 2015)LP,RP123,143371,089,0001,079,04049784982
OGB WikiKG 2 (Hu et al.,2020)LP2,500,60453517,137,18116,109,182429,456598,543
WD50KNC46,164526222.5634600 (N)4600 (N)4600 (N)
oFB15k-237 (Albooyeh et al.,2020)OOS-LP11k/1395/1395234292,173193,49044,60154,082
oYAGO 3-10OOS-LP117k/2960/2959371,086,416988,12447,11251,180
", + "bbox": [ + 176, + 188, + 823, + 301 + ], + "page_idx": 14 + }, + { + "type": "table", + "img_path": "images/e4b2ed7a3ffbc403405ada034a7db9e773a1891ece205c03384a48302e1a112e.jpg", + "table_caption": [ + "Table 9: Inductive relation prediction dataset statistics. Facts denote the size of the input graph while queries denote the triples to be predicted. Training sets contain all queries as facts. Note that in validation and test we receive a new graph disjoint from the training one, and queries are sent against this new inference graph (hence the number of entities and facts for validation and test is the same). " + ], + "table_footnote": [], + "table_body": "
DatasetRelationsTrainValidationTest
EntityQueryFactsEntityQueryFactEntityQueryFacts
FB15k-237v11832.0004,2454,2451,5002061,9931,5002051,993
v22033,0009,7399,7392,0004694,1452.0004784,145
v32184,00017,98617,9863.0008667,4063,0008657,406
Vv42225,00027,20327,2033,5001,41611,7143,5001,42411,714
WN18RRv192,7465,4105,4109221851,6189221881,618
v2106,95415,26215,2622,9234114,0112.9234414,011
v31112.07825,90125,9015,0845386,3275.0846056.327
V493,8617,9407,9407,2081,39412,3347,2081,42912.334
NELL-995v1143,1034,6874,687225101833225100833
v2882,5648,2198,2194,9374594,5864,9374764,586
v31424,64716,39316,3934,9218118.0484,9218098.048
v4772,0927,5467,5463,2947167,0733,2947317,073
", + "bbox": [ + 174, + 402, + 823, + 589 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "NodePiece is implemented in Python using igraph library (licensed under GNU GPL 2) for computing centrality measures and perform basic tokenization. Downstream tasks employ NodePiece in conjunction with PyTorch (Paszke et al., 2019) (BSD-style license), PyKEEN (Ali et al., 2021) (MIT License), and PyTorch-Geometric (Fey & Lenssen, 2019) (MIT License). We ran experiments on a machine with one RTX 8000 GPU and 64 GB RAM. The OGB WikiKG 2 experiments were executed on a single Tesla V100 16 GB VRAM and 64 GB RAM. All used datasets are available under open licenses. ", + "bbox": [ + 173, + 612, + 825, + 710 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "For all downstream tasks and datasets we employ the deterministic anchor selection strategy where $40 \\%$ of the total number of anchors $| A |$ are nodes with top PPR scores, $40 \\%$ are top degree nodes, and remaining $20 \\%$ are selected randomly. All anchor sets are non-overlapping and disjoint, i.e., if some top degree nodes have already been selected with the PPR policy, they will be skipped in favor of next nodes in the sorted list. The choice for this strategy is motivated in Appendix C. ", + "bbox": [ + 174, + 717, + 825, + 786 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "A.1 DATASETS ", + "text_level": 1, + "bbox": [ + 176, + 811, + 290, + 825 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Details on the datasets for transductive link prediction, out-of-sample link prediction, relation prediction and node classification are collected in Table 8. The inductive link prediction benchmark introduced by Teru et al. (2020) includes 3 graphs, FB15k-237, WN18RR, and NELL-995, each has 4 different splits that vary in the number of unique relations, number of nodes and triples at training and inference time. Full dataset statistics is provided in Table 9. FB15k-237 and most splits of NELL-995 can be considered as relation-rich graphs while WN18RR is a sparse graph with few relation types. ", + "bbox": [ + 174, + 840, + 825, + 924 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "A.2 TRANSDUCTIVE LINK PREDICTION", + "text_level": 1, + "bbox": [ + 176, + 103, + 462, + 118 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "The optimizer is Adam for all experiments. As RotatE is a scoring function in the complex space, the reported embedding dimensions are a sum of real and imaginary dimensions, e.g., 1000d means that both real and imaginary vectors are 500d. ", + "bbox": [ + 174, + 128, + 825, + 171 + ], + "page_idx": 15 + }, + { + "type": "table", + "img_path": "images/72533995f6259bb7362949f59129ef5b26f5bf3bc567f3c765f49da247faab96.jpg", + "table_caption": [ + "Table 10: NodePiece hyperparameters for transductive link prediction experiments " + ], + "table_footnote": [], + "table_body": "
ParameterFB15k-237WN18RRCoDEx-LYAGO 3-10OGB WikiKG 2
# Anchors, |A|100050070001000020000
# Anchors per node,k2050202020
Relational context, m1546512
Vocabulary dim, d200200200200200
Batch size512512256512512
Learning rate0.00050.00050.00050.000250.0001
Epochs400600120600300k (steps)
Encoder typeMLPMLPMLPMLPMLP
Encoder dim400400400400400
Encoder layers22222
Encoder dropout0.10.10.10.10.1
Loss functionBCENSSALBCENSSALNSSAL
Margin-15-5050
# Negative samples=20110128
Label smoothing0.410.311
Training time, hours75.5262311
", + "bbox": [ + 179, + 210, + 818, + 470 + ], + "page_idx": 15 + }, + { + "type": "table", + "img_path": "images/f229425c5d0a78b96ed1bbb7ffe4a74b4bb0a1f0245d57c7a445a7da74ced102.jpg", + "table_caption": [ + "Table 11: RotatE hyperparameters for transductive link prediction experiments. CoDEx-L and YAGO 3-10 also list the hyperparameters (after the symbol / ) for smaller models (reported in Table 3) of the same parameter budget as NodePiece " + ], + "table_footnote": [], + "table_body": "
ParameterFB15k-237WN18RRCoDEx-LYAGO 3-10
Embedding dim, d200010001000 /501000 /40
Batch size1024512512/5121024 /512
Loss functionNSSALNSSALNSSALNSSAL
Margin9625/924/15
# Negative samples2561024100 /100400/100
", + "bbox": [ + 253, + 545, + 745, + 646 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "A.3 RELATION PREDICTION ", + "text_level": 1, + "bbox": [ + 174, + 670, + 383, + 684 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Configurations (Table 12) for the compared models are almost identical to those of the transductive link prediction experiment. We mostly reduce the number of epochs and negative samples as models converge faster on this task. ", + "bbox": [ + 176, + 696, + 823, + 738 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "A.4 NODE CLASSIFICATION ", + "text_level": 1, + "bbox": [ + 176, + 756, + 382, + 770 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "In this experiment (Table 13), NodePiece is used at the initial step to bootstrap a node embeddings matrix which is then sent to the CompGCN graph encoder. In contrast, CompGCN and MLP baselines use directly a trained node embedding matrix as their initial input. ", + "bbox": [ + 174, + 781, + 825, + 824 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "A.5 OUT-OF-SAMPLE LINK PREDICTION", + "text_level": 1, + "bbox": [ + 176, + 842, + 467, + 856 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "The set of NodePiece hyperparameters (Table 14) is similar to the set of the transductive experiments except the scoring function (DistMult), encoder function (Transformer), and number of epochs as the model converges faster. We do not provide a setup for the baseline oDistMult on oYAGO 3-10 as the model was not able to pre-process the dataset on a machine with 256 GB RAM. Reported training times for NodePiece models exclude evaluation. Training times of the baseline oDistMult were not reported by its authors. ", + "bbox": [ + 174, + 867, + 823, + 924 + ], + "page_idx": 15 + }, + { + "type": "table", + "img_path": "images/06ad9e51338b1e263a90d332b91766b7b98d9a903d1b973369405d573ef2ba4c.jpg", + "table_caption": [ + "Table 12: Hyperparameters for relation prediction experiments. The content is largely identical to Table 10, only changed parameters are listed " + ], + "table_footnote": [], + "table_body": "
NodePiece+RotatERotatE
ParameterFB15k-237WN18RRYAGO 3-10FB15k-237WN18RRYAGO 3-10
Batch size512512512512512512
Epochs201507150150150
Loss functionNSSALNSSALNSSALNSSALNSSALNSSAL
Margin151225935
# Negative samples202020202020
Training time,min253025281057
", + "bbox": [ + 176, + 140, + 821, + 273 + ], + "page_idx": 16 + }, + { + "type": "table", + "img_path": "images/3532eef79cae102c3518e4b75def6d2e27bc162245d39cf556c83d446c8eedbd.jpg", + "table_caption": [ + "Table 13: Hyperparameters for node classification experiments " + ], + "table_footnote": [], + "table_body": "
ParameterNodePiece + CompGCNCompGCNMLP
# Anchors, |Al50=
# Anchors per node, k10=
Relational context,m5=1
Vocabulary dim, d100100100
Batch size512512512
Learning rate0.0010.0010.001
Epochs400040004000
NodePiece encoderMLP--
NodePiece encoder dim2001
NodePiece encoder layers2
NodePiece encoder dropout0.1==
GNN (MLP) layers333
GNN (MLP) dropout0.50.50.5
Loss functionBCEBCEBCE
Label smoothing0.10.10.1
Training time, hours14226
", + "bbox": [ + 238, + 332, + 759, + 592 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "", + "bbox": [ + 176, + 638, + 823, + 666 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "A.6 DEPLOYMENT IN REAL-WORLD DYNAMIC KNOWLEDGE GRAPHS ", + "bbox": [ + 174, + 707, + 669, + 722 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "NodePiece, on account of its compositional representation, can be applied to dynamic real-world knowledge graphs where nodes are added and removed over time. That is, training on a graph snapshot we can obtain the embeddings of new nodes without re-computing and updating representations of every other node in the graph. This is valuable in settings where there are latency requirements such as many online services. For example, if a user creates an account on a social media service and begins liking content (represented as a “like” edge in the social network graph between the user and the content), it would be desirable to make future content recommendations rapidly reflect this new data without waiting for the next batched retraining to update the users embedding. In order to reduce latency the entire embedding matrix can be materialized and cached ahead of time, updating embeddings as new nodes and edges are added. The parameter efficiency and compositionality of NodePeice means that for large real-world graphs NodePiece subsumes what would before have been a complex system of a large-scale embedding framework like Pytorch-BigGraph (Lerer et al., 2019), an OOV embedding method (e.g,. ERAvg) and a shallow embedding method (e.g., RotatE). ", + "bbox": [ + 174, + 743, + 825, + 924 + ], + "page_idx": 16 + }, + { + "type": "table", + "img_path": "images/9c7a2bc4cb2aeacc29c9bdc69d70ca3599d377ea0fa0a3c9a8f3b403f092be84.jpg", + "table_caption": [ + "Table 14: Hyperparameters for out-of-sample link prediction experiments. The content is largely identical to Table 10, only changed parameters are listed " + ], + "table_footnote": [], + "table_body": "
ParameterNodePiece+DistMultoDistMult
oFB15k-237oYAGO 3-10
# Anchors, |A|100010000
# Anchors per node,k2020=
Relational context, m155=
Vocabulary dim,d200200200
Batch size2562561000
Learning rate0.00050.00050.01
Epochs40401000
NodePiece encoderTransformerTransformer-
NodePiece encoder dim512512■
NodePiece encoder layers22
NodePiece encoder dropout0.10.1
Loss functionSoftplusSoftplusSoftplus
# Negative samples551
Training time,hours28
", + "bbox": [ + 250, + 138, + 746, + 392 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "B LIMITATIONS AND FUTURE WORK ", + "text_level": 1, + "bbox": [ + 174, + 421, + 495, + 438 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Our experimental results demonstrate the promise of using NodePeice to significantly reduce the parameter complexity of node embeddings. While it is difficult to prove, we also hypothesize that the parameters required by NodePeice to maintain the same level of performance (as the graph scales) increase sublinearly according to the size of the graph. The intuition for this is twofold. First, the number of unique anchor combinations of size $k$ that can be encoded increases according to $\\textstyle { \\binom { | A | } { k } }$ (i.e. $\\mathcal { O } ( | A | ^ { k } ) )$ if randomly sampled — if the sampling is done via nearest neighbor anchor selection then the number of unique permutations is expected to increase polynomially. Second, increasing the size of the graph will only require sublinear increase in the number of anchors in order to maintain the same average node-anchor distance. Although proving causality is difficult, we believe that maintaining hashing uniqueness and node-anchor distances stable will be sufficient to maintain equivalent performance. ", + "bbox": [ + 174, + 457, + 825, + 612 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "C ANCHOR SELECTION STRATEGIES ", + "text_level": 1, + "bbox": [ + 174, + 637, + 493, + 654 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Here, we provide more details as to anchor configurations ( $k$ nearest from total $A$ anchors) and anchor distances. Recall that there exist several ways to select the total set of anchors $A$ as stated in Section 3.1, i.e., random or centrality-measure based. Then, $k$ anchors per node can be chosen either as $k$ nearest (default NodePiece mode) or $k$ random anchors. Figure 3 depicts the effect of those strategies on the distribution of anchor distances (number of hops between a target node and its anchors). We use the configurations used in the main experiments, i.e., 1000 anchors and 20 anchors per node for FB15k-237, and 500 anchors with 50 anchors per node on WN18RR. ", + "bbox": [ + 174, + 672, + 825, + 770 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "First, we observe that PPR, degree, and mixed ( $40 \\%$ PPR, $40 \\%$ degree, $20 \\%$ random) strategies generally skew the distribution towards smaller anchor distances compared to random strategies. This fact supports the hypothesis that deterministic strategies improve the chances to find an anchor in a closer $l$ -hop neighborhood of a target node. Second, varying the way of selecting $k$ anchors per node between nearest (left column) and random (right column), we also observe the skew of a distribution of anchor distances. ", + "bbox": [ + 174, + 776, + 823, + 861 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Next, we fix the anchor selection strategy to the mix, fix the number of anchors per node (50 for WN18RR and 20 for FB15k-237), and vary a total number of anchors $A$ (50 to 1000 for WN18RR and 20 to 1000 for FB15k-237) along with the method of sampling $k$ anchors per node, i.e., nearest and random. Figure 4 shows that increasing the total number of anchors together with $k$ nearest anchors again skews the distribution of anchor distances towards smaller values and, hence, to higher probabilities of finding anchors in a closer neighborhood of a target node. ", + "bbox": [ + 174, + 867, + 823, + 924 + ], + "page_idx": 17 + }, + { + "type": "image", + "img_path": "images/057e081baea2ec49fd398141c91fb1aaf275530faf7f731a530bcf6c764d7cbc.jpg", + "image_caption": [ + "Figure 3: Distribution of anchor distances under various anchor selection strategies. Top-bottom: mixed, degree-based, PPR-based, random. For each dataset, left: selecting $k$ nearest anchors, right: $k$ random anchors. (a) Selecting a fixed 500/50 configuration on WN18RR; (b) Selecting a fixed 1000/20 configuration on FB15k237. Generally, all strategies except random ones skew the distributions towards nearest anchors. " + ], + "image_footnote": [], + "bbox": [ + 174, + 108, + 828, + 382 + ], + "page_idx": 18 + }, + { + "type": "image", + "img_path": "images/4d1a1400fc473a8e8f824414e521480429db385e107068b82d278aee5c8aac3c.jpg", + "image_caption": [ + "Figure 4: Distribution of anchor distances under the fixed mix anchor selection strategy when varying the total number of anchors $A$ (50–1000 for WN18RR, 20–1000 for $\\mathbf { F b 1 5 k - 2 3 7 }$ ). For each dataset, left column - $k$ nearest anchors, right - $k$ random anchors. On both graphs, increase in $A$ with the nearest anchors always leads to shorter anchor distances. " + ], + "image_footnote": [], + "bbox": [ + 173, + 489, + 828, + 762 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 859, + 823, + 888 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "We would recommend using centrality-based strategies to select $A$ with $k$ nearest anchors per node if anchor distances and probability of finding anchors in a closer neighborhood are of higher importance. ", + "bbox": [ + 173, + 895, + 823, + 924 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Finally, we fix the anchor selection strategy as mix, obtain nearest anchors per node, and under this setup study average anchor distances varying $k$ - the number of anchors per node in various combinations of total anchors $A$ . The results presented on Figure 5 suggest that sparser graphs (like WN18RR) benefit more from increasing the number of anchors $A$ , i.e., the delta between distances is much larger than that of dense FB15k-237. The difference in distances on Figure 5 might also explain the performance on Figure 8, i.e., generally, smaller $A / k$ configurations like 25/5 are inferior on sparser graphs but perform competitively on denser ones. ", + "bbox": [ + 173, + 103, + 825, + 202 + ], + "page_idx": 19 + }, + { + "type": "image", + "img_path": "images/5c83762ad70aceab22b28829e29e8a4217f2f51bb030fca7fc3d9b6ffed58232.jpg", + "image_caption": [ + "Figure 5: Average node-anchor distances when varying the total number of anchors $A$ from 25 to 1000 and $k$ nearest anchors per node from 5 to 50. Note that on a sparser WN18RR the gap between min and max values is much wider than of denser FB15k-237. Signs of saturation suggest that further increasing $A$ is not beneficial. " + ], + "image_footnote": [], + "bbox": [ + 187, + 248, + 799, + 433 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "D EMBEDDING VISUALIZATIONS ", + "text_level": 1, + "bbox": [ + 176, + 532, + 462, + 547 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "To further study learned representations of anchors and capabilities of the encoder, we build tSNE and UMAP projections from subsamples of FB15k-237 and WN18RR based on trained models from the transductive link prediction experiments (hyperparameters listed in Table A). ", + "bbox": [ + 174, + 564, + 823, + 606 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "For FB15k-237, we randomly sample 1000 entities (out of total 15K) and find their top-100 most common anchors. The anchor embeddings are extracted from the learned tensor while 1000 entity embeddings are obtained through the NodePiece encoder. Similarly for WN18RR, we sample 4000 entities (out of total 40K) keeping their top-100 most common anchors. As we use the RotatE decoder that assumes entities and anchors are modeled in a complex space, we visualize their real parts (e.g., first 100 dimensions out of 200). ", + "bbox": [ + 174, + 612, + 825, + 695 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Recalling that link prediction performance of NodePiece $^ +$ RotatE retains $80 \\%$ of the state of the art models performance, the results on Fig. 6 and Fig. 7 demonstrate that (1) NodePiece encoder is able to reconstruct clusters of similar entities; (2) anchors are well-scattered among communities. Albeit entity embeddings are built as a composition of $k$ anchors, it can be seen that all communities have \"specialized\" nearby anchors. On a higher level, common anchors tend to be well-scattered in the space. Less common anchors, as seen on FB15k-237 and Fig. 6, tend to group together. However, thanks to the non-linear nature of the NodePiece encoder, resulting entity embeddings still form different clusters and communities not concentrated around one point. We believe this is the effect of a compositional encoder and plan to investigate this phenomenon further. ", + "bbox": [ + 174, + 703, + 825, + 829 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "E TRANSDUCTIVE LINK PREDICTION RESULTS: MRR ", + "text_level": 1, + "bbox": [ + 174, + 849, + 638, + 866 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "In addition to Figure 2 that presents Hits $@ 1 0$ , we report variations of mean reciprocal rank (MRR) depending on combinations of $A$ and $k$ on Figure 8 from the same set of experiments. On sparser WN18RR, smaller $A / k$ combinations like 25/5 or 50/10 struggle with more precise predictions like ", + "bbox": [ + 176, + 882, + 825, + 924 + ], + "page_idx": 19 + }, + { + "type": "image", + "img_path": "images/1af61e1e10b58335056878a25e13f8e5248e69fd3b55b7b56b177dfd7f48b627.jpg", + "image_caption": [ + "Figure 6: tSNE (left) and UMAP (right) projections of 1000 encoded entities sampled randomly from FB $1 5 \\mathrm { k }$ -237 and their top 100 most common anchors. " + ], + "image_footnote": [], + "bbox": [ + 176, + 104, + 820, + 328 + ], + "page_idx": 20 + }, + { + "type": "image", + "img_path": "images/8957842fac60ab0a93c114967577de387a5575380933e8686ecfe472945834f4.jpg", + "image_caption": [ + "Figure 7: tSNE (left) and UMAP (right) projections of 4000 encoded entities sampled randomly from WN18RR and their top 100 most common anchors. " + ], + "image_footnote": [], + "bbox": [ + 176, + 388, + 818, + 616 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Hits $@ 1$ which is captured by low values of MRR. Starting from 500/10, the WN18RR performance starts to saturate. On the other hand, on denser FB15k-237, the difference between minimum and maximum MRR is less than 4 points, and performance exhibits signs of saturation already at 50/10. ", + "bbox": [ + 174, + 685, + 825, + 728 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "F DATASETS CONSTRUCTION ", + "text_level": 1, + "bbox": [ + 176, + 751, + 433, + 767 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "F.1 NODE CLASSIFICATION: WD50K NC", + "text_level": 1, + "bbox": [ + 174, + 785, + 475, + 799 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "The original WD50K (Galkin et al., 2020) contains a triple-only KG version on which we base a new dataset for semi-supervised multi-class multi-label node classification. First, we remove all triples containing Wikidata properties P31 (instance of ) and P279 (subclass of ) as they already contain class information. We then remove nodes that became disconnected after removing those edges. Third, using SPARQL queries, for each remaining node in a graph, we extract a 3-hop class hierarchy of Wikidata classes and their superclasses. We only keep class labels that occur at least 50 times in the training set. Then, we sample $10 \\%$ of nodes with labels for validation and $10 \\%$ for test, and of remaining $80 \\%$ we sample a set of nodes for the semi-supervised setup, i.e., we keep only $5 \\%$ and ", + "bbox": [ + 174, + 811, + 825, + 924 + ], + "page_idx": 20 + }, + { + "type": "image", + "img_path": "images/5a8194a73210c9ecf515ecad0695a36c6cc94619e6c07cb6c8a88372a7ca6b56.jpg", + "image_caption": [ + "Figure 8: Combinations of total anchors $A$ and anchors per node. Denser FB15k-237 saturates faster on smaller $A$ while sparse WN18RR saturates at around 500 anchors. MRR metric captures all ranks. Note that performance gap on FB15k-237 is very small indicating that saturation has occurred already with small anchor configurations. " + ], + "image_footnote": [], + "bbox": [ + 178, + 101, + 818, + 291 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "$10 \\%$ of those nodes. The resulting graph has 46k nodes, 526 distinct relation types, and 465 class labels. ", + "bbox": [ + 174, + 386, + 823, + 414 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "F.2 OUF-OF-SAMPLE LINK PREDICTION: OYAGO 3-10", + "text_level": 1, + "bbox": [ + 173, + 431, + 570, + 445 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "For sampling the out-of-sample version of a bigger YAGO 3-10 we largely follow the same original procedure described in Section 4 of (Albooyeh et al., 2020). We first merge the train, validation and test triples from the original dataset for transductive link prediction. Then, from all entities appearing in at least two triples, we randomly sample $5 \\%$ of nodes to be the out-of-sample entities for validation and $5 \\%$ for test. All triples containing the out-of-sample entities on subject or object positions are put into validation or test, respectively, as edges that connect an unseen entity with the seen graph. ", + "bbox": [ + 174, + 457, + 825, + 541 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "G NODE CLASSIFICATION: TRAINING CURVES ", + "text_level": 1, + "bbox": [ + 174, + 560, + 578, + 577 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Figure 9 depicts train and validation values of Hard Accuracy and PRC-AUC metrics for all the compared models on WD50K NC with $5 \\%$ of labeled nodes. The NodePiece model has only 50 total anchors with 10 nearest anchors per node, and 5 unique relation types in the relational context. The performance on the dataset with $10 \\%$ of nodes is almost the same, so we report the charts only on $5 \\%$ dataset. By the generalization gap we understand the delta between training and validation values. ", + "bbox": [ + 174, + 592, + 825, + 661 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "The MLP baseline quickly overfits but fails to generalize on the validation. The generalization gap of CompGCN is smaller compared to MLP but is still significant, i.e., validation performance is $2 { - } 3 \\times$ smaller than train. Finally, the NodePiece-enabled model has the smallest generalization gaps, especially along the Hard Accuracy metric where the validation performance is very close to that of train. Similarly, the gap on PRC-AUC is smaller than 10 points. ", + "bbox": [ + 174, + 669, + 825, + 738 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "As shown in the ablation study in Table 6, it appears that explicit node embeddings do not contribute to the classification performance. Hence, the baseline models tend to be overparameterized where learnable node embeddings add noise, while the NodePiece model has only a few anchors (or no anchors at all when using only the relational context), much fewer parameters, and therefore generalizes better. This hypothesis also explains the observation that the node classification performance does not improve when increasing $A / k$ anchor configurations. ", + "bbox": [ + 174, + 746, + 825, + 829 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "H PROOFS ", + "text_level": 1, + "bbox": [ + 174, + 849, + 276, + 864 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Proposition 2. The nearest-anchor encoder with $\\binom { | A | } { k }$ anchors and $| m |$ subsampled relations, can be considered a $\\pi$ -SGD approximation of $( k + | m | )$ -ary Janossy pooling with a canonical ordering induced by the anchor distances. ", + "bbox": [ + 176, + 877, + 823, + 922 + ], + "page_idx": 21 + }, + { + "type": "image", + "img_path": "images/0e115ed23e92c50662171a6d4f54a45b264834f88d611131451b294b96f29048.jpg", + "image_caption": [ + "Figure 9: Generalization gap on WD50K $5 \\%$ labeled nodes). NodePiece-based model has observably smaller generalization gaps compared to the baselines. " + ], + "image_footnote": [], + "bbox": [ + 173, + 98, + 825, + 299 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Proof. ", + "text_level": 1, + "bbox": [ + 174, + 369, + 217, + 383 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "We begin by providing the definition of Janossy pooling as it was presented in the original paper Murphy et al. (2019). ", + "bbox": [ + 173, + 391, + 825, + 421 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Definition 1 (Janossy pooling). Let $\\mathbb { H } ^ { \\cup }$ be the union of all anchors and relations. Consider a function $\\vec { f } : \\mathbb { N } \\times \\mathbb { H } ^ { \\cup } \\times \\mathbb { R } ^ { d } \\stackrel { \\left. } { \\right. } \\mathbb { F }$ on variable-length but finite sequences h, parameterized by $\\pmb { \\theta } ^ { ( f ) } \\in \\mathbb { R } ^ { d }$ , $d > 0$ . A permutation-invariant function $\\overline { { \\overline { { f } } } } : \\mathbb { N } \\times \\mathbb { H } ^ { \\cup } \\times \\mathbb { R } ^ { d } \\mathbb { F }$ is the Janossy function associated with $\\vec { \\boldsymbol { f } }$ if ", + "bbox": [ + 173, + 425, + 826, + 489 + ], + "page_idx": 22 + }, + { + "type": "equation", + "img_path": "images/539e3091c4c847890a2d10a79cee3dee8ba8e43a7a20263dde83af7cfa17fe5e.jpg", + "text": "$$\n\\overline { { \\overline { { f } } } } ( | { \\bf h } | , { \\bf h } ; \\theta ^ { ( f ) } ) = \\frac { 1 } { | { \\bf h } | ! } \\sum _ { \\pi \\in \\Pi _ { | { \\bf h } | } } \\vec { f } ( | { \\bf h } | , { \\bf h } _ { \\pi } ; \\theta ^ { ( f ) } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 344, + 493, + 651, + 534 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "where $\\Pi _ { | \\mathbf { h } | }$ is the set of all permutations of the integers 1 to $| \\mathbf { h } |$ , and $\\mathbf { h } _ { \\pi }$ represents a particular reordering of the elements of sequence h according to $\\pi \\in \\Pi _ { | \\mathbf { h } | }$ . We refer the operation used to construct $\\overline { { \\overline { { f } } } }$ from $\\vec { \\boldsymbol { f } }$ as Janossy pooling. ", + "bbox": [ + 174, + 541, + 825, + 590 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "While Janossy pooling provides a simple approach to construct permutation-invariant functions from arbitrary permutation sensitive functions, it is computationally intractable due to the need to sum over all computations. Three general strategies proposed under this framework to overcome this combinatorial challenge: canonical orderings, $k$ -ary Janossy pooling, and $\\pi$ -SGD approximations. ", + "bbox": [ + 174, + 603, + 825, + 660 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "A very effective way of reducing the complexity is to constrain the permutations to a canonical ordering that is independent of a specific adjacency matrix ordering over a given graph. More precisely, one defines as a function CANONICAL : $\\mathbb { H } ^ { \\cup } \\mathbb { H } ^ { \\cup }$ such that CANONICAL $( { \\bf h } ) =$ $\\mathrm { C A N O N I C A L } ( \\mathbf { h } _ { \\pi } ) \\forall \\pi \\in \\Pi _ { | \\mathbf { h } | }$ and only considers functions $\\vec { \\boldsymbol { f } }$ based on the composition ${ \\vec { f } } =$ $\\mathrm { C A N O N I C A L } \\circ \\vec { f } ^ { \\prime }$ (Murphy et al., 2019). In the case of NodePiece we are able to define this ordering for the anchors according to their distance to the target node. Assuming that the number of relations is fixed or grows at slow rate throughout the life-cycle of a graph we can define an arbitrary ordering for relations as a canonical ordering for the relational context. However, since anchors can be equidistant such a canonical ordering does fully satisfy permutation invariance. We propose a trivial relaxation of the original definition of canonical orderings simply requiring that an ordering greatly reduce the number of unique permutations since in practice an exact canonical ordering is rarely feasible. Specifically, $| \\{ \\mathrm { C A } \\mathrm { \\hat { N } O N I C A L } ( \\mathbf { h } _ { \\pi } ) \\forall \\pi \\in \\Pi _ { | \\mathbf { h } | } \\} | \\ll | \\{ ( \\mathbf { h } _ { \\pi } ) \\forall \\pi \\in \\Pi _ { | \\mathbf { h } | } \\} |$ . ", + "bbox": [ + 173, + 665, + 826, + 842 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "To further reduce the number of permutations we can truncate our ordered sequence $\\mathbf { h }$ . This is known as $k$ -ary Janossy pooling pooling (Definition 2) and is implicitly performed by the NodePeice algorithm by varying the anchor per node parameter, $k$ , and the size of the relational context, $| m |$ . ", + "bbox": [ + 174, + 847, + 825, + 890 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Definition 2 ( $k$ -ary Janossy pooling). Fix $k \\in \\mathbb N$ . For any sequence h, define $\\downarrow _ { k }$ (h) as its projection to a length $k$ sequence; in particular, if $| { \\bf h } | \\ge k$ , we keep the first $k$ elements. Then, a $k$ -ary permutation-invariant Janossy function $\\overline { { \\overline { { f } } } }$ is given by ", + "bbox": [ + 173, + 895, + 823, + 925 + ], + "page_idx": 22 + }, + { + "type": "table", + "img_path": "images/482bb994d645327a8a8e978c2370cc40f76f25ee2a7e9572c1ffbd4652addb36.jpg", + "table_caption": [ + "Table 15: Relation prediction results. $| V |$ denotes vocabulary size (anchors $^ +$ relations). " + ], + "table_footnote": [], + "table_body": "
FB15k-237WN18RRYAGO 3-10
|V|MRRH@10|V|MRRH@10|VIMRRH@10
RotatE15k +0.5k0.9050.97940k +220.7740.897123k +740.9090.992
NodePiece +RotatE1k + 0.5k0.8740.971500 +220.7610.98510k + 740.9510.997
- no rel. context1k + 0.5k0.8760.968500+220.5410.95810k + 740.8980.993
- no distances1k + 0.5k0.8770.970500+220.7460.97510k + 740.9430.997
- no anchors,rels only0+ 0.5k0.8730.9710+220.5450.9470+740.9510.998
", + "bbox": [ + 176, + 125, + 831, + 232 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 256, + 521, + 272 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/891a57716add874166094a50d1382fb19692b8e13514375821b2f4c415f98b14.jpg", + "text": "$$\n\\overline { { \\overline { { f } } } } ( | { \\bf h } | , { \\bf h } ; \\pmb { \\theta } ^ { ( f ) } ) = \\frac { 1 } { | { \\bf h } | ! } \\sum _ { \\pi \\in \\Pi _ { | { \\bf h } | } } \\vec { f } ( | { \\bf h } | , \\downarrow _ { k } ( { \\bf h } _ { \\pi } ) ; \\pmb { \\theta } ^ { ( f ) } ) .\n$$", + "text_format": "latex", + "bbox": [ + 330, + 280, + 666, + 320 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Since an imperfect truncated canonical ordering may still result in a potentially intractable number of permutations, we use permutation sampling also known as $\\pi$ -SGD to learn arbitrary functions that approximate $( k + | m | )$ -ary Janossy pooling. This is done by randomly ordering anchors that are equidistant resulting in a uniform sampling of possible permutations during training and evaluation. For more details on the formal definition of $\\pi$ -SGD we point the reader to the original paper (Murphy et al., 2019). ", + "bbox": [ + 173, + 335, + 825, + 419 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "I RELATION PREDICTION ", + "text_level": 1, + "bbox": [ + 174, + 440, + 400, + 457 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Setup. We conduct the relation prediction experiment on the same FB15k-237, WN18RR, and YAGO 3-10 datasets. While link prediction deals with entities, the relation prediction model has to rank a correct relation given a (head, ?, tail) query. We report MRR and Hits $@ 1 0$ in the filtered setting as evaluation metrics. Similar to the link prediction configuration, we use NodePiece $^ { + 2 }$ -layer MLP and compare against RotatE of the same total parameter count. ", + "bbox": [ + 174, + 472, + 825, + 542 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Discussion. The reported results (Table 15) demonstrate a competitive performance of NodePiecebased models with reduced vocabulary sizes bringing more than $9 7 \\%$ Hits $@ 1 0$ across graphs of different sizes. In the case of WN18RR and YAGO 3-10, NodePiece models with fewer anchors even slightly improve the accuracy upon the shallow embedding baseline. The ablation study suggests that on dense graphs with a reasonable amount of unique relations having explicit learnable node embeddings might not be needed at all for this task. That is, we see that on $\\mathrm { F B } 1 5 \\mathrm { k } { - } 2 3 7$ and YAGO 3-10 the NodePiece hashes comprised only of the relational context deliver the same performance without any performance drop confirming the findings from the previous experiment. ", + "bbox": [ + 173, + 549, + 825, + 661 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "J INDUCTIVE LINK PREDICTION ", + "text_level": 1, + "bbox": [ + 176, + 683, + 457, + 698 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Setup. Nodes in the inference graphs do not have any associated feature vectors which makes this benchmark very relevant for graph representation learning. Importantly, the set of relation types in the inference graphs is a subset of those seen in the training set. Since the relation embedding matrix can be learned on the training graph, we therefore have a uniform method for constructing node representations both on seen and unseen graphs. As an encoder we try MLP and Transformer. ", + "bbox": [ + 174, + 713, + 825, + 784 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Evaluation Protocol. Following the original work (Teru et al., 2020), we employ a filtered setting and rank each triple against 50 random negative triples reporting the Hits $@ 1 0$ metric. This setup is motivated by computational complexity of GraIL at inference time, while NodePiece $^ +$ CompGCN is as fast in the inductive inference as in the transductive regime reported in other experiments. ", + "bbox": [ + 174, + 790, + 825, + 847 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Discussion. For each KG, there are 4 splits of increasing size of train and inference nodes and edges. The empirical results on this spectrum of various sizes demonstrates interesting scalability properties of NodePiece in inductive settings. Without anchor nodes, the NodePiece vocabulary size is independent of the number of nodes and edges, depending only on the number of relation types. On dense relation-rich graphs such a vocabulary is enough to yield very competitive performance. ", + "bbox": [ + 174, + 853, + 825, + 924 + ], + "page_idx": 23 + }, + { + "type": "table", + "img_path": "images/f40740d16ec0a5117f0a5370b1cf31b6efa8f4f028fb4405c88c5c188da31c3e.jpg", + "table_caption": [ + "Table 16: Hyperparameters for inductive link prediction experiments. Entries are shared among 4 splits of each graph if not particularly specified. V1 | V2 | V3 | V4 otherwise. " + ], + "table_footnote": [], + "table_body": "
ParameterFB15k-237WN18RRNELL-995
# Anchors, |A|
# Anchors per node,k=
Relational context, m1241414134161416
Vocabulary dim, d100100100
Batch size512512512
Learning rate0.00010.00010.0001
Num negatives323232
Epochs2500 V1|2000 rest590|2000|210丨20002000
NodePiece encoderMLPMLPMLPIMLPIMLP|Trf
NodePiece encoder dim200200200
NodePiece encoder layers222
NodePiece encoder dropout0.10.10.1
CompGCNlayers3316161103141313
CompGCN attentionyesyesyes
CompGCN dropout0.10.10.210.110.110.1
Loss functionNSSALNSSALNSSAL
Margin25115115125151151512015|20|30120
Training time, hours2141161191118|41106151818
", + "bbox": [ + 174, + 140, + 823, + 415 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "K UNIQUENESS OF NODE HASHES ", + "text_level": 1, + "bbox": [ + 176, + 438, + 475, + 454 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "NodePiece represents nodes as a sequence of tokens and a natural question in this context is how unique such sequences can be in light of different anchor selection and tokenization strategies. ", + "bbox": [ + 174, + 469, + 823, + 498 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Assuming the input graph is a single connected component, when sampling $| A |$ total anchors and selecting randomly $k$ anchors per node, the number of possible hash combinations is bounded by $\\textstyle { \\binom { | A | } { k } }$ . In this scenario, uniqueness of hashes is achieved by having this number bigger than the number $N$ of nodes in a graph, $\\binom { | A | } { k } > N$ , and this happens with high probability for any reasonably large $| A |$ , eg, $\\binom { 5 0 } { 2 0 }$ encodes about $4 . 7 \\cdot 1 0 ^ { 1 3 }$ combinations which covers all existing public KGs combined. ", + "bbox": [ + 174, + 505, + 826, + 585 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "In the deterministic selection of nearest anchors per node we do not have such guarantees. Nevertheless, additional sequences of relational contexts and anchor distances help to obtain more unique hashes. Collisions are possible in highly regular graphs like Wordnet, but real-world KGs like Wikidata and DBpedia do not exhibit such a regular structure. Similar to homonyms whose meaning depends on the surrounding context in a sentence, we hypothesize that adding message passing layers (that can be seen as encoding of a neighboring context for a given node) on top of NodePiece hashes might further improve the diversity of node representations. That said, the future work research agenda might include proving tighter theoretical bounds on hashes uniqueness and developing new anchor sampling and tokenization strategies. 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Assuming parameterization with vectors, atoms are mapped to", + "type": "text" + }, + { + "bbox": [ + 414, + 499, + 420, + 508 + ], + "score": 0.82, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 498, + 505, + 511 + ], + "score": 1.0, + "content": "-dimensional vectors", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 507, + 506, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 216, + 522 + ], + "score": 1.0, + "content": "through shallow encoders", + "type": "text" + }, + { + "bbox": [ + 216, + 508, + 277, + 520 + ], + "score": 0.93, + "content": "f _ { n } : n \\to \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 507, + 297, + 522 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 297, + 508, + 356, + 520 + ], + "score": 0.92, + "content": "f _ { r } : r \\to \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 507, + 506, + 522 + ], + "score": 1.0, + "content": "which scale linearly to the number", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 519, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 265, + 533 + ], + "score": 1.0, + "content": "of nodes and edge types2, i.e., having", + "type": "text" + }, + { + "bbox": [ + 266, + 520, + 297, + 532 + ], + "score": 0.92, + "content": "O ( | N | )", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 519, + 506, + 533 + ], + "score": 1.0, + "content": "space complexity of the entity embedding matrix.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 530, + 506, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 506, + 544 + ], + "score": 1.0, + "content": "Albeit efficient on small conventional benchmarking datasets based on Freebase (Toutanova & Chen,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 541, + 506, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 135, + 555 + ], + "score": 1.0, + "content": "2015)", + "type": "text" + }, + { + "bbox": [ + 135, + 542, + 160, + 552 + ], + "score": 0.6, + "content": "( \\sim 1 5 \\mathrm { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 541, + 344, + 555 + ], + "score": 1.0, + "content": "nodes) and WordNet (Dettmers et al., 2018)", + "type": "text" + }, + { + "bbox": [ + 345, + 542, + 370, + 553 + ], + "score": 0.53, + "content": "- 4 0 \\mathrm { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 541, + 506, + 555 + ], + "score": 1.0, + "content": "nodes), training on larger graphs", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 552, + 507, + 567 + ], + "spans": [ + { + "bbox": [ + 104, + 552, + 507, + 567 + ], + "score": 1.0, + "content": "(e.g., YAGO 3-10 (Mahdisoltani et al., 2015) of 120K nodes) becomes computationally challenging.", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 563, + 506, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 506, + 576 + ], + "score": 1.0, + "content": "Scaling it further up to larger subsets (Hu et al., 2020; 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The OOV issue was resolved with the ability to build infinite combinations with a finite", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 658, + 505, + 670 + ], + "spans": [ + { + "bbox": [ + 105, + 658, + 505, + 670 + ], + "score": 1.0, + "content": "vocabulary enabled by subword units. 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Such a shallow lookup results in a linear", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 276, + 469, + 288 + ], + "spans": [ + { + "bbox": [ + 141, + 276, + 469, + 288 + ], + "score": 1.0, + "content": "growth of memory consumption for storing the embedding matrix and incurs high", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 286, + 469, + 299 + ], + "spans": [ + { + "bbox": [ + 141, + 286, + 469, + 299 + ], + "score": 1.0, + "content": "computational costs when working with real-world KGs. Drawing parallels with", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 298, + 469, + 310 + ], + "spans": [ + { + "bbox": [ + 141, + 298, + 469, + 310 + ], + "score": 1.0, + "content": "subword tokenization commonly used in NLP, we explore the landscape of more", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 308, + 471, + 322 + ], + "spans": [ + { + "bbox": [ + 141, + 308, + 471, + 322 + ], + "score": 1.0, + "content": "parameter-efficient node embedding strategies. To this end, we propose NodePiece,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 319, + 471, + 332 + ], + "spans": [ + { + "bbox": [ + 141, + 319, + 471, + 332 + ], + "score": 1.0, + "content": "an anchor-based approach to learn a fixed-size entity vocabulary. In NodePiece,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 330, + 470, + 342 + ], + "spans": [ + { + "bbox": [ + 141, + 330, + 470, + 342 + ], + "score": 1.0, + "content": "a vocabulary of subword/sub-entity units is constructed from anchor nodes in a", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 342, + 469, + 353 + ], + "spans": [ + { + "bbox": [ + 141, + 342, + 469, + 353 + ], + "score": 1.0, + "content": "graph with known relation types. Given such a fixed-size vocabulary, it is possible", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 353, + 469, + 364 + ], + "spans": [ + { + "bbox": [ + 141, + 353, + 469, + 364 + ], + "score": 1.0, + "content": "to bootstrap an encoding and embedding for any entity, including those unseen", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 142, + 363, + 469, + 375 + ], + "spans": [ + { + "bbox": [ + 142, + 363, + 469, + 375 + ], + "score": 1.0, + "content": "during training. Experiments show that NodePiece performs competitively in node", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 142, + 375, + 469, + 386 + ], + "spans": [ + { + "bbox": [ + 142, + 375, + 469, + 386 + ], + "score": 1.0, + "content": "classification, link prediction, and relation prediction tasks while retaining less than", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 142, + 385, + 470, + 398 + ], + "spans": [ + { + "bbox": [ + 142, + 385, + 161, + 396 + ], + "score": 0.85, + "content": "10 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 385, + 382, + 398 + ], + "score": 1.0, + "content": "of explicit nodes in a graph as anchors and often having", + "type": "text" + }, + { + "bbox": [ + 382, + 385, + 398, + 396 + ], + "score": 0.29, + "content": "1 0 \\mathrm { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 385, + 470, + 398 + ], + "score": 1.0, + "content": "fewer parameters.", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 396, + 469, + 408 + ], + "spans": [ + { + "bbox": [ + 141, + 396, + 469, + 408 + ], + "score": 1.0, + "content": "To this end, we show that a NodePiece-enabled model outperforms existing shallow", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 141, + 406, + 438, + 420 + ], + "spans": [ + { + "bbox": [ + 141, + 406, + 337, + 420 + ], + "score": 1.0, + "content": "models on a large OGB WikiKG 2 graph having", + "type": "text" + }, + { + "bbox": [ + 337, + 407, + 359, + 417 + ], + "score": 0.7, + "content": "\\mathrm { \\sim } 7 0 \\mathrm { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 406, + 438, + 420 + ], + "score": 1.0, + "content": "fewer parameters1.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 15, + "bbox_fs": [ + 141, + 252, + 471, + 420 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 440, + 205, + 452 + ], + "lines": [ + { + "bbox": [ + 105, + 439, + 208, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 208, + 456 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 106, + 465, + 505, + 608 + ], + "lines": [ + { + "bbox": [ + 106, + 465, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 106, + 465, + 505, + 478 + ], + "score": 1.0, + "content": "Representation learning tasks on knowledge graphs (KGs) often require a parameterization of each", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 476, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 476, + 505, + 488 + ], + "score": 1.0, + "content": "unique atom in the graph with a vector or matrix. 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Assuming parameterization with vectors, atoms are mapped to", + "type": "text" + }, + { + "bbox": [ + 414, + 499, + 420, + 508 + ], + "score": 0.82, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 498, + 505, + 511 + ], + "score": 1.0, + "content": "-dimensional vectors", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 507, + 506, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 216, + 522 + ], + "score": 1.0, + "content": "through shallow encoders", + "type": "text" + }, + { + "bbox": [ + 216, + 508, + 277, + 520 + ], + "score": 0.93, + "content": "f _ { n } : n \\to \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 507, + 297, + 522 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 297, + 508, + 356, + 520 + ], + "score": 0.92, + "content": "f _ { r } : r \\to \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 507, + 506, + 522 + ], + "score": 1.0, + "content": "which scale linearly to the number", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 519, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 265, + 533 + ], + "score": 1.0, + "content": "of nodes and edge types2, i.e., having", + "type": "text" + }, + { + "bbox": [ + 266, + 520, + 297, + 532 + ], + "score": 0.92, + "content": "O ( | N | )", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 519, + 506, + 533 + ], + "score": 1.0, + "content": "space complexity of the entity embedding matrix.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 530, + 506, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 506, + 544 + ], + "score": 1.0, + "content": "Albeit efficient on small conventional benchmarking datasets based on Freebase (Toutanova & Chen,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 541, + 506, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 135, + 555 + ], + "score": 1.0, + "content": "2015)", + "type": "text" + }, + { + "bbox": [ + 135, + 542, + 160, + 552 + ], + "score": 0.6, + "content": "( \\sim 1 5 \\mathrm { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 541, + 344, + 555 + ], + "score": 1.0, + "content": "nodes) and WordNet (Dettmers et al., 2018)", + "type": "text" + }, + { + "bbox": [ + 345, + 542, + 370, + 553 + ], + "score": 0.53, + "content": "- 4 0 \\mathrm { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 541, + 506, + 555 + ], + "score": 1.0, + "content": "nodes), training on larger graphs", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 552, + 507, + 567 + ], + "spans": [ + { + "bbox": [ + 104, + 552, + 507, + 567 + ], + "score": 1.0, + "content": "(e.g., YAGO 3-10 (Mahdisoltani et al., 2015) of 120K nodes) becomes computationally challenging.", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 563, + 506, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 506, + 576 + ], + "score": 1.0, + "content": "Scaling it further up to larger subsets (Hu et al., 2020; 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This holds for different em-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 533, + 507, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 507, + 546 + ], + "score": 1.0, + "content": "bedding families, e.g., translational (Sun et al., 2019), tensor factorization (Lacroix et al., 2018),", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 544, + 507, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 507, + 557 + ], + "score": 1.0, + "content": "convolutional (Dettmers et al., 2018), and hyperbolic (Chami et al., 2020; Balazevic et al., 2019).", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 555, + 507, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 507, + 568 + ], + "score": 1.0, + "content": "The same applies to relation-aware graph neural network (GNN) encoders (Schlichtkrull et al., 2018;", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 567, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 567, + 505, + 578 + ], + "score": 1.0, + "content": "Vashishth et al., 2020) who still initialize each node with a learned embedding or feature vector before", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 577, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 506, + 590 + ], + "score": 1.0, + "content": "message passing. Furthermore, shallow encoding is also used in higher-order KG structures such as", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 588, + 506, + 601 + ], + "spans": [ + { + "bbox": [ + 106, + 588, + 506, + 601 + ], + "score": 1.0, + "content": "hypergraphs (Fatemi et al., 2020) and hyper-relational graphs (Rosso et al., 2020; Galkin et al., 2020).", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 599, + 507, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 507, + 612 + ], + "score": 1.0, + "content": "NodePiece can be used as a drop-in replacement of the embedding lookup with any of those models.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 488, + 507, + 612 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 615, + 505, + 671 + ], + "lines": [ + { + "bbox": [ + 106, + 616, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 506, + 628 + ], + "score": 1.0, + "content": "Distillation and compression. Several recent techniques for reducing memory footprint of embed-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 627, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 506, + 639 + ], + "score": 1.0, + "content": "ding matrices follow successful applications of distilling large language models in NLP (Sanh et al.,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 638, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 506, + 651 + ], + "score": 1.0, + "content": "2019), i.e., distillation (Wang et al., 2020; Zhu et al., 2020) into low-dimensional counterparts, and", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 649, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 506, + 662 + ], + "score": 1.0, + "content": "compression of trained matrices into discrete codes (Sachan, 2020). 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A recent line of work such as KG-BERT (Yao et al.,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 309, + 506, + 323 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 506, + 323 + ], + "score": 1.0, + "content": "2019), MLMLM (Clouâtre et al., 2021), BLP (Daza et al., 2021) utilize entity descriptions passed", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 320, + 506, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 506, + 334 + ], + "score": 1.0, + "content": "through a language model (LM) encoder as entity embeddings suitable for link prediction. We would", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 332, + 506, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 332, + 506, + 345 + ], + "score": 1.0, + "content": "like to emphasize that such approaches are rather orthogonal to NodePiece. Textual features are", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 342, + 505, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 505, + 356 + ], + "score": 1.0, + "content": "mostly available in Wikipedia-derived KGs like Wikidata but are often missing in domain-specific", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 354, + 506, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 506, + 366 + ], + "score": 1.0, + "content": "graphs like social networks and product graphs. We therefore assume textual features are not available", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 366, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 505, + 378 + ], + "score": 1.0, + "content": "and rather learn node representations based on their spatial characteristics. Still, textual features can", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 375, + 472, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 472, + 389 + ], + "score": 1.0, + "content": "be easily added by concatenating NodePiece-encoded features with LM-produced features.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 107, + 393, + 505, + 470 + ], + "lines": [ + { + "bbox": [ + 106, + 393, + 505, + 406 + ], + "spans": [ + { + "bbox": [ + 106, + 393, + 505, + 406 + ], + "score": 1.0, + "content": "Out-of-sample representation learning. This task focuses on predictions involving previously", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 403, + 506, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 506, + 417 + ], + "score": 1.0, + "content": "unseen, or out-of-sample, entities that attach to a known KG with a few edges. These new edges are", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 415, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 505, + 428 + ], + "score": 1.0, + "content": "then utilized as a context to compute its embedding. 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On graphs, such strategies", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "might employ centrality measures like degree centrality or Personalized PageRank (Page et al., 1999).", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "score": 1.0, + "content": "However, in our preliminary experiments, we found random anchor selection to be as effective as", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 82, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 96 + ], + "score": 1.0, + "content": "centrality-based strategies. A choice for deterministic strategies might be justified when optimizing", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "for certain task-specific topological characteristics, e.g., degree and PPR strategies indeed skew", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "the distribution of shortest anchor distances towards smaller values thus increasing chances to find", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "anchors in 2- or 3-hop neighborhood of any node (we provide more evidence for that in Appendix C).", + "type": "text", + "cross_page": true + } + ], + "index": 3 + } + ], + "index": 38, + "bbox_fs": [ + 105, + 676, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 127 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 96 + ], + "score": 1.0, + "content": "centrality-based strategies. A choice for deterministic strategies might be justified when optimizing", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "for certain task-specific topological characteristics, e.g., degree and PPR strategies indeed skew", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "the distribution of shortest anchor distances towards smaller values thus increasing chances to find", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "anchors in 2- or 3-hop neighborhood of any node (we provide more evidence for that in Appendix C).", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "title", + "bbox": [ + 108, + 140, + 225, + 151 + ], + "lines": [ + { + "bbox": [ + 105, + 139, + 226, + 152 + ], + "spans": [ + { + "bbox": [ + 105, + 139, + 226, + 152 + ], + "score": 1.0, + "content": "3.2 NODE TOKENIZATION", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 107, + 160, + 505, + 204 + ], + "lines": [ + { + "bbox": [ + 105, + 160, + 506, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 193, + 173 + ], + "score": 1.0, + "content": "Once the vocabulary", + "type": "text" + }, + { + "bbox": [ + 194, + 161, + 245, + 171 + ], + "score": 0.91, + "content": "V = A + R", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 160, + 351, + 173 + ], + "score": 1.0, + "content": "is constructed, each node", + "type": "text" + }, + { + "bbox": [ + 351, + 163, + 359, + 171 + ], + "score": 0.75, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 160, + 506, + 173 + ], + "score": 1.0, + "content": "can be hashed (or tokenized) into a", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 171, + 506, + 184 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 140, + 183 + ], + "score": 0.67, + "content": "h a s h ( n )", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 171, + 176, + 184 + ], + "score": 1.0, + "content": "using 1)", + "type": "text" + }, + { + "bbox": [ + 177, + 172, + 183, + 182 + ], + "score": 0.76, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 171, + 373, + 184 + ], + "score": 1.0, + "content": "nearest anchors and their discrete distances; 2)", + "type": "text" + }, + { + "bbox": [ + 374, + 173, + 384, + 182 + ], + "score": 0.74, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 171, + 506, + 184 + ], + "score": 1.0, + "content": "immediate outgoing relations", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 182, + 505, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 225, + 194 + ], + "score": 1.0, + "content": "from the relational context of", + "type": "text" + }, + { + "bbox": [ + 225, + 185, + 232, + 192 + ], + "score": 0.73, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 182, + 403, + 194 + ], + "score": 1.0, + "content": ". Since anchor nodes are concrete nodes in", + "type": "text" + }, + { + "bbox": [ + 403, + 183, + 412, + 192 + ], + "score": 0.8, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 182, + 505, + 194 + ], + "score": 1.0, + "content": ", they get hashed in the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 194, + 257, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 257, + 205 + ], + "score": 1.0, + "content": "same way as other non-anchor nodes.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 106, + 210, + 505, + 293 + ], + "lines": [ + { + "bbox": [ + 106, + 210, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 106, + 210, + 215, + 222 + ], + "score": 1.0, + "content": "Anchors per node. Given", + "type": "text" + }, + { + "bbox": [ + 216, + 210, + 230, + 222 + ], + "score": 0.93, + "content": "| A |", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 210, + 505, + 222 + ], + "score": 1.0, + "content": "anchor nodes, it is impractical to use all of them for encoding each", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 221, + 505, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 221, + 202, + 234 + ], + "score": 1.0, + "content": "node. Instead, we select", + "type": "text" + }, + { + "bbox": [ + 203, + 222, + 209, + 231 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 221, + 505, + 234 + ], + "score": 1.0, + "content": "anchors per node and describe two possible strategies for that, i.e., random", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 232, + 504, + 244 + ], + "spans": [ + { + "bbox": [ + 106, + 232, + 442, + 244 + ], + "score": 1.0, + "content": "and deterministic. The basic random strategy uniformly samples an unordered set of", + "type": "text" + }, + { + "bbox": [ + 442, + 233, + 449, + 242 + ], + "score": 0.83, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 232, + 504, + 244 + ], + "score": 1.0, + "content": "anchors from", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 241, + 507, + 261 + ], + "spans": [ + { + "bbox": [ + 106, + 245, + 115, + 255 + ], + "score": 0.68, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 116, + 241, + 153, + 261 + ], + "score": 1.0, + "content": "yielding", + "type": "text" + }, + { + "bbox": [ + 153, + 243, + 172, + 258 + ], + "score": 0.91, + "content": "\\binom { | A | } { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 241, + 450, + 261 + ], + "score": 1.0, + "content": "possible combinations. To avoid collisions when hashing the nodes,", + "type": "text" + }, + { + "bbox": [ + 450, + 245, + 464, + 257 + ], + "score": 0.9, + "content": "| A |", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 241, + 482, + 261 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 482, + 245, + 489, + 255 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 241, + 507, + 261 + ], + "score": 1.0, + "content": "are", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 254, + 505, + 269 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 505, + 269 + ], + "score": 1.0, + "content": "to be chosen according to the lower bound on possible combinations that is defined by the total", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 102, + 264, + 509, + 287 + ], + "spans": [ + { + "bbox": [ + 102, + 264, + 198, + 287 + ], + "score": 1.0, + "content": "number of nodes, e.g.,", + "type": "text" + }, + { + "bbox": [ + 198, + 267, + 246, + 282 + ], + "score": 0.94, + "content": "\\bar { \\binom { | A | } { k } } \\geq | N |", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 264, + 509, + 287 + ], + "score": 1.0, + "content": ". Note that running depth-first search (DFS) to random anchors at", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 280, + 504, + 293 + ], + "spans": [ + { + "bbox": [ + 106, + 280, + 278, + 293 + ], + "score": 1.0, + "content": "inference time is inefficient and, therefore,", + "type": "text" + }, + { + "bbox": [ + 278, + 281, + 312, + 293 + ], + "score": 0.31, + "content": "h a s h ( n )", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 280, + 504, + 293 + ], + "score": 1.0, + "content": "of the random strategy has to be pre-computed.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 297, + 505, + 374 + ], + "lines": [ + { + "bbox": [ + 106, + 297, + 507, + 310 + ], + "spans": [ + { + "bbox": [ + 106, + 297, + 402, + 310 + ], + "score": 1.0, + "content": "On the other hand, the deterministic strategy selects an ordered sequence of", + "type": "text" + }, + { + "bbox": [ + 403, + 298, + 410, + 308 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 297, + 507, + 310 + ], + "score": 1.0, + "content": "nearest anchors. Hence,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 309, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 106, + 309, + 358, + 320 + ], + "score": 1.0, + "content": "the anchors can be obtained via breadth-first search (BFS) in the", + "type": "text" + }, + { + "bbox": [ + 358, + 309, + 363, + 318 + ], + "score": 0.7, + "content": "l", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 309, + 448, + 320 + ], + "score": 1.0, + "content": "-hop neighborhood of", + "type": "text" + }, + { + "bbox": [ + 449, + 311, + 456, + 318 + ], + "score": 0.78, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 309, + 505, + 320 + ], + "score": 1.0, + "content": "at inference", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 320, + 505, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 505, + 332 + ], + "score": 1.0, + "content": "time (or pre-computed for speed reasons). However, the combinatorial bound is not applicable in", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 331, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 331, + 505, + 342 + ], + "score": 1.0, + "content": "this strategy and we need more discriminative signals to avoid hash collisions since nearby nodes", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 341, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 505, + 354 + ], + "score": 1.0, + "content": "will have similar anchors (we elaborate on the uniqueness issue in Appendix K). Such signals have", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 352, + 506, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 506, + 365 + ], + "score": 1.0, + "content": "to better ground anchors to the underlying graph structure, and we accomplish that using anchor", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 363, + 308, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 308, + 375 + ], + "score": 1.0, + "content": "distances3 and relational context described below.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 108, + 380, + 503, + 414 + ], + "lines": [ + { + "bbox": [ + 105, + 379, + 504, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 504, + 394 + ], + "score": 1.0, + "content": "A node residing in a disconnected component is assigned with an auxiliary [DISCONNECTED]", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 392, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 106, + 392, + 505, + 404 + ], + "score": 1.0, + "content": "token or can be turned into an anchor. However, the majority of existing KGs are graphs with one", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 403, + 492, + 415 + ], + "spans": [ + { + "bbox": [ + 106, + 403, + 492, + 415 + ], + "score": 1.0, + "content": "large connected component with very few disconnected nodes, such that this effect is negligible.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 419, + 505, + 476 + ], + "lines": [ + { + "bbox": [ + 106, + 417, + 505, + 432 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 278, + 432 + ], + "score": 1.0, + "content": "Anchor Distances. Given a target node", + "type": "text" + }, + { + "bbox": [ + 278, + 421, + 285, + 429 + ], + "score": 0.75, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 417, + 350, + 432 + ], + "score": 1.0, + "content": "and an anchor", + "type": "text" + }, + { + "bbox": [ + 351, + 421, + 360, + 430 + ], + "score": 0.85, + "content": "a _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 417, + 478, + 432 + ], + "score": 1.0, + "content": ", we define anchor distance", + "type": "text" + }, + { + "bbox": [ + 478, + 420, + 505, + 431 + ], + "score": 0.89, + "content": "z _ { a _ { i } } \\in", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 430, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 106, + 430, + 154, + 442 + ], + "score": 1.0, + "content": "[0; diameter", + "type": "text" + }, + { + "bbox": [ + 155, + 430, + 173, + 442 + ], + "score": 0.74, + "content": "\\cdot ( G ) ]", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 430, + 408, + 442 + ], + "score": 1.0, + "content": "as an integer denoting the shortest path distance between", + "type": "text" + }, + { + "bbox": [ + 408, + 432, + 418, + 441 + ], + "score": 0.84, + "content": "a _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 430, + 437, + 442 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 437, + 432, + 444, + 440 + ], + "score": 0.77, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 430, + 505, + 442 + ], + "score": 1.0, + "content": "in the original", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 440, + 506, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 133, + 454 + ], + "score": 1.0, + "content": "graph", + "type": "text" + }, + { + "bbox": [ + 133, + 442, + 142, + 451 + ], + "score": 0.72, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 440, + 311, + 454 + ], + "score": 1.0, + "content": ". Note that when tokenizing an anchor", + "type": "text" + }, + { + "bbox": [ + 311, + 443, + 322, + 453 + ], + "score": 0.85, + "content": "a _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 440, + 506, + 454 + ], + "score": 1.0, + "content": "with the deterministic strategy, the nearest", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 452, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 184, + 464 + ], + "score": 1.0, + "content": "anchor among top-", + "type": "text" + }, + { + "bbox": [ + 184, + 453, + 191, + 462 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 452, + 232, + 464 + ], + "score": 1.0, + "content": "is always", + "type": "text" + }, + { + "bbox": [ + 233, + 454, + 244, + 464 + ], + "score": 0.85, + "content": "a _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 452, + 505, + 464 + ], + "score": 1.0, + "content": "itself with distance 0. We then map each integer to a learnable", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 107, + 463, + 426, + 478 + ], + "spans": [ + { + "bbox": [ + 107, + 465, + 113, + 474 + ], + "score": 0.8, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 463, + 194, + 478 + ], + "score": 1.0, + "content": "-dimensional vector", + "type": "text" + }, + { + "bbox": [ + 194, + 464, + 252, + 477 + ], + "score": 0.92, + "content": "f _ { z } : z _ { a _ { i } } \\to \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 463, + 426, + 478 + ], + "score": 1.0, + "content": "akin to relative distance encoding scheme.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 106, + 481, + 506, + 559 + ], + "lines": [ + { + "bbox": [ + 105, + 480, + 506, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 506, + 494 + ], + "score": 1.0, + "content": "Relational Context. We also leverage the multi-relational nature of an underlying KG. 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Given", + "type": "text" + }, + { + "bbox": [ + 216, + 210, + 230, + 222 + ], + "score": 0.93, + "content": "| A |", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 210, + 505, + 222 + ], + "score": 1.0, + "content": "anchor nodes, it is impractical to use all of them for encoding each", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 221, + 505, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 221, + 202, + 234 + ], + "score": 1.0, + "content": "node. 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To avoid collisions when hashing the nodes,", + "type": "text" + }, + { + "bbox": [ + 450, + 245, + 464, + 257 + ], + "score": 0.9, + "content": "| A |", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 241, + 482, + 261 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 482, + 245, + 489, + 255 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 241, + 507, + 261 + ], + "score": 1.0, + "content": "are", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 254, + 505, + 269 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 505, + 269 + ], + "score": 1.0, + "content": "to be chosen according to the lower bound on possible combinations that is defined by the total", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 102, + 264, + 509, + 287 + ], + "spans": [ + { + "bbox": [ + 102, + 264, + 198, + 287 + ], + "score": 1.0, + "content": "number of nodes, e.g.,", + "type": "text" + }, + { + "bbox": [ + 198, + 267, + 246, + 282 + ], + "score": 0.94, + "content": "\\bar { \\binom { | A | } { k } } \\geq | N |", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 264, + 509, + 287 + ], + "score": 1.0, + "content": ". Note that running depth-first search (DFS) to random anchors at", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 280, + 504, + 293 + ], + "spans": [ + { + "bbox": [ + 106, + 280, + 278, + 293 + ], + "score": 1.0, + "content": "inference time is inefficient and, therefore,", + "type": "text" + }, + { + "bbox": [ + 278, + 281, + 312, + 293 + ], + "score": 0.31, + "content": "h a s h ( n )", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 280, + 504, + 293 + ], + "score": 1.0, + "content": "of the random strategy has to be pre-computed.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 12, + "bbox_fs": [ + 102, + 210, + 509, + 293 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 297, + 505, + 374 + ], + "lines": [ + { + "bbox": [ + 106, + 297, + 507, + 310 + ], + "spans": [ + { + "bbox": [ + 106, + 297, + 402, + 310 + ], + "score": 1.0, + "content": "On the other hand, the deterministic strategy selects an ordered sequence of", + "type": "text" + }, + { + "bbox": [ + 403, + 298, + 410, + 308 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 297, + 507, + 310 + ], + "score": 1.0, + "content": "nearest anchors. Hence,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 309, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 106, + 309, + 358, + 320 + ], + "score": 1.0, + "content": "the anchors can be obtained via breadth-first search (BFS) in the", + "type": "text" + }, + { + "bbox": [ + 358, + 309, + 363, + 318 + ], + "score": 0.7, + "content": "l", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 309, + 448, + 320 + ], + "score": 1.0, + "content": "-hop neighborhood of", + "type": "text" + }, + { + "bbox": [ + 449, + 311, + 456, + 318 + ], + "score": 0.78, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 309, + 505, + 320 + ], + "score": 1.0, + "content": "at inference", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 320, + 505, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 505, + 332 + ], + "score": 1.0, + "content": "time (or pre-computed for speed reasons). However, the combinatorial bound is not applicable in", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 331, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 331, + 505, + 342 + ], + "score": 1.0, + "content": "this strategy and we need more discriminative signals to avoid hash collisions since nearby nodes", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 341, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 505, + 354 + ], + "score": 1.0, + "content": "will have similar anchors (we elaborate on the uniqueness issue in Appendix K). Such signals have", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 352, + 506, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 506, + 365 + ], + "score": 1.0, + "content": "to better ground anchors to the underlying graph structure, and we accomplish that using anchor", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 363, + 308, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 308, + 375 + ], + "score": 1.0, + "content": "distances3 and relational context described below.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 297, + 507, + 375 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 380, + 503, + 414 + ], + "lines": [ + { + "bbox": [ + 105, + 379, + 504, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 504, + 394 + ], + "score": 1.0, + "content": "A node residing in a disconnected component is assigned with an auxiliary [DISCONNECTED]", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 392, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 106, + 392, + 505, + 404 + ], + "score": 1.0, + "content": "token or can be turned into an anchor. However, the majority of existing KGs are graphs with one", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 403, + 492, + 415 + ], + "spans": [ + { + "bbox": [ + 106, + 403, + 492, + 415 + ], + "score": 1.0, + "content": "large connected component with very few disconnected nodes, such that this effect is negligible.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 379, + 505, + 415 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 419, + 505, + 476 + ], + "lines": [ + { + "bbox": [ + 106, + 417, + 505, + 432 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 278, + 432 + ], + "score": 1.0, + "content": "Anchor Distances. Given a target node", + "type": "text" + }, + { + "bbox": [ + 278, + 421, + 285, + 429 + ], + "score": 0.75, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 417, + 350, + 432 + ], + "score": 1.0, + "content": "and an anchor", + "type": "text" + }, + { + "bbox": [ + 351, + 421, + 360, + 430 + ], + "score": 0.85, + "content": "a _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 417, + 478, + 432 + ], + "score": 1.0, + "content": ", we define anchor distance", + "type": "text" + }, + { + "bbox": [ + 478, + 420, + 505, + 431 + ], + "score": 0.89, + "content": "z _ { a _ { i } } \\in", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 430, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 106, + 430, + 154, + 442 + ], + "score": 1.0, + "content": "[0; diameter", + "type": "text" + }, + { + "bbox": [ + 155, + 430, + 173, + 442 + ], + "score": 0.74, + "content": "\\cdot ( G ) ]", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 430, + 408, + 442 + ], + "score": 1.0, + "content": "as an integer denoting the shortest path distance between", + "type": "text" + }, + { + "bbox": [ + 408, + 432, + 418, + 441 + ], + "score": 0.84, + "content": "a _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 430, + 437, + 442 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 437, + 432, + 444, + 440 + ], + "score": 0.77, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 430, + 505, + 442 + ], + "score": 1.0, + "content": "in the original", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 440, + 506, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 133, + 454 + ], + "score": 1.0, + "content": "graph", + "type": "text" + }, + { + "bbox": [ + 133, + 442, + 142, + 451 + ], + "score": 0.72, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 440, + 311, + 454 + ], + "score": 1.0, + "content": ". Note that when tokenizing an anchor", + "type": "text" + }, + { + "bbox": [ + 311, + 443, + 322, + 453 + ], + "score": 0.85, + "content": "a _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 440, + 506, + 454 + ], + "score": 1.0, + "content": "with the deterministic strategy, the nearest", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 452, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 184, + 464 + ], + "score": 1.0, + "content": "anchor among top-", + "type": "text" + }, + { + "bbox": [ + 184, + 453, + 191, + 462 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 452, + 232, + 464 + ], + "score": 1.0, + "content": "is always", + "type": "text" + }, + { + "bbox": [ + 233, + 454, + 244, + 464 + ], + "score": 0.85, + "content": "a _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 452, + 505, + 464 + ], + "score": 1.0, + "content": "itself with distance 0. We then map each integer to a learnable", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 107, + 463, + 426, + 478 + ], + "spans": [ + { + "bbox": [ + 107, + 465, + 113, + 474 + ], + "score": 0.8, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 463, + 194, + 478 + ], + "score": 1.0, + "content": "-dimensional vector", + "type": "text" + }, + { + "bbox": [ + 194, + 464, + 252, + 477 + ], + "score": 0.92, + "content": "f _ { z } : z _ { a _ { i } } \\to \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 463, + 426, + 478 + ], + "score": 1.0, + "content": "akin to relative distance encoding scheme.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 417, + 506, + 478 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 481, + 506, + 559 + ], + "lines": [ + { + "bbox": [ + 105, + 480, + 506, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 506, + 494 + ], + "score": 1.0, + "content": "Relational Context. We also leverage the multi-relational nature of an underlying KG. Commonly4,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 492, + 506, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 492, + 248, + 505 + ], + "score": 1.0, + "content": "the amount of unique edge types in", + "type": "text" + }, + { + "bbox": [ + 248, + 493, + 257, + 502 + ], + "score": 0.83, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 492, + 506, + 505 + ], + "score": 1.0, + "content": "is orders of magnitude smaller than the total number of nodes,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 503, + 506, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 125, + 516 + ], + "score": 1.0, + "content": "i.e.,", + "type": "text" + }, + { + "bbox": [ + 125, + 503, + 171, + 515 + ], + "score": 0.92, + "content": "| R | \\ll | N |", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 503, + 332, + 516 + ], + "score": 1.0, + "content": ". This fact allows to include the entire", + "type": "text" + }, + { + "bbox": [ + 332, + 503, + 346, + 515 + ], + "score": 0.9, + "content": "| R |", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 503, + 469, + 516 + ], + "score": 1.0, + "content": "in the NodePiece vocabulary", + "type": "text" + }, + { + "bbox": [ + 470, + 504, + 486, + 515 + ], + "score": 0.88, + "content": "V _ { N P }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 503, + 506, + 516 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 515, + 506, + 526 + ], + "spans": [ + { + "bbox": [ + 106, + 515, + 506, + 526 + ], + "score": 1.0, + "content": "further featurize each node with a unique relational context. We construct a relational context of a", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 524, + 507, + 539 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 128, + 539 + ], + "score": 1.0, + "content": "node", + "type": "text" + }, + { + "bbox": [ + 129, + 527, + 136, + 535 + ], + "score": 0.75, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 524, + 262, + 539 + ], + "score": 1.0, + "content": "by randomly sampling a set of", + "type": "text" + }, + { + "bbox": [ + 262, + 527, + 272, + 535 + ], + "score": 0.77, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 524, + 478, + 539 + ], + "score": 1.0, + "content": "immediate unique outgoing relations starting from", + "type": "text" + }, + { + "bbox": [ + 478, + 528, + 485, + 536 + ], + "score": 0.65, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 524, + 507, + 539 + ], + "score": 1.0, + "content": ", i.e.,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 535, + 506, + 549 + ], + "spans": [ + { + "bbox": [ + 106, + 536, + 210, + 548 + ], + "score": 0.91, + "content": "r c o n _ { n } = \\mathsf { \\bar { \\{ } } r _ { j } \\mathsf { \\} } ^ { m } \\subseteq \\mathsf { \\bar { \\mathcal { N } } } _ { r } ( n ) \\mathsf { \\bar { \\Psi } }", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 535, + 239, + 549 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 239, + 536, + 267, + 548 + ], + "score": 0.92, + "content": "\\mathcal { N } _ { r } ( n )", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 535, + 506, + 549 + ], + "score": 1.0, + "content": "denotes all outgoing relation types. Due to a non-uniform", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 547, + 501, + 560 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 195, + 560 + ], + "score": 1.0, + "content": "degree distribution, if", + "type": "text" + }, + { + "bbox": [ + 195, + 547, + 249, + 559 + ], + "score": 0.92, + "content": "| { \\mathcal { N } } _ { r } ( n ) | < m", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 547, + 432, + 560 + ], + "score": 1.0, + "content": ", we add auxiliary [PAD] tokens to complete", + "type": "text" + }, + { + "bbox": [ + 433, + 549, + 457, + 558 + ], + "score": 0.83, + "content": "r c o n _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 547, + 486, + 560 + ], + "score": 1.0, + "content": "to size", + "type": "text" + }, + { + "bbox": [ + 487, + 549, + 496, + 557 + ], + "score": 0.71, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 497, + 547, + 501, + 560 + ], + "score": 1.0, + "content": ".", + 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While MLP is", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 168, + 504, + 181 + ], + "spans": [ + { + "bbox": [ + 105, + 168, + 504, + 181 + ], + "score": 1.0, + "content": "faster and better scales to graphs with more edges, Transformer is slower but requires less trainable", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 179, + 506, + 192 + ], + "spans": [ + { + "bbox": [ + 105, + 179, + 506, + 192 + ], + "score": 1.0, + "content": "parameters. 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The permutation-invariant", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 321, + 505, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 505, + 333 + ], + "score": 1.0, + "content": "nature of the nearest-neighbor encoding scheme combined with the lack of transductive features such", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 332, + 504, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 332, + 504, + 344 + ], + "score": 1.0, + "content": "as node-specific embeddings mean that NodePiece can be used for inductive learning tasks as well.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 107, + 348, + 505, + 404 + ], + "lines": [ + { + "bbox": [ + 106, + 349, + 505, + 361 + ], + "spans": [ + { + "bbox": [ + 106, + 349, + 227, + 361 + ], + "score": 1.0, + "content": "With a fixed-size vocabulary", + "type": "text" + }, + { + "bbox": [ + 228, + 349, + 247, + 360 + ], + "score": 0.89, + "content": "V _ { N P }", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 349, + 505, + 361 + ], + "score": 1.0, + "content": ", the overall complexity and parameter budget of downstream", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 360, + 505, + 372 + ], + "spans": [ + { + "bbox": [ + 106, + 360, + 505, + 372 + ], + "score": 1.0, + "content": "models are largely defined by the complexity of the encoder and its inductive biases. By design, the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 371, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 106, + 371, + 505, + 383 + ], + "score": 1.0, + "content": "NodePiece smaller vocabulary - larger encoder framework is similar to various Transformer-based", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 381, + 506, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 506, + 394 + ], + "score": 1.0, + "content": "language models (Qiu et al., 2020) whose vocabulary size remains rather stable with the encoder", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 392, + 379, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 379, + 406 + ], + "score": 1.0, + "content": "being the most important part responsible for the final performance.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23 + }, + { + "type": "title", + "bbox": [ + 108, + 420, + 200, + 432 + ], + "lines": [ + { + "bbox": [ + 105, + 419, + 201, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 201, + 434 + ], + "score": 1.0, + "content": "4 EXPERIMENTS", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 444, + 505, + 522 + ], + "lines": [ + { + "bbox": [ + 106, + 444, + 506, + 457 + ], + "spans": [ + { + "bbox": [ + 106, + 444, + 506, + 457 + ], + "score": 1.0, + "content": "We design the experimental program not seeking to outperform the best existing approaches but to", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 455, + 507, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 507, + 469 + ], + "score": 1.0, + "content": "show the versatility of NodePiece on a variety of KG-related tasks: transductive, inductive, out-of-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 466, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 505, + 478 + ], + "score": 1.0, + "content": "sample link prediction, and node classification (with relation prediction results in Appendix I). With", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 477, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 505, + 489 + ], + "score": 1.0, + "content": "this desiderata, we formulate the following research questions: RQ 1) Is it necessary to map each", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 488, + 506, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 506, + 500 + ], + "score": 1.0, + "content": "node to a unique vector for an acceptable performance on KG tasks?; RQ 2) What is the effect of", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 498, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 506, + 513 + ], + "score": 1.0, + "content": "hashing features?; RQ 3) Is there an optimal number of anchors per node, after which diminishing", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 510, + 222, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 222, + 522 + ], + "score": 1.0, + "content": "returns hit the performance?", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30 + }, + { + "type": "title", + "bbox": [ + 108, + 534, + 280, + 545 + ], + "lines": [ + { + "bbox": [ + 106, + 534, + 282, + 547 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 282, + 547 + ], + "score": 1.0, + "content": "4.1 TRANSDUCTIVE LINK PREDICTION", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 106, + 555, + 505, + 621 + ], + "lines": [ + { + "bbox": [ + 106, + 555, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 505, + 567 + ], + "score": 1.0, + "content": "Setup. We run experiments on five KGs of different sizes (Appendix A.1) varying the total number", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 565, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 165, + 578 + ], + "score": 1.0, + "content": "of nodes from", + "type": "text" + }, + { + "bbox": [ + 165, + 566, + 189, + 577 + ], + "score": 0.61, + "content": "\\mathord { \\sim } 1 5 \\mathrm { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 565, + 200, + 578 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 200, + 566, + 228, + 577 + ], + "score": 0.81, + "content": "{ \\sim } 2 . 5 \\mathbf { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 565, + 505, + 578 + ], + "score": 1.0, + "content": ". As a baseline, we compare to RotatE (Sun et al., 2019) that remains", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 577, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 505, + 590 + ], + "score": 1.0, + "content": "one of state-of-the-art shallow embedding models for transductive link prediction tasks. To balance", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 587, + 506, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 506, + 601 + ], + "score": 1.0, + "content": "with NodePiece, RotatE operates on a graph with added inverse edges as well. We report MRR with", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 599, + 506, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 124, + 612 + ], + "score": 1.0, + "content": "Hits", + "type": "text" + }, + { + "bbox": [ + 124, + 599, + 144, + 609 + ], + "score": 0.62, + "content": "@ 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 599, + 506, + 612 + ], + "score": 1.0, + "content": "in the filtered (Bordes et al., 2013) setting as evaluation metrics, and count parameters for", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 609, + 497, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 497, + 623 + ], + "score": 1.0, + "content": "all models. On larger KGs, we also compare to a smaller RotatE with a similar parameter budget.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 37.5 + }, + { + "type": "text", + "bbox": [ + 107, + 626, + 505, + 693 + ], + "lines": [ + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "score": 1.0, + "content": "In this task, NodePiece is equipped with a 2-layer MLP encoder. For a fair comparison, we also adopt", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 637, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 506, + 651 + ], + "score": 1.0, + "content": "the RotatE scoring function as a link prediction decoder. As to the NodePiece configuration, we", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 648, + 504, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 282, + 662 + ], + "score": 1.0, + "content": "generally keep the number of anchors below", + "type": "text" + }, + { + "bbox": [ + 282, + 649, + 301, + 659 + ], + "score": 0.87, + "content": "10 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 648, + 480, + 662 + ], + "score": 1.0, + "content": "of total nodes in respective graphs. We select", + "type": "text" + }, + { + "bbox": [ + 481, + 649, + 504, + 659 + ], + "score": 0.31, + "content": "1 \\mathrm { k } / 2 0", + "type": "inline_equation" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 659, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 506, + 673 + ], + "score": 1.0, + "content": "for FB15k-237 (i.e., total 1000 anchors and 20 anchors per tokenized node) with 15 unique outgoing", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 671, + 505, + 682 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 400, + 682 + ], + "score": 1.0, + "content": "relations in the relational context; 500/50 with 4 relations for WN18RR;", + "type": "text" + }, + { + "bbox": [ + 400, + 671, + 425, + 681 + ], + "score": 0.69, + "content": "7 \\mathrm { k } / 2 0", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 671, + 505, + 682 + ], + "score": 1.0, + "content": "with 6 relations for", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 681, + 506, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 151, + 694 + ], + "score": 1.0, + "content": "CoDEx-L;", + "type": "text" + }, + { + "bbox": [ + 151, + 682, + 180, + 692 + ], + "score": 0.42, + "content": "1 0 \\mathrm { k } / 2 0", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 681, + 506, + 694 + ], + "score": 1.0, + "content": "with 5 relations for YAGO 3-10. Other hyperparameters are listed in Appendix A.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 43.5 + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 442, + 712 + ], + "score": 1.0, + "content": "Discussion. 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Some performance loss is", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 720, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 505, + 734 + ], + "score": 1.0, + "content": "expected due to the compositional and compressive nature of entity tokenization. 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In our experiments, we probe two basic encoders: 1) MLP that takes as input a", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 142, + 507, + 160 + ], + "spans": [ + { + "bbox": [ + 104, + 142, + 211, + 160 + ], + "score": 1.0, + "content": "concatenated hash vector", + "type": "text" + }, + { + "bbox": [ + 212, + 145, + 259, + 156 + ], + "score": 0.9, + "content": "\\mathbb { R } ^ { 1 \\times ( k + m ) d }", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 142, + 348, + 160 + ], + "score": 1.0, + "content": "projecting it down to", + "type": "text" + }, + { + "bbox": [ + 348, + 145, + 361, + 156 + ], + "score": 0.73, + "content": "\\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 142, + 507, + 160 + ], + "score": 1.0, + "content": "; 2) Transformer encoder (Vaswani", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 155, + 507, + 172 + ], + "spans": [ + { + "bbox": [ + 104, + 155, + 399, + 172 + ], + "score": 1.0, + "content": "et al., 2017) with average pooling that takes as input an original sequence", + "type": "text" + }, + { + "bbox": [ + 399, + 156, + 442, + 168 + ], + "score": 0.92, + "content": "\\mathbb { R } ^ { ( k + m ) \\times d }", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 155, + 507, + 172 + ], + "score": 1.0, + "content": ". While MLP is", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 168, + 504, + 181 + ], + "spans": [ + { + "bbox": [ + 105, + 168, + 504, + 181 + ], + "score": 1.0, + "content": "faster and better scales to graphs with more edges, Transformer is slower but requires less trainable", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 179, + 506, + 192 + ], + "spans": [ + { + "bbox": [ + 105, + 179, + 506, + 192 + ], + "score": 1.0, + "content": "parameters. As the two encoders were chosen to illustrate the general applicability of the whole", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 191, + 463, + 203 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 463, + 203 + ], + "score": 1.0, + "content": "approach, we leave a study of even more efficient and effective encoders for future work.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6, + "bbox_fs": [ + 104, + 119, + 507, + 203 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 207, + 505, + 252 + ], + "lines": [ + { + "bbox": [ + 105, + 207, + 505, + 220 + ], + "spans": [ + { + "bbox": [ + 105, + 207, + 505, + 220 + ], + "score": 1.0, + "content": "While the nearest-neighbor hashing function has a greater number of collisions, its non-arbitrary", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 219, + 505, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 505, + 230 + ], + "score": 1.0, + "content": "mapping means that it is effectively permutation invariant. 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The nearest-anchor encoder with", + "type": "text" + }, + { + "bbox": [ + 307, + 254, + 327, + 269 + ], + "score": 0.91, + "content": "\\textstyle { \\binom { | A | } { k } }", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 253, + 381, + 270 + ], + "score": 1.0, + "content": "anchors and", + "type": "text" + }, + { + "bbox": [ + 381, + 256, + 396, + 268 + ], + "score": 0.89, + "content": "| m |", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 253, + 505, + 270 + ], + "score": 1.0, + "content": "subsampled relations, can", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 267, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 171, + 282 + ], + "score": 1.0, + "content": "be considered a", + "type": "text" + }, + { + "bbox": [ + 172, + 270, + 179, + 279 + ], + "score": 0.62, + "content": "\\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 267, + 274, + 282 + ], + "score": 1.0, + "content": "-SGD approximation of", + "type": "text" + }, + { + "bbox": [ + 275, + 268, + 314, + 280 + ], + "score": 0.91, + "content": "( k + | m | )", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 267, + 505, + 282 + ], + "score": 1.0, + "content": "-ary Janossy pooling with a canonical ordering", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 279, + 239, + 291 + ], + "spans": [ + { + "bbox": [ + 106, + 279, + 239, + 291 + ], + "score": 1.0, + "content": "induced by the anchor distances.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 253, + 505, + 291 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 298, + 505, + 343 + ], + "lines": [ + { + "bbox": [ + 105, + 299, + 505, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 191, + 311 + ], + "score": 1.0, + "content": "Janossy pooling with", + "type": "text" + }, + { + "bbox": [ + 191, + 300, + 198, + 309 + ], + "score": 0.75, + "content": "\\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 299, + 505, + 311 + ], + "score": 1.0, + "content": "-SGD can be used to learn a permutation-invariant function from a broad class", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 310, + 505, + 322 + ], + "spans": [ + { + "bbox": [ + 106, + 310, + 505, + 322 + ], + "score": 1.0, + "content": "of permutation-sensitve functions such as MLPs (Murphy et al., 2019). The permutation-invariant", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 321, + 505, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 505, + 333 + ], + "score": 1.0, + "content": "nature of the nearest-neighbor encoding scheme combined with the lack of transductive features such", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 332, + 504, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 332, + 504, + 344 + ], + "score": 1.0, + "content": "as node-specific embeddings mean that NodePiece can be used for inductive learning tasks as well.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 299, + 505, + 344 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 348, + 505, + 404 + ], + "lines": [ + { + "bbox": [ + 106, + 349, + 505, + 361 + ], + "spans": [ + { + "bbox": [ + 106, + 349, + 227, + 361 + ], + "score": 1.0, + "content": "With a fixed-size vocabulary", + "type": "text" + }, + { + "bbox": [ + 228, + 349, + 247, + 360 + ], + "score": 0.89, + "content": "V _ { N P }", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 349, + 505, + 361 + ], + "score": 1.0, + "content": ", the overall complexity and parameter budget of downstream", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 360, + 505, + 372 + ], + "spans": [ + { + "bbox": [ + 106, + 360, + 505, + 372 + ], + "score": 1.0, + "content": "models are largely defined by the complexity of the encoder and its inductive biases. By design, the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 371, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 106, + 371, + 505, + 383 + ], + "score": 1.0, + "content": "NodePiece smaller vocabulary - larger encoder framework is similar to various Transformer-based", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 381, + 506, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 506, + 394 + ], + "score": 1.0, + "content": "language models (Qiu et al., 2020) whose vocabulary size remains rather stable with the encoder", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 392, + 379, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 379, + 406 + ], + "score": 1.0, + "content": "being the most important part responsible for the final performance.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 349, + 506, + 406 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 420, + 200, + 432 + ], + "lines": [ + { + "bbox": [ + 105, + 419, + 201, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 201, + 434 + ], + "score": 1.0, + "content": "4 EXPERIMENTS", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 444, + 505, + 522 + ], + "lines": [ + { + "bbox": [ + 106, + 444, + 506, + 457 + ], + "spans": [ + { + "bbox": [ + 106, + 444, + 506, + 457 + ], + "score": 1.0, + "content": "We design the experimental program not seeking to outperform the best existing approaches but to", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 455, + 507, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 507, + 469 + ], + "score": 1.0, + "content": "show the versatility of NodePiece on a variety of KG-related tasks: transductive, inductive, out-of-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 466, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 505, + 478 + ], + "score": 1.0, + "content": "sample link prediction, and node classification (with relation prediction results in Appendix I). With", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 477, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 505, + 489 + ], + "score": 1.0, + "content": "this desiderata, we formulate the following research questions: RQ 1) Is it necessary to map each", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 488, + 506, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 506, + 500 + ], + "score": 1.0, + "content": "node to a unique vector for an acceptable performance on KG tasks?; RQ 2) What is the effect of", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 498, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 506, + 513 + ], + "score": 1.0, + "content": "hashing features?; RQ 3) Is there an optimal number of anchors per node, after which diminishing", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 510, + 222, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 222, + 522 + ], + "score": 1.0, + "content": "returns hit the performance?", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 444, + 507, + 522 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 534, + 280, + 545 + ], + "lines": [ + { + "bbox": [ + 106, + 534, + 282, + 547 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 282, + 547 + ], + "score": 1.0, + "content": "4.1 TRANSDUCTIVE LINK PREDICTION", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 106, + 555, + 505, + 621 + ], + "lines": [ + { + "bbox": [ + 106, + 555, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 505, + 567 + ], + "score": 1.0, + "content": "Setup. We run experiments on five KGs of different sizes (Appendix A.1) varying the total number", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 565, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 165, + 578 + ], + "score": 1.0, + "content": "of nodes from", + "type": "text" + }, + { + "bbox": [ + 165, + 566, + 189, + 577 + ], + "score": 0.61, + "content": "\\mathord { \\sim } 1 5 \\mathrm { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 565, + 200, + 578 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 200, + 566, + 228, + 577 + ], + "score": 0.81, + "content": "{ \\sim } 2 . 5 \\mathbf { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 565, + 505, + 578 + ], + "score": 1.0, + "content": ". As a baseline, we compare to RotatE (Sun et al., 2019) that remains", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 577, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 505, + 590 + ], + "score": 1.0, + "content": "one of state-of-the-art shallow embedding models for transductive link prediction tasks. To balance", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 587, + 506, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 506, + 601 + ], + "score": 1.0, + "content": "with NodePiece, RotatE operates on a graph with added inverse edges as well. We report MRR with", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 599, + 506, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 124, + 612 + ], + "score": 1.0, + "content": "Hits", + "type": "text" + }, + { + "bbox": [ + 124, + 599, + 144, + 609 + ], + "score": 0.62, + "content": "@ 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 599, + 506, + 612 + ], + "score": 1.0, + "content": "in the filtered (Bordes et al., 2013) setting as evaluation metrics, and count parameters for", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 609, + 497, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 497, + 623 + ], + "score": 1.0, + "content": "all models. On larger KGs, we also compare to a smaller RotatE with a similar parameter budget.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 555, + 506, + 623 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 626, + 505, + 693 + ], + "lines": [ + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "score": 1.0, + "content": "In this task, NodePiece is equipped with a 2-layer MLP encoder. For a fair comparison, we also adopt", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 637, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 506, + 651 + ], + "score": 1.0, + "content": "the RotatE scoring function as a link prediction decoder. As to the NodePiece configuration, we", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 648, + 504, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 282, + 662 + ], + "score": 1.0, + "content": "generally keep the number of anchors below", + "type": "text" + }, + { + "bbox": [ + 282, + 649, + 301, + 659 + ], + "score": 0.87, + "content": "10 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 648, + 480, + 662 + ], + "score": 1.0, + "content": "of total nodes in respective graphs. We select", + "type": "text" + }, + { + "bbox": [ + 481, + 649, + 504, + 659 + ], + "score": 0.31, + "content": "1 \\mathrm { k } / 2 0", + "type": "inline_equation" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 659, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 506, + 673 + ], + "score": 1.0, + "content": "for FB15k-237 (i.e., total 1000 anchors and 20 anchors per tokenized node) with 15 unique outgoing", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 671, + 505, + 682 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 400, + 682 + ], + "score": 1.0, + "content": "relations in the relational context; 500/50 with 4 relations for WN18RR;", + "type": "text" + }, + { + "bbox": [ + 400, + 671, + 425, + 681 + ], + "score": 0.69, + "content": "7 \\mathrm { k } / 2 0", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 671, + 505, + 682 + ], + "score": 1.0, + "content": "with 6 relations for", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 681, + 506, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 151, + 694 + ], + "score": 1.0, + "content": "CoDEx-L;", + "type": "text" + }, + { + "bbox": [ + 151, + 682, + 180, + 692 + ], + "score": 0.42, + "content": "1 0 \\mathrm { k } / 2 0", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 681, + 506, + 694 + ], + "score": 1.0, + "content": "with 5 relations for YAGO 3-10. Other hyperparameters are listed in Appendix A.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 43.5, + "bbox_fs": [ + 105, + 626, + 506, + 694 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 442, + 712 + ], + "score": 1.0, + "content": "Discussion. Generally, the results suggest that a fixed-size NodePiece vocabulary of", + "type": "text" + }, + { + "bbox": [ + 442, + 699, + 468, + 709 + ], + "score": 0.89, + "content": "{ < } 1 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "of nodes", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 710, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 140, + 721 + ], + "score": 1.0, + "content": "sustains", + "type": "text" + }, + { + "bbox": [ + 140, + 710, + 173, + 720 + ], + "score": 0.88, + "content": "80 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 711, + 202, + 721 + ], + "score": 1.0, + "content": "of Hits", + "type": "text" + }, + { + "bbox": [ + 202, + 710, + 221, + 720 + ], + "score": 0.7, + "content": "@ 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 711, + 505, + 721 + ], + "score": 1.0, + "content": "compared to 10x larger best shallow models. Some performance loss is", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 720, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 505, + 734 + ], + "score": 1.0, + "content": "expected due to the compositional and compressive nature of entity tokenization. On smaller graphs", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 363, + 506, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 506, + 375 + ], + "score": 1.0, + "content": "(Table 2), parameter saving might not be well pronounced due to the overall small number of nodes to", + "type": "text", + "cross_page": true + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 374, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 106, + 374, + 414, + 385 + ], + "score": 1.0, + "content": "embed. Still, taking even as few as 500 nodes as anchors on WN18RR retains", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 414, + 374, + 433, + 384 + ], + "score": 0.85, + "content": "90 \\%", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 434, + 374, + 505, + 385 + ], + "score": 1.0, + "content": "of the best model", + "type": "text", + "cross_page": true + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 384, + 506, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 506, + 397 + ], + "score": 1.0, + "content": "performance. On bigger graphs (Table 3), parameter efficiency is more pronounced, i.e., on YAGO", + "type": "text", + "cross_page": true + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 395, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 283, + 408 + ], + "score": 1.0, + "content": "3-10, a RotatE model of comparable size is", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 283, + 396, + 334, + 407 + ], + "score": 0.57, + "content": "2 0 \\mathrm { H i t s } @ 1 0", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 334, + 395, + 506, + 408 + ], + "score": 1.0, + "content": "points worse than a NodePiece-based one.", + "type": "text", + "cross_page": true + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 406, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 505, + 419 + ], + "score": 1.0, + "content": "This observation can be attributed to the fact the shrinking shallow models results in shrinking the", + "type": "text", + "cross_page": true + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 417, + 507, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 507, + 431 + ], + "score": 1.0, + "content": "embedding dimension of each node (20d for RotatE) which is inefficient on small parameter budgets.", + "type": "text", + "cross_page": true + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 429, + 506, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 506, + 441 + ], + "score": 1.0, + "content": "In contrast, a small fixed-size vocabulary allows for larger anchor embedding dimensions (100d for", + "type": "text", + "cross_page": true + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 439, + 452, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 452, + 452 + ], + "score": 1.0, + "content": "NodePiece with RotatE) since most of the parameter budget is defined by the encoder.", + "type": "text", + "cross_page": true + } + ], + "index": 18 + } + ], + "index": 48, + "bbox_fs": [ + 106, + 698, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 108, + 122, + 504, + 203 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 80, + 505, + 114 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 80, + 504, + 92 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 329, + 92 + ], + "score": 1.0, + "content": "Table 2: Transductive link prediction on smaller KGs.", + "type": "text" + }, + { + "bbox": [ + 329, + 81, + 336, + 92 + ], + "score": 0.71, + "content": "\\dagger", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 80, + 488, + 92 + ], + "score": 1.0, + "content": "results taken from (Sun et al., 2019).", + "type": "text" + }, + { + "bbox": [ + 489, + 80, + 504, + 92 + ], + "score": 0.79, + "content": "| V |", + "type": "inline_equation" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 91, + 505, + 103 + ], + "spans": [ + { + "bbox": [ + 106, + 91, + 237, + 103 + ], + "score": 1.0, + "content": "denotes vocabulary size (anchors", + "type": "text" + }, + { + "bbox": [ + 238, + 93, + 246, + 101 + ], + "score": 0.73, + "content": "^ +", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 91, + 288, + 103 + ], + "score": 1.0, + "content": "relations),", + "type": "text" + }, + { + "bbox": [ + 288, + 91, + 301, + 101 + ], + "score": 0.31, + "content": "\\# \\mathrm { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 91, + 447, + 103 + ], + "score": 1.0, + "content": "is a total parameter count (millions).", + "type": "text" + }, + { + "bbox": [ + 447, + 92, + 457, + 101 + ], + "score": 0.76, + "content": "\\%", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 91, + 505, + 103 + ], + "score": 1.0, + "content": "denotes the", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 102, + 286, + 114 + ], + "spans": [ + { + "bbox": [ + 106, + 102, + 124, + 114 + ], + "score": 1.0, + "content": "Hits", + "type": "text" + }, + { + "bbox": [ + 124, + 102, + 144, + 113 + ], + "score": 0.75, + "content": "@ 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 102, + 286, + 114 + ], + "score": 1.0, + "content": "ratio based on the strongest model.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "table_body", + "bbox": [ + 108, + 122, + 504, + 203 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 122, + 504, + 203 + ], + "spans": [ + { + "bbox": [ + 108, + 122, + 504, + 203 + ], + "score": 0.979, + "html": "
FB15k-237WN18RR
IV#P (M)MRRH@10%|VI#P (M)MRRH@10%
RotatE15k + 0.5k290.338t0.533t10040k +22410.476t0.571t100
NodePiece + RotatE1k + 0.5k3.20.2560.42079500+224.40.4030.51590
- no rel. context1k + 0.5k20.2580.42580500+224.20.2660.46581
- no distances1k + 0.5k3.20.2540.42179500+224.40.3910.51089
- no anchors,rels only0+ 0.5k1.40.2040.355670+220.30.0110.0190.3
", + "type": "table", + "image_path": "66b4734cb79ba1f9abda07094b916b95ba912ecc19fe62733588476f0e008b89.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 108, + 122, + 504, + 149.0 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 108, + 149.0, + 504, + 176.0 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 108, + 176.0, + 504, + 203.0 + ], + "spans": [], + "index": 5 + } + ] + } + ], + "index": 2.5 + }, + { + "type": "table", + "bbox": [ + 108, + 248, + 504, + 338 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 217, + 503, + 240 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 105, + 217, + 505, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 505, + 230 + ], + "score": 1.0, + "content": "Table 3: Transductive link prediction on bigger KGs. The same denotation as in Table 2. Second", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 228, + 380, + 240 + ], + "spans": [ + { + "bbox": [ + 106, + 228, + 380, + 240 + ], + "score": 1.0, + "content": "RotatE has a similar parameter budget as a NodePiece-based model.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5 + }, + { + "type": "table_body", + "bbox": [ + 108, + 248, + 504, + 338 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 108, + 248, + 504, + 338 + ], + "spans": [ + { + "bbox": [ + 108, + 248, + 504, + 338 + ], + "score": 0.971, + "html": "
CoDEx-LYAGO 3-10
IVI#P(M)MRRH@10%|VI#P (M)MRRH@10%
RotatE (500d)77k + 138770.2580.387100123k + 741230.495t0.670t100
RotatE (20d)77k + 1383.80.1960.32283123k + 744.80.1210.26239
NodePiece+RotatE7k +1383.60.1900.3138110k +744.10.2470.48873
- no rel. context7k +1383.10.2010.3328610k+743.70.2490.48272
- no distances7k +1383.60.1790.3027810k + 744.10.2500.49173
- no anchors,rels only0+1380.60.0630.121310+740.50.0250.0416
", + "type": "table", + "image_path": "f90816be4d022610ef598e181c863797306c91412226cf8126b7b575de0a482e.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 108, + 248, + 504, + 278.0 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 108, + 278.0, + 504, + 308.0 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 108, + 308.0, + 504, + 338.0 + ], + "spans": [], + "index": 10 + } + ] + } + ], + "index": 7.75 + }, + { + "type": "text", + "bbox": [ + 106, + 362, + 505, + 451 + ], + "lines": [ + { + "bbox": [ + 106, + 363, + 506, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 506, + 375 + ], + "score": 1.0, + "content": "(Table 2), parameter saving might not be well pronounced due to the overall small number of nodes to", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 374, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 106, + 374, + 414, + 385 + ], + "score": 1.0, + "content": "embed. Still, taking even as few as 500 nodes as anchors on WN18RR retains", + "type": "text" + }, + { + "bbox": [ + 414, + 374, + 433, + 384 + ], + "score": 0.85, + "content": "90 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 374, + 505, + 385 + ], + "score": 1.0, + "content": "of the best model", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 384, + 506, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 506, + 397 + ], + "score": 1.0, + "content": "performance. On bigger graphs (Table 3), parameter efficiency is more pronounced, i.e., on YAGO", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 395, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 283, + 408 + ], + "score": 1.0, + "content": "3-10, a RotatE model of comparable size is", + "type": "text" + }, + { + "bbox": [ + 283, + 396, + 334, + 407 + ], + "score": 0.57, + "content": "2 0 \\mathrm { H i t s } @ 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 395, + 506, + 408 + ], + "score": 1.0, + "content": "points worse than a NodePiece-based one.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 406, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 505, + 419 + ], + "score": 1.0, + "content": "This observation can be attributed to the fact the shrinking shallow models results in shrinking the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 417, + 507, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 507, + 431 + ], + "score": 1.0, + "content": "embedding dimension of each node (20d for RotatE) which is inefficient on small parameter budgets.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 429, + 506, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 506, + 441 + ], + "score": 1.0, + "content": "In contrast, a small fixed-size vocabulary allows for larger anchor embedding dimensions (100d for", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 439, + 452, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 452, + 452 + ], + "score": 1.0, + "content": "NodePiece with RotatE) since most of the parameter budget is defined by the encoder.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 106, + 456, + 505, + 545 + ], + "lines": [ + { + "bbox": [ + 105, + 456, + 506, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 506, + 469 + ], + "score": 1.0, + "content": "We further study the effect of different anchor selection combinations (Fig. 2). On WN18RR, fewer", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 467, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 257, + 479 + ], + "score": 1.0, + "content": "anchors with fewer anchors per node", + "type": "text" + }, + { + "bbox": [ + 258, + 467, + 288, + 479 + ], + "score": 0.91, + "content": "( | A | / k )", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 468, + 505, + 479 + ], + "score": 1.0, + "content": "yield relatively low accuracy but starting from 50/20", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 109, + 478, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 109, + 478, + 136, + 489 + ], + "score": 0.85, + "content": "( \\sim 0 . 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 479, + 147, + 491 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 148, + 479, + 164, + 489 + ], + "score": 0.28, + "content": "4 0 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 479, + 273, + 491 + ], + "score": 1.0, + "content": "nodes in the graph) the Hits", + "type": "text" + }, + { + "bbox": [ + 274, + 479, + 293, + 489 + ], + "score": 0.7, + "content": "@ 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 479, + 506, + 491 + ], + "score": 1.0, + "content": "performance starts to saturate. On FB15k-237, as few", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 489, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 506, + 502 + ], + "score": 1.0, + "content": "as 25 anchors already exhibit the signs of saturation where a further increase to 500 or 1000 anchors", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 500, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 505, + 514 + ], + "score": 1.0, + "content": "only marginally improves the performance. We hypothesize such a difference can be explained by", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "score": 1.0, + "content": "graph density, e.g., WN18RR is a sparse graph with a diameter of 23 and average anchor distance", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 522, + 507, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 507, + 535 + ], + "score": 1.0, + "content": "of 6 hops; while FB15k-237 is a denser graph with an average anchor distance of 2-3. Hence, on a", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 533, + 445, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 445, + 546 + ], + "score": 1.0, + "content": "sparse graph with longer distances, it takes more anchors to properly encode a node.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 107, + 550, + 505, + 594 + ], + "lines": [ + { + "bbox": [ + 105, + 550, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 285, + 562 + ], + "score": 1.0, + "content": "However, more precise predictions (e.g., Hits", + "type": "text" + }, + { + "bbox": [ + 285, + 551, + 299, + 560 + ], + "score": 0.57, + "content": "@ 1", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 550, + 505, + 562 + ], + "score": 1.0, + "content": ") reflected in the MRR metric (see Appendix E) still", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 561, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 422, + 573 + ], + "score": 1.0, + "content": "remain a challenging task for small vocabulary NodePiece setups, and bigger", + "type": "text" + }, + { + "bbox": [ + 422, + 561, + 447, + 573 + ], + "score": 0.92, + "content": "| A | / k", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 561, + 505, + 573 + ], + "score": 1.0, + "content": "combinations", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "score": 1.0, + "content": "alleviate this issue. We also observe that diminishing returns, which make further vocabulary increase", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 583, + 406, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 325, + 595 + ], + "score": 1.0, + "content": "less rewarding, start to appear from anchor set sizes of", + "type": "text" + }, + { + "bbox": [ + 326, + 583, + 345, + 593 + ], + "score": 0.88, + "content": "{ \\sim } 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 583, + 406, + 595 + ], + "score": 1.0, + "content": "of total nodes.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28.5 + }, + { + "type": "text", + "bbox": [ + 107, + 600, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 600, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 505, + 613 + ], + "score": 1.0, + "content": "Ablations. In the ablation study, we measure the impact of relational context and anchor distances", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "score": 1.0, + "content": "on link prediction (Table 2). Removing relational context and anchor distances does not tangibly", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 620, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 505, + 635 + ], + "score": 1.0, + "content": "affect the denser FB15k-237 data but does impair the accuracy on the sparser WN18RR. Pushing", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 633, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 506, + 645 + ], + "score": 1.0, + "content": "vocabulary sizes to the limit, we also investigate NodePiece behavior in the absence of anchors at all,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 645, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 645, + 374, + 655 + ], + "score": 1.0, + "content": "i.e., when hashes are defined only by the relational context of size", + "type": "text" + }, + { + "bbox": [ + 374, + 645, + 384, + 654 + ], + "score": 0.59, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 645, + 505, + 655 + ], + "score": 1.0, + "content": ". Interestingly, this still yields", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 312, + 667 + ], + "score": 1.0, + "content": "fair performance on FB15k-237 with just 7 points", + "type": "text" + }, + { + "bbox": [ + 312, + 655, + 351, + 666 + ], + "score": 0.46, + "content": "\\mathrm { H i t s } @ 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "drop, but drops to zero the WN18RR", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "performance. The fact that node embeddings might not be at all necessary but relations are more", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 677, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 689 + ], + "score": 1.0, + "content": "important supports the recent findings of Teru et al. (2020) that relies only on relations seen in a", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 688, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 506, + 700 + ], + "score": 1.0, + "content": "small subgraph around a target node. However, at this point, it seems to be a virtue of graphs with a", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 322, + 711 + ], + "score": 1.0, + "content": "diverse set of unique relations. That is, FB15k-237 has", + "type": "text" + }, + { + "bbox": [ + 323, + 699, + 339, + 709 + ], + "score": 0.52, + "content": "2 0 \\mathrm { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "more unique relations than WN18RR and", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 709, + 506, + 721 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 721 + ], + "score": 1.0, + "content": "resulting hashes have more diverse combinations of relations which lead to more discriminative node", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 721, + 466, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 466, + 732 + ], + "score": 1.0, + "content": "representations. Additionally, we visualize anchor embedding projections in Appendix D.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 36.5 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 310, + 762 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 108, + 122, + 504, + 203 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 80, + 505, + 114 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 80, + 504, + 92 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 329, + 92 + ], + "score": 1.0, + "content": "Table 2: Transductive link prediction on smaller KGs.", + "type": "text" + }, + { + "bbox": [ + 329, + 81, + 336, + 92 + ], + "score": 0.71, + "content": "\\dagger", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 80, + 488, + 92 + ], + "score": 1.0, + "content": "results taken from (Sun et al., 2019).", + "type": "text" + }, + { + "bbox": [ + 489, + 80, + 504, + 92 + ], + "score": 0.79, + "content": "| V |", + "type": "inline_equation" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 91, + 505, + 103 + ], + "spans": [ + { + "bbox": [ + 106, + 91, + 237, + 103 + ], + "score": 1.0, + "content": "denotes vocabulary size (anchors", + "type": "text" + }, + { + "bbox": [ + 238, + 93, + 246, + 101 + ], + "score": 0.73, + "content": "^ +", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 91, + 288, + 103 + ], + "score": 1.0, + "content": "relations),", + "type": "text" + }, + { + "bbox": [ + 288, + 91, + 301, + 101 + ], + "score": 0.31, + "content": "\\# \\mathrm { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 91, + 447, + 103 + ], + "score": 1.0, + "content": "is a total parameter count (millions).", + "type": "text" + }, + { + "bbox": [ + 447, + 92, + 457, + 101 + ], + "score": 0.76, + "content": "\\%", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 91, + 505, + 103 + ], + "score": 1.0, + "content": "denotes the", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 102, + 286, + 114 + ], + "spans": [ + { + "bbox": [ + 106, + 102, + 124, + 114 + ], + "score": 1.0, + "content": "Hits", + "type": "text" + }, + { + "bbox": [ + 124, + 102, + 144, + 113 + ], + "score": 0.75, + "content": "@ 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 102, + 286, + 114 + ], + "score": 1.0, + "content": "ratio based on the strongest model.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "table_body", + "bbox": [ + 108, + 122, + 504, + 203 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 122, + 504, + 203 + ], + "spans": [ + { + "bbox": [ + 108, + 122, + 504, + 203 + ], + "score": 0.979, + "html": "
FB15k-237WN18RR
IV#P (M)MRRH@10%|VI#P (M)MRRH@10%
RotatE15k + 0.5k290.338t0.533t10040k +22410.476t0.571t100
NodePiece + RotatE1k + 0.5k3.20.2560.42079500+224.40.4030.51590
- no rel. context1k + 0.5k20.2580.42580500+224.20.2660.46581
- no distances1k + 0.5k3.20.2540.42179500+224.40.3910.51089
- no anchors,rels only0+ 0.5k1.40.2040.355670+220.30.0110.0190.3
", + "type": "table", + "image_path": "66b4734cb79ba1f9abda07094b916b95ba912ecc19fe62733588476f0e008b89.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 108, + 122, + 504, + 149.0 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 108, + 149.0, + 504, + 176.0 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 108, + 176.0, + 504, + 203.0 + ], + "spans": [], + "index": 5 + } + ] + } + ], + "index": 2.5 + }, + { + "type": "table", + "bbox": [ + 108, + 248, + 504, + 338 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 217, + 503, + 240 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 105, + 217, + 505, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 505, + 230 + ], + "score": 1.0, + "content": "Table 3: Transductive link prediction on bigger KGs. The same denotation as in Table 2. Second", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 228, + 380, + 240 + ], + "spans": [ + { + "bbox": [ + 106, + 228, + 380, + 240 + ], + "score": 1.0, + "content": "RotatE has a similar parameter budget as a NodePiece-based model.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5 + }, + { + "type": "table_body", + "bbox": [ + 108, + 248, + 504, + 338 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 108, + 248, + 504, + 338 + ], + "spans": [ + { + "bbox": [ + 108, + 248, + 504, + 338 + ], + "score": 0.971, + "html": "
CoDEx-LYAGO 3-10
IVI#P(M)MRRH@10%|VI#P (M)MRRH@10%
RotatE (500d)77k + 138770.2580.387100123k + 741230.495t0.670t100
RotatE (20d)77k + 1383.80.1960.32283123k + 744.80.1210.26239
NodePiece+RotatE7k +1383.60.1900.3138110k +744.10.2470.48873
- no rel. context7k +1383.10.2010.3328610k+743.70.2490.48272
- no distances7k +1383.60.1790.3027810k + 744.10.2500.49173
- no anchors,rels only0+1380.60.0630.121310+740.50.0250.0416
", + "type": "table", + "image_path": "f90816be4d022610ef598e181c863797306c91412226cf8126b7b575de0a482e.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 108, + 248, + 504, + 278.0 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 108, + 278.0, + 504, + 308.0 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 108, + 308.0, + 504, + 338.0 + ], + "spans": [], + "index": 10 + } + ] + } + ], + "index": 7.75 + }, + { + "type": "text", + "bbox": [ + 106, + 362, + 505, + 451 + ], + "lines": [], + "index": 14.5, + "bbox_fs": [ + 105, + 363, + 507, + 452 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 456, + 505, + 545 + ], + "lines": [ + { + "bbox": [ + 105, + 456, + 506, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 506, + 469 + ], + "score": 1.0, + "content": "We further study the effect of different anchor selection combinations (Fig. 2). On WN18RR, fewer", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 467, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 257, + 479 + ], + "score": 1.0, + "content": "anchors with fewer anchors per node", + "type": "text" + }, + { + "bbox": [ + 258, + 467, + 288, + 479 + ], + "score": 0.91, + "content": "( | A | / k )", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 468, + 505, + 479 + ], + "score": 1.0, + "content": "yield relatively low accuracy but starting from 50/20", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 109, + 478, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 109, + 478, + 136, + 489 + ], + "score": 0.85, + "content": "( \\sim 0 . 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 479, + 147, + 491 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 148, + 479, + 164, + 489 + ], + "score": 0.28, + "content": "4 0 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 479, + 273, + 491 + ], + "score": 1.0, + "content": "nodes in the graph) the Hits", + "type": "text" + }, + { + "bbox": [ + 274, + 479, + 293, + 489 + ], + "score": 0.7, + "content": "@ 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 479, + 506, + 491 + ], + "score": 1.0, + "content": "performance starts to saturate. On FB15k-237, as few", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 489, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 506, + 502 + ], + "score": 1.0, + "content": "as 25 anchors already exhibit the signs of saturation where a further increase to 500 or 1000 anchors", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 500, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 505, + 514 + ], + "score": 1.0, + "content": "only marginally improves the performance. We hypothesize such a difference can be explained by", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "score": 1.0, + "content": "graph density, e.g., WN18RR is a sparse graph with a diameter of 23 and average anchor distance", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 522, + 507, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 507, + 535 + ], + "score": 1.0, + "content": "of 6 hops; while FB15k-237 is a denser graph with an average anchor distance of 2-3. Hence, on a", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 533, + 445, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 445, + 546 + ], + "score": 1.0, + "content": "sparse graph with longer distances, it takes more anchors to properly encode a node.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 456, + 507, + 546 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 550, + 505, + 594 + ], + "lines": [ + { + "bbox": [ + 105, + 550, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 285, + 562 + ], + "score": 1.0, + "content": "However, more precise predictions (e.g., Hits", + "type": "text" + }, + { + "bbox": [ + 285, + 551, + 299, + 560 + ], + "score": 0.57, + "content": "@ 1", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 550, + 505, + 562 + ], + "score": 1.0, + "content": ") reflected in the MRR metric (see Appendix E) still", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 561, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 422, + 573 + ], + "score": 1.0, + "content": "remain a challenging task for small vocabulary NodePiece setups, and bigger", + "type": "text" + }, + { + "bbox": [ + 422, + 561, + 447, + 573 + ], + "score": 0.92, + "content": "| A | / k", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 561, + 505, + 573 + ], + "score": 1.0, + "content": "combinations", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "score": 1.0, + "content": "alleviate this issue. We also observe that diminishing returns, which make further vocabulary increase", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 583, + 406, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 325, + 595 + ], + "score": 1.0, + "content": "less rewarding, start to appear from anchor set sizes of", + "type": "text" + }, + { + "bbox": [ + 326, + 583, + 345, + 593 + ], + "score": 0.88, + "content": "{ \\sim } 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 583, + 406, + 595 + ], + "score": 1.0, + "content": "of total nodes.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 550, + 505, + 595 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 600, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 600, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 505, + 613 + ], + "score": 1.0, + "content": "Ablations. In the ablation study, we measure the impact of relational context and anchor distances", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "score": 1.0, + "content": "on link prediction (Table 2). Removing relational context and anchor distances does not tangibly", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 620, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 505, + 635 + ], + "score": 1.0, + "content": "affect the denser FB15k-237 data but does impair the accuracy on the sparser WN18RR. Pushing", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 633, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 506, + 645 + ], + "score": 1.0, + "content": "vocabulary sizes to the limit, we also investigate NodePiece behavior in the absence of anchors at all,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 645, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 645, + 374, + 655 + ], + "score": 1.0, + "content": "i.e., when hashes are defined only by the relational context of size", + "type": "text" + }, + { + "bbox": [ + 374, + 645, + 384, + 654 + ], + "score": 0.59, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 645, + 505, + 655 + ], + "score": 1.0, + "content": ". Interestingly, this still yields", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 312, + 667 + ], + "score": 1.0, + "content": "fair performance on FB15k-237 with just 7 points", + "type": "text" + }, + { + "bbox": [ + 312, + 655, + 351, + 666 + ], + "score": 0.46, + "content": "\\mathrm { H i t s } @ 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "drop, but drops to zero the WN18RR", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "performance. The fact that node embeddings might not be at all necessary but relations are more", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 677, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 689 + ], + "score": 1.0, + "content": "important supports the recent findings of Teru et al. (2020) that relies only on relations seen in a", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 688, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 506, + 700 + ], + "score": 1.0, + "content": "small subgraph around a target node. However, at this point, it seems to be a virtue of graphs with a", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 322, + 711 + ], + "score": 1.0, + "content": "diverse set of unique relations. That is, FB15k-237 has", + "type": "text" + }, + { + "bbox": [ + 323, + 699, + 339, + 709 + ], + "score": 0.52, + "content": "2 0 \\mathrm { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "more unique relations than WN18RR and", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 709, + 506, + 721 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 721 + ], + "score": 1.0, + "content": "resulting hashes have more diverse combinations of relations which lead to more discriminative node", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 721, + 466, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 466, + 732 + ], + "score": 1.0, + "content": "representations. Additionally, we visualize anchor embedding projections in Appendix D.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 36.5, + "bbox_fs": [ + 105, + 600, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 177, + 79, + 433, + 178 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 177, + 79, + 433, + 178 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 177, + 79, + 433, + 178 + ], + "spans": [ + { + "bbox": [ + 177, + 79, + 433, + 178 + ], + "score": 0.97, + "type": "image", + "image_path": "8880db12bcff05dfd8a2c907eaa983fcd983dd4d64311541239dc80378c547a9.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 177, + 79, + 433, + 112.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 177, + 112.0, + 433, + 145.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 177, + 145.0, + 433, + 178.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 188, + 503, + 210 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 187, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 267, + 200 + ], + "score": 1.0, + "content": "Figure 2: Combinations of total anchors", + "type": "text" + }, + { + "bbox": [ + 267, + 189, + 276, + 198 + ], + "score": 0.7, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 187, + 505, + 200 + ], + "score": 1.0, + "content": "and anchors per node. Denser FB15k-237 saturates faster", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 199, + 386, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 199, + 150, + 211 + ], + "score": 1.0, + "content": "on smaller", + "type": "text" + }, + { + "bbox": [ + 151, + 199, + 160, + 209 + ], + "score": 0.73, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 199, + 386, + 211 + ], + "score": 1.0, + "content": "while sparse WN18RR saturates at around 500 anchors.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "title", + "bbox": [ + 108, + 232, + 211, + 243 + ], + "lines": [ + { + "bbox": [ + 105, + 231, + 213, + 245 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 213, + 245 + ], + "score": 1.0, + "content": "4.1.1 OGB WIKIKG 2", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 108, + 251, + 353, + 405 + ], + "lines": [ + { + "bbox": [ + 106, + 250, + 354, + 264 + ], + "spans": [ + { + "bbox": [ + 106, + 250, + 354, + 264 + ], + "score": 1.0, + "content": "To measure the benefits of NodePiece on large-scale KGs, we", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 262, + 354, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 354, + 275 + ], + "score": 1.0, + "content": "run a link prediction experiment on OGB WikiKG 2 (Hu et al.,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 272, + 354, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 285, + 285 + ], + "score": 1.0, + "content": "2020), a subset of Wikidata that consists of", + "type": "text" + }, + { + "bbox": [ + 285, + 273, + 309, + 284 + ], + "score": 0.54, + "content": "2 . 5 \\mathbf { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 272, + 354, + 285 + ], + "score": 1.0, + "content": "nodes and", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 283, + 353, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 283, + 353, + 297 + ], + "score": 1.0, + "content": "16M edges. 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Gener-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 338, + 354, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 354, + 352 + ], + "score": 1.0, + "content": "ally, such a NodePiece configuration can be paired with any", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 349, + 355, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 355, + 363 + ], + "score": 1.0, + "content": "link prediction decoder and we chose a non-parametric Au-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 361, + 353, + 372 + ], + "spans": [ + { + "bbox": [ + 106, + 361, + 353, + 372 + ], + "score": 1.0, + "content": "toSF (Zhang et al., 2020b) as one of the strongest decoders", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 371, + 354, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 264, + 384 + ], + "score": 1.0, + "content": "on this graph. 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Compared to the best reported shallow approaches,", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 14.5 + }, + { + "type": "table", + "bbox": [ + 362, + 296, + 502, + 389 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 361, + 264, + 504, + 287 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 360, + 263, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 360, + 263, + 505, + 277 + ], + "score": 1.0, + "content": "Table 4: Test MRR and parameter", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 360, + 276, + 473, + 288 + ], + "spans": [ + { + "bbox": [ + 360, + 276, + 473, + 288 + ], + "score": 1.0, + "content": "budget on OGB WikiKG 2.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 13.5 + }, + { + "type": "table_body", + "bbox": [ + 362, + 296, + 502, + 389 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 362, + 296, + 502, + 389 + ], + "spans": [ + { + "bbox": [ + 362, + 296, + 502, + 389 + ], + "score": 0.973, + "html": "
Model#ParamsMRR
NP + AutoSF - rel. context6.9M 5.9M0.570 ±0.003 0.592 ±0.003
- anc.dists - no anchors6.9M 1.3M0.570 ±0.004 0.476 ±0.001
AutoSF500M
PairRE0.546 ±0.005
500M0.521 ±0.003
RotatE1250M0.433 ±0.002
TransE1250M0.426 ±0.003
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We believe this result shows the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 426, + 506, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 506, + 439 + ], + "score": 1.0, + "content": "effectiveness of NodePiece on large KGs with a dramatic parameter size reduction without significant", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 438, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 505, + 450 + ], + "score": 1.0, + "content": "performance losses. Ablations report the duality of a relational context, i.e., removing it from hashes", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 449, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 106, + 449, + 505, + 460 + ], + "score": 1.0, + "content": "leads to even higher MRR scores. 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The unique feature of this benchmark compared to other evaluated tasks is that training and", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 556, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 106, + 556, + 505, + 568 + ], + "score": 1.0, + "content": "inference graphs are disjoint, i.e., inference at validation and test time is performed on a completely", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 564, + 507, + 580 + ], + "spans": [ + { + "bbox": [ + 104, + 564, + 507, + 580 + ], + "score": 1.0, + "content": "new graph comprised of new entities, and link prediction involves only entities unseen during training.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 578, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 106, + 578, + 505, + 590 + ], + "score": 1.0, + "content": "As inference graphs are disconnected from training ones, learning anchors from the training graph is", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 588, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 106, + 588, + 299, + 600 + ], + "score": 1.0, + "content": "useless, so node hashes are built only using the", + "type": "text" + }, + { + "bbox": [ + 299, + 589, + 309, + 598 + ], + "score": 0.74, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 310, + 588, + 505, + 600 + ], + "score": 1.0, + "content": "-sized relational context. On top of the obtained", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 598, + 506, + 612 + ], + "spans": [ + { + "bbox": [ + 104, + 598, + 506, + 612 + ], + "score": 1.0, + "content": "NodePiece features we then employ a relational message passing GNN, CompGCN (Vashishth et al.,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 609, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 506, + 623 + ], + "score": 1.0, + "content": "2020), with RotatE (Sun et al., 2019) as a scoring function for triples. More details on the setup and", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 621, + 371, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 371, + 633 + ], + "score": 1.0, + "content": "best hyperparameters for NodePiece are presented in Appendix J.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 37.5 + }, + { + "type": "text", + "bbox": [ + 107, + 637, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 105, + 637, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 254, + 650 + ], + "score": 1.0, + "content": "Baselines. We compare NodePiece", + "type": "text" + }, + { + "bbox": [ + 255, + 639, + 264, + 648 + ], + "score": 0.75, + "content": "^ +", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 637, + 505, + 650 + ], + "score": 1.0, + "content": "CompGCN with two families of models applicable in the", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 648, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 506, + 662 + ], + "score": 1.0, + "content": "inductive setting, i.e., rule-based methods, Neural LP (Yang et al., 2017), DRUM (Sadeghian et al.,", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 659, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 506, + 673 + ], + "score": 1.0, + "content": "2019), RuleN (Meilicke et al., 2018), and GNNs: GraIL (Teru et al., 2020) and recently proposed", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 670, + 333, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 333, + 683 + ], + "score": 1.0, + "content": "Neural Bellman-Ford Nets (NBFNet) (Zhu et al., 2021).", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 43.5 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "Discussion. 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Model#ParamsMRR
NP + AutoSF - rel. context6.9M 5.9M0.570 ±0.003 0.592 ±0.003
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AutoSF500M
PairRE0.546 ±0.005
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Best results are in bold, second best are", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 91, + 492, + 103 + ], + "spans": [ + { + "bbox": [ + 106, + 91, + 154, + 102 + ], + "score": 1.0, + "content": "underlined.", + "type": "text" + }, + { + "bbox": [ + 154, + 92, + 161, + 103 + ], + "score": 0.71, + "content": "\\dagger", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 91, + 492, + 102 + ], + "score": 1.0, + "content": "results taken from Teru et al. (2020). NBFNet results taken from Zhu et al. (2021).", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table_body", + "bbox": [ + 108, + 111, + 503, + 196 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 111, + 503, + 196 + ], + "spans": [ + { + "bbox": [ + 108, + 111, + 503, + 196 + ], + "score": 0.979, + "html": "
ClassMethodFB15k-237WN18RRNELL-995
V1V2V3V4v1V2V3V4V1V2V3V4
PathNeural LP †0.5290.5890.5290.5590.7440.6890.4620.6710.4080.7870.8270.806
DRUM+0.5290.5870.5290.5590.7440.6890.4620.6710.1940.7860.8270.806
RuleN†0.4980.7780.8770.8560.8090.7820.5340.7160.5350.8180.7730.614
GNNGraIL +0.6420.8180.8280.8930.8250.7870.5840.7340.5950.9330.9140.732
NBFNet0.8340.9490.9510.9600.9480.9050.8930.890-、--
NP + CompGCN0.8730.9390.9440.9490.8300.8860.7850.8070.8900.9010.9360.893
", + "type": "table", + "image_path": "f48ec65dc1b122e598bade987edd3f0babf946e3daa4bfe57adda5b94d6d061e.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 108, + 111, + 503, + 139.33333333333334 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 108, + 139.33333333333334, + 503, + 167.66666666666669 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 108, + 167.66666666666669, + 503, + 196.00000000000003 + ], + "spans": [], + "index": 4 + } + ] + } + ], + "index": 1.75 + }, + { + "type": "table", + "bbox": [ + 108, + 243, + 504, + 325 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 208, + 504, + 231 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 106, + 208, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 106, + 208, + 253, + 221 + ], + "score": 1.0, + "content": "Table 6: Node classification results.", + "type": "text" + }, + { + "bbox": [ + 253, + 208, + 268, + 221 + ], + "score": 0.87, + "content": "| V |", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 208, + 403, + 221 + ], + "score": 1.0, + "content": "denotes vocabulary size (anchors", + "type": "text" + }, + { + "bbox": [ + 403, + 210, + 412, + 219 + ], + "score": 0.77, + "content": "^ +", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 208, + 505, + 221 + ], + "score": 1.0, + "content": "relations), #P is a total", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 219, + 218, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 218, + 231 + ], + "score": 1.0, + "content": "parameter count (millions).", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5 + }, + { + "type": "table_caption", + "bbox": [ + 284, + 240, + 354, + 249 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 283, + 239, + 354, + 250 + ], + "spans": [ + { + "bbox": [ + 283, + 239, + 314, + 250 + ], + "score": 1.0, + "content": "WD50K", + "type": "text" + }, + { + "bbox": [ + 326, + 239, + 354, + 250 + ], + "score": 1.0, + "content": "labeled)", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "table_caption", + "bbox": [ + 405, + 240, + 479, + 249 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 405, + 240, + 479, + 249 + ], + "spans": [], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "table_body", + "bbox": [ + 108, + 243, + 504, + 325 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 108, + 243, + 504, + 325 + ], + "spans": [ + { + "bbox": [ + 108, + 243, + 504, + 325 + ], + "score": 0.972, + "html": "
WD50K (5%labeled)WD50K(10%labeled)
|VI#P (M)ROC-AUCPRC-AUCHard AccROC-AUCPRC-AUCHard Acc
MLP46k + 1k4.10.5030.0160.0010.5100.0170.002
CompGCN46k + 1k4.40.8360.2800.1760.8340.2650.161
NodePiece +GNN50+1k0.750.9810.4430.5130.9810.4500.516
- no rel. context50+1k0.640.9820.4460.5340.9820.4490.530
- no distances50+1k0.740.9810.4480.5160.9810.4480.513
- no anchors,rels only0+ 1k0.540.9840.4530.5320.9840.4560.533
", + "type": "table", + "image_path": "dbbe4b623335aa6f2ceb8d498783c88373c9358c87ce5cb140612386961ccd3b.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 108, + 243, + 504, + 270.3333333333333 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 108, + 270.3333333333333, + 504, + 297.66666666666663 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 108, + 297.66666666666663, + 504, + 324.99999999999994 + ], + "spans": [], + "index": 11 + } + ] + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 107, + 347, + 505, + 381 + ], + "lines": [ + { + "bbox": [ + 106, + 347, + 506, + 361 + ], + "spans": [ + { + "bbox": [ + 106, + 347, + 506, + 361 + ], + "score": 1.0, + "content": "CompGCN outperforms GraIL by a large margin in all (except one) experiments and is competitive", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 358, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 505, + 371 + ], + "score": 1.0, + "content": "to NBFNet on relation-rich FB15k-237 splits. As expected, NodePiece features are less efficient on", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 369, + 506, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 506, + 383 + ], + "score": 1.0, + "content": "sparse graphs (like WN18RR with few unique relations) but still outperform topology-based GraIL.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13 + }, + { + "type": "title", + "bbox": [ + 108, + 396, + 231, + 407 + ], + "lines": [ + { + "bbox": [ + 105, + 394, + 233, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 233, + 410 + ], + "score": 1.0, + "content": "4.3 NODE CLASSIFICATION", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 106, + 417, + 505, + 495 + ], + "lines": [ + { + "bbox": [ + 105, + 417, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 505, + 430 + ], + "score": 1.0, + "content": "Setup. Due to the lack of established node classification datasets on multi-relational KGs, we design", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 429, + 505, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 505, + 440 + ], + "score": 1.0, + "content": "a multi-class multi-label task based on a triple version of a recent WD50K (Galkin et al., 2020)", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 438, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 505, + 453 + ], + "score": 1.0, + "content": "extracted from Wikidata. The pre-processing steps are described in Appendix F, and the final graph", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 451, + 504, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 504, + 462 + ], + "score": 1.0, + "content": "consists of 46K nodes and 222K edges. The task belongs to the family of transductive (the whole", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 462, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 505, + 474 + ], + "score": 1.0, + "content": "graph is seen during training) semi-supervised (only a fraction of nodes are labeled) problems, where", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 472, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 505, + 485 + ], + "score": 1.0, + "content": "labels are 465 classes as seen in Wikidata. In a semi-supervised mode, we test the models on a graph", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 482, + 470, + 497 + ], + "spans": [ + { + "bbox": [ + 104, + 482, + 126, + 497 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 126, + 484, + 141, + 494 + ], + "score": 0.85, + "content": "5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 482, + 159, + 497 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 159, + 484, + 179, + 494 + ], + "score": 0.86, + "content": "10 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 482, + 470, + 497 + ], + "score": 1.0, + "content": "of labeled nodes. Node features have to be learned as node embeddings.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 500, + 505, + 567 + ], + "lines": [ + { + "bbox": [ + 106, + 501, + 505, + 512 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 505, + 512 + ], + "score": 1.0, + "content": "As baselines, we compare to a 2-layer MLP and CompGCN (Vashishth et al., 2020) in a full-batch", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 512, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 506, + 523 + ], + "score": 1.0, + "content": "mode which is one of the strongest GNN encoders for multi-relational KGs. Both baselines learn a", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "score": 1.0, + "content": "full entity vocabulary. We report ROC-AUC, PRC-AUC, and Hard Accuracy metrics as commonly", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 531, + 507, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 507, + 547 + ], + "score": 1.0, + "content": "done in standard graph benchmarks like OGB (Hu et al., 2020). For PRC-AUC and Hard Accuracy,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 544, + 506, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 506, + 557 + ], + "score": 1.0, + "content": "we binarize predicted logits using a threshold of 0.5. Hard Accuracy corresponds to the exact match", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 554, + 457, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 457, + 568 + ], + "score": 1.0, + "content": "of a predicted sparse 465-dimensional vector to a sparse 465-dimensional labels vector.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 25.5 + }, + { + "type": "text", + "bbox": [ + 107, + 572, + 505, + 616 + ], + "lines": [ + { + "bbox": [ + 105, + 572, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 505, + 584 + ], + "score": 1.0, + "content": "NodePiece is configured to have only 50 anchors and use 10 nearest anchors per node with 5 unique", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 583, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 505, + 595 + ], + "score": 1.0, + "content": "relations in the relational context. The dimensionality of anchors and relations is the same as in the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "score": 1.0, + "content": "baseline CompGCN. Each epoch, we first materialize all entity embeddings through the NodePiece", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 605, + 448, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 448, + 617 + ], + "score": 1.0, + "content": "encoder and then send the materialized matrix to CompGCN with the class predictor.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30.5 + }, + { + "type": "text", + "bbox": [ + 106, + 622, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 622, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 212, + 634 + ], + "score": 1.0, + "content": "Discussion. Surprisingly,", + "type": "text" + }, + { + "bbox": [ + 213, + 622, + 245, + 633 + ], + "score": 0.62, + "content": "{ \\sim } 1 0 0 0 \\mathrm { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 622, + 505, + 634 + ], + "score": 1.0, + "content": "vocabulary reduction ratio (50 anchors against 46k for shallow", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 633, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 505, + 646 + ], + "score": 1.0, + "content": "models) greatly outperforms the baselines (Table 6). MLP, as expected, is not able to cope with", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 644, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 506, + 657 + ], + "score": 1.0, + "content": "the task producing random predictions. CompGCN, in turn, outperforms MLP demonstrating non-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 654, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 104, + 654, + 506, + 669 + ], + "score": 1.0, + "content": "random outputs as seen by the ROC-AUC score of 0.836 and higher PRC-AUC and Hard Accuracy", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "score": 1.0, + "content": "metrics. Still, a NodePiece-equipped CompGCN with 50 anchors reaches even higher ROC-AUC of", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 351, + 690 + ], + "score": 1.0, + "content": "0.98 with considerable improvements along other metrics, i.e.,", + "type": "text" + }, + { + "bbox": [ + 352, + 677, + 381, + 687 + ], + "score": 0.71, + "content": "+ 1 6 - 1 9", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 676, + 469, + 690 + ], + "score": 1.0, + "content": "PRC-AUC points and", + "type": "text" + }, + { + "bbox": [ + 469, + 677, + 480, + 687 + ], + "score": 0.6, + "content": "3 \\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "boost", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "along the hardest accuracy metric. We attribute such a noticeable performance difference to better", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 698, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 713 + ], + "score": 1.0, + "content": "generalization capabilities of the NodePiece model. That is, a generalization gap between training", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 263, + 722 + ], + "score": 1.0, + "content": "and validation metrics of the NodePiece", + "type": "text" + }, + { + "bbox": [ + 263, + 711, + 272, + 720 + ], + "score": 0.67, + "content": "^ +", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "CompGCN is much smaller compared to the baselines who", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "score": 1.0, + "content": "overfit rather heavily ( cf. the training curves in the Appendix G). The effect remains after increasing", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 37.5 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 300, + 750, + 309, + 761 + ], + "spans": [ + { + "bbox": [ + 300, + 750, + 309, + 761 + ], + "score": 1.0, + "content": "8", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 108, + 111, + 503, + 196 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 80, + 504, + 103 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 79, + 505, + 93 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 285, + 93 + ], + "score": 1.0, + "content": "Table 5: Inductive link prediction results,", + "type": "text" + }, + { + "bbox": [ + 285, + 80, + 324, + 91 + ], + "score": 0.42, + "content": "\\mathrm { H i t s } @ 1 0 .", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 79, + 505, + 93 + ], + "score": 1.0, + "content": ". Best results are in bold, second best are", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 91, + 492, + 103 + ], + "spans": [ + { + "bbox": [ + 106, + 91, + 154, + 102 + ], + "score": 1.0, + "content": "underlined.", + "type": "text" + }, + { + "bbox": [ + 154, + 92, + 161, + 103 + ], + "score": 0.71, + "content": "\\dagger", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 91, + 492, + 102 + ], + "score": 1.0, + "content": "results taken from Teru et al. (2020). NBFNet results taken from Zhu et al. (2021).", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table_body", + "bbox": [ + 108, + 111, + 503, + 196 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 111, + 503, + 196 + ], + "spans": [ + { + "bbox": [ + 108, + 111, + 503, + 196 + ], + "score": 0.979, + "html": "
ClassMethodFB15k-237WN18RRNELL-995
V1V2V3V4v1V2V3V4V1V2V3V4
PathNeural LP †0.5290.5890.5290.5590.7440.6890.4620.6710.4080.7870.8270.806
DRUM+0.5290.5870.5290.5590.7440.6890.4620.6710.1940.7860.8270.806
RuleN†0.4980.7780.8770.8560.8090.7820.5340.7160.5350.8180.7730.614
GNNGraIL +0.6420.8180.8280.8930.8250.7870.5840.7340.5950.9330.9140.732
NBFNet0.8340.9490.9510.9600.9480.9050.8930.890-、--
NP + CompGCN0.8730.9390.9440.9490.8300.8860.7850.8070.8900.9010.9360.893
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WD50K (5%labeled)WD50K(10%labeled)
|VI#P (M)ROC-AUCPRC-AUCHard AccROC-AUCPRC-AUCHard Acc
MLP46k + 1k4.10.5030.0160.0010.5100.0170.002
CompGCN46k + 1k4.40.8360.2800.1760.8340.2650.161
NodePiece +GNN50+1k0.750.9810.4430.5130.9810.4500.516
- no rel. context50+1k0.640.9820.4460.5340.9820.4490.530
- no distances50+1k0.740.9810.4480.5160.9810.4480.513
- no anchors,rels only0+ 1k0.540.9840.4530.5320.9840.4560.533
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As expected, NodePiece features are less efficient on", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 369, + 506, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 506, + 383 + ], + "score": 1.0, + "content": "sparse graphs (like WN18RR with few unique relations) but still outperform topology-based GraIL.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 347, + 506, + 383 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 396, + 231, + 407 + ], + "lines": [ + { + "bbox": [ + 105, + 394, + 233, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 233, + 410 + ], + "score": 1.0, + "content": "4.3 NODE CLASSIFICATION", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 106, + 417, + 505, + 495 + ], + "lines": [ + { + "bbox": [ + 105, + 417, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 505, + 430 + ], + "score": 1.0, + "content": "Setup. Due to the lack of established node classification datasets on multi-relational KGs, we design", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 429, + 505, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 505, + 440 + ], + "score": 1.0, + "content": "a multi-class multi-label task based on a triple version of a recent WD50K (Galkin et al., 2020)", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 438, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 505, + 453 + ], + "score": 1.0, + "content": "extracted from Wikidata. The pre-processing steps are described in Appendix F, and the final graph", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 451, + 504, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 504, + 462 + ], + "score": 1.0, + "content": "consists of 46K nodes and 222K edges. The task belongs to the family of transductive (the whole", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 462, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 505, + 474 + ], + "score": 1.0, + "content": "graph is seen during training) semi-supervised (only a fraction of nodes are labeled) problems, where", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 472, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 505, + 485 + ], + "score": 1.0, + "content": "labels are 465 classes as seen in Wikidata. In a semi-supervised mode, we test the models on a graph", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 482, + 470, + 497 + ], + "spans": [ + { + "bbox": [ + 104, + 482, + 126, + 497 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 126, + 484, + 141, + 494 + ], + "score": 0.85, + "content": "5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 482, + 159, + 497 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 159, + 484, + 179, + 494 + ], + "score": 0.86, + "content": "10 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 482, + 470, + 497 + ], + "score": 1.0, + "content": "of labeled nodes. Node features have to be learned as node embeddings.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 19, + "bbox_fs": [ + 104, + 417, + 505, + 497 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 500, + 505, + 567 + ], + "lines": [ + { + "bbox": [ + 106, + 501, + 505, + 512 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 505, + 512 + ], + "score": 1.0, + "content": "As baselines, we compare to a 2-layer MLP and CompGCN (Vashishth et al., 2020) in a full-batch", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 512, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 506, + 523 + ], + "score": 1.0, + "content": "mode which is one of the strongest GNN encoders for multi-relational KGs. Both baselines learn a", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "score": 1.0, + "content": "full entity vocabulary. We report ROC-AUC, PRC-AUC, and Hard Accuracy metrics as commonly", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 531, + 507, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 507, + 547 + ], + "score": 1.0, + "content": "done in standard graph benchmarks like OGB (Hu et al., 2020). For PRC-AUC and Hard Accuracy,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 544, + 506, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 506, + 557 + ], + "score": 1.0, + "content": "we binarize predicted logits using a threshold of 0.5. Hard Accuracy corresponds to the exact match", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 554, + 457, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 457, + 568 + ], + "score": 1.0, + "content": "of a predicted sparse 465-dimensional vector to a sparse 465-dimensional labels vector.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 501, + 507, + 568 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 572, + 505, + 616 + ], + "lines": [ + { + "bbox": [ + 105, + 572, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 505, + 584 + ], + "score": 1.0, + "content": "NodePiece is configured to have only 50 anchors and use 10 nearest anchors per node with 5 unique", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 583, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 505, + 595 + ], + "score": 1.0, + "content": "relations in the relational context. The dimensionality of anchors and relations is the same as in the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "score": 1.0, + "content": "baseline CompGCN. Each epoch, we first materialize all entity embeddings through the NodePiece", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 605, + 448, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 448, + 617 + ], + "score": 1.0, + "content": "encoder and then send the materialized matrix to CompGCN with the class predictor.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 572, + 505, + 617 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 622, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 622, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 212, + 634 + ], + "score": 1.0, + "content": "Discussion. Surprisingly,", + "type": "text" + }, + { + "bbox": [ + 213, + 622, + 245, + 633 + ], + "score": 0.62, + "content": "{ \\sim } 1 0 0 0 \\mathrm { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 622, + 505, + 634 + ], + "score": 1.0, + "content": "vocabulary reduction ratio (50 anchors against 46k for shallow", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 633, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 505, + 646 + ], + "score": 1.0, + "content": "models) greatly outperforms the baselines (Table 6). MLP, as expected, is not able to cope with", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 644, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 506, + 657 + ], + "score": 1.0, + "content": "the task producing random predictions. CompGCN, in turn, outperforms MLP demonstrating non-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 654, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 104, + 654, + 506, + 669 + ], + "score": 1.0, + "content": "random outputs as seen by the ROC-AUC score of 0.836 and higher PRC-AUC and Hard Accuracy", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "score": 1.0, + "content": "metrics. Still, a NodePiece-equipped CompGCN with 50 anchors reaches even higher ROC-AUC of", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 351, + 690 + ], + "score": 1.0, + "content": "0.98 with considerable improvements along other metrics, i.e.,", + "type": "text" + }, + { + "bbox": [ + 352, + 677, + 381, + 687 + ], + "score": 0.71, + "content": "+ 1 6 - 1 9", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 676, + 469, + 690 + ], + "score": 1.0, + "content": "PRC-AUC points and", + "type": "text" + }, + { + "bbox": [ + 469, + 677, + 480, + 687 + ], + "score": 0.6, + "content": "3 \\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "boost", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "along the hardest accuracy metric. We attribute such a noticeable performance difference to better", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 698, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 713 + ], + "score": 1.0, + "content": "generalization capabilities of the NodePiece model. That is, a generalization gap between training", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 263, + 722 + ], + "score": 1.0, + "content": "and validation metrics of the NodePiece", + "type": "text" + }, + { + "bbox": [ + 263, + 711, + 272, + 720 + ], + "score": 0.67, + "content": "^ +", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "CompGCN is much smaller compared to the baselines who", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "score": 1.0, + "content": "overfit rather heavily ( cf. the training curves in the Appendix G). The effect remains after increasing", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 212, + 505, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 212, + 232, + 225 + ], + "score": 1.0, + "content": "the number of labeled nodes to", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 233, + 213, + 252, + 223 + ], + "score": 0.9, + "content": "10 \\%", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 253, + 212, + 505, + 225 + ], + "score": 1.0, + "content": ". Even with 50 anchors, the overall performance is saturated as", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 222, + 408, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 408, + 236 + ], + "score": 1.0, + "content": "the further increase of the vocabulary size did not bring any improvements.", + "type": "text", + "cross_page": true + } + ], + "index": 6 + } + ], + "index": 37.5, + "bbox_fs": [ + 104, + 622, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 108, + 111, + 504, + 192 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 80, + 504, + 103 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 80, + 505, + 93 + ], + "spans": [ + { + "bbox": [ + 106, + 80, + 262, + 93 + ], + "score": 1.0, + "content": "Table 7: Out-of-sample link prediction.", + "type": "text" + }, + { + "bbox": [ + 263, + 81, + 269, + 92 + ], + "score": 0.68, + "content": "\\dagger", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 80, + 456, + 93 + ], + "score": 1.0, + "content": "results are taken from (Albooyeh et al., 2020).", + "type": "text" + }, + { + "bbox": [ + 457, + 80, + 471, + 92 + ], + "score": 0.78, + "content": "| V |", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 80, + 505, + 93 + ], + "score": 1.0, + "content": "denotes", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 91, + 419, + 103 + ], + "spans": [ + { + "bbox": [ + 106, + 91, + 207, + 103 + ], + "score": 1.0, + "content": "vocabulary size (anchors", + "type": "text" + }, + { + "bbox": [ + 207, + 93, + 215, + 101 + ], + "score": 0.76, + "content": "^ +", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 91, + 419, + 103 + ], + "score": 1.0, + "content": "relations), #P is a total parameter count (millions).", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table_body", + "bbox": [ + 108, + 111, + 504, + 192 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 111, + 504, + 192 + ], + "spans": [ + { + "bbox": [ + 108, + 111, + 504, + 192 + ], + "score": 0.978, + "html": "
oFB15k-2370YAGO 3-10 (117k)
|VI#P (M)MRRH@10%|VI#P (M)MRRH@10%
oDistMult-ERAvg11k + 0.5k2.40.256t0.420t100117k + 7423.40OM0OM-
NodePiece +DistMult1k + 0.5k10.2060.3728810k+742.70.1330.261100
- no rel. context1k + 0.5k10.1730.3297810k +742.70.1250.24594
- no distances1k + 0.5k10.2080.3728810k + 742.70.1330.26099
- no anchors,rels only0+ 0.5k0.80.0690.127300+740.70.0150.0176
", + "type": "table", + "image_path": "07c20997c24e66437e8a429e50adb5f396b8a72b678b575f26f49a5e4438c66e.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 108, + 111, + 504, + 138.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 108, + 138.0, + 504, + 165.0 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 108, + 165.0, + 504, + 192.0 + ], + "spans": [], + "index": 4 + } + ] + } + ], + "index": 1.75 + }, + { + "type": "text", + "bbox": [ + 106, + 213, + 504, + 235 + ], + "lines": [ + { + "bbox": [ + 105, + 212, + 505, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 212, + 232, + 225 + ], + "score": 1.0, + "content": "the number of labeled nodes to", + "type": "text" + }, + { + "bbox": [ + 233, + 213, + 252, + 223 + ], + "score": 0.9, + "content": "10 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 212, + 505, + 225 + ], + "score": 1.0, + "content": ". Even with 50 anchors, the overall performance is saturated as", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 222, + 408, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 408, + 236 + ], + "score": 1.0, + "content": "the further increase of the vocabulary size did not bring any improvements.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5 + }, + { + "type": "text", + "bbox": [ + 107, + 240, + 505, + 285 + ], + "lines": [ + { + "bbox": [ + 105, + 241, + 505, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 505, + 253 + ], + "score": 1.0, + "content": "Ablations. We probe setups where NodePiece hashes use only anchors or only relational context, and", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 251, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 505, + 265 + ], + "score": 1.0, + "content": "find they both deliver a similar performance. Following the previous experiments on dense graphs", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 263, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 505, + 275 + ], + "score": 1.0, + "content": "with lots of unique relations, it appears that node classification can be performed rather accurately", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 274, + 435, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 435, + 286 + ], + "score": 1.0, + "content": "based only on the node relational context which is captured by NodePiece hashes.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8.5 + }, + { + "type": "title", + "bbox": [ + 108, + 299, + 284, + 310 + ], + "lines": [ + { + "bbox": [ + 106, + 299, + 285, + 311 + ], + "spans": [ + { + "bbox": [ + 106, + 299, + 285, + 311 + ], + "score": 1.0, + "content": "4.4 OUT-OF-SAMPLE LINK PREDICTION", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 320, + 505, + 375 + ], + "lines": [ + { + "bbox": [ + 106, + 320, + 505, + 332 + ], + "spans": [ + { + "bbox": [ + 106, + 320, + 505, + 332 + ], + "score": 1.0, + "content": "Setup. In the out-of-sample setup, validation and test splits contain unseen entities that arrive with", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 330, + 506, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 506, + 344 + ], + "score": 1.0, + "content": "a few edges connected to the seen nodes. For this experiment, we use the out-of-sample FB15k-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 341, + 506, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 506, + 354 + ], + "score": 1.0, + "content": "237 split (oFB15k-237) as designed in Albooyeh et al. (2020). We do not employ their version of", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 352, + 507, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 507, + 366 + ], + "score": 1.0, + "content": "WN18RR as the split contains too many disconnected entities and components in the train graph.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 364, + 496, + 377 + ], + "spans": [ + { + "bbox": [ + 106, + 364, + 496, + 377 + ], + "score": 1.0, + "content": "Instead, using the authors script, we sample a much bigger out-of-sample version of YAGO 3-10.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 380, + 505, + 436 + ], + "lines": [ + { + "bbox": [ + 105, + 380, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 505, + 393 + ], + "score": 1.0, + "content": "As a baseline, we compare to oDistMult (Albooyeh et al., 2020) which aggregates embeddings", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 391, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 505, + 405 + ], + "score": 1.0, + "content": "of all seen neighboring nodes around the unseen one (akin to 1-layer message passing with mean", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 402, + 506, + 417 + ], + "spans": [ + { + "bbox": [ + 104, + 402, + 506, + 417 + ], + "score": 1.0, + "content": "aggregator). We adopt the same evaluation protocol - given an unseen node with its connecting", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 414, + 505, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 505, + 427 + ], + "score": 1.0, + "content": "edges, we mask one of the edges and predict its tail or head using the rest of the edges, repeating this", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 425, + 428, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 339, + 438 + ], + "score": 1.0, + "content": "procedure for each edge. We report filtered MRR and Hits", + "type": "text" + }, + { + "bbox": [ + 340, + 425, + 360, + 435 + ], + "score": 0.73, + "content": "@ 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 425, + 428, + 438 + ], + "score": 1.0, + "content": "as main metrics.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 442, + 505, + 496 + ], + "lines": [ + { + "bbox": [ + 106, + 441, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 505, + 453 + ], + "score": 1.0, + "content": "NodePiece enables traditional transductive-only models to perform inductive inference as both seen", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 453, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 106, + 453, + 505, + 465 + ], + "score": 1.0, + "content": "and unseen nodes are tokenized using the same vocabulary. For a smaller oFB15k-237 the NodePiece", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 463, + 504, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 168, + 475 + ], + "score": 1.0, + "content": "vocabulary has", + "type": "text" + }, + { + "bbox": [ + 169, + 464, + 192, + 474 + ], + "score": 0.34, + "content": "1 \\mathrm { k } / 2 0", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 463, + 475, + 475 + ], + "score": 1.0, + "content": "configuration with 15 relations, while in a bigger oYAGO 3-10 we use", + "type": "text" + }, + { + "bbox": [ + 475, + 464, + 504, + 474 + ], + "score": 0.48, + "content": "1 0 \\mathrm { k } / 2 0", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 474, + 507, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 507, + 487 + ], + "score": 1.0, + "content": "with 5 relations. For this task, we apply a transformer encoder instead of MLP. For a fair comparison,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 486, + 292, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 292, + 497 + ], + "score": 1.0, + "content": "we use DistMult as a scoring function as well.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 502, + 505, + 612 + ], + "lines": [ + { + "bbox": [ + 106, + 502, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 106, + 502, + 453, + 514 + ], + "score": 1.0, + "content": "Discussion. The results in Table 7 show that a simple NodePiece-based model retains", + "type": "text" + }, + { + "bbox": [ + 453, + 502, + 478, + 513 + ], + "score": 0.88, + "content": "{ \\sim } 9 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 502, + 505, + 514 + ], + "score": 1.0, + "content": "of the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 513, + 505, + 526 + ], + "spans": [ + { + "bbox": [ + 106, + 513, + 505, + 526 + ], + "score": 1.0, + "content": "baseline performance on oFB15k-237, but achieved faster and computationally inexpensive compared", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 524, + 506, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 506, + 537 + ], + "score": 1.0, + "content": "to oDistMult. Moreover, while oDistMult is tailored specifically for the out-of-sample task, we", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 534, + 506, + 548 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 506, + 548 + ], + "score": 1.0, + "content": "did not do any task-specific modifications to the NodePiece-enabled model as it is inductive by", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 546, + 506, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 506, + 559 + ], + "score": 1.0, + "content": "design. Furthermore, oDistMult is not able to scale to a bigger oYAGO 3-10 on a 256 GB RAM", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 558, + 506, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 506, + 570 + ], + "score": 1.0, + "content": "machine due to the out of memory crash. Conversely, a NodePiece-equipped model has the same", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 568, + 505, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 505, + 580 + ], + "score": 1.0, + "content": "computational requirements as in other tasks and converges rather quickly (40 epochs). Performed", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 578, + 505, + 593 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 505, + 593 + ], + "score": 1.0, + "content": "ablations underline the importance of having both anchors and relational context for tokenizing", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 590, + 506, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 506, + 603 + ], + "score": 1.0, + "content": "unseen entities. We elaborate more on possible inference strategies for transductive and inductive", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 600, + 200, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 200, + 613 + ], + "score": 1.0, + "content": "tasks in Appendix A.6.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 31.5 + }, + { + "type": "title", + "bbox": [ + 108, + 629, + 195, + 642 + ], + "lines": [ + { + "bbox": [ + 104, + 626, + 198, + 646 + ], + "spans": [ + { + "bbox": [ + 104, + 626, + 198, + 646 + ], + "score": 1.0, + "content": "5 CONCLUSION", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 654, + 505, + 731 + ], + "lines": [ + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "In this paper, we have introduced NodePiece, a compositional approach for representing nodes in", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "score": 1.0, + "content": "multi-relational graphs with a fixed-size vocabulary. Similar to subword units, NodePiece allows to", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 677, + 504, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 504, + 689 + ], + "score": 1.0, + "content": "tokenize every node as a combination of anchors and relations where the number of anchors can be", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 145, + 699 + ], + "score": 0.86, + "content": "1 0 { - } 1 0 0 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "smaller than the total number of nodes. We show that in some tasks, node embeddings are", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 698, + 507, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 507, + 712 + ], + "score": 1.0, + "content": "not even necessary for getting an acceptable accuracy thanks to a rich set of relation types. 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In the out-of-sample setup, validation and test splits contain unseen entities that arrive with", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 330, + 506, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 506, + 344 + ], + "score": 1.0, + "content": "a few edges connected to the seen nodes. For this experiment, we use the out-of-sample FB15k-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 341, + 506, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 506, + 354 + ], + "score": 1.0, + "content": "237 split (oFB15k-237) as designed in Albooyeh et al. (2020). We do not employ their version of", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 352, + 507, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 507, + 366 + ], + "score": 1.0, + "content": "WN18RR as the split contains too many disconnected entities and components in the train graph.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 364, + 496, + 377 + ], + "spans": [ + { + "bbox": [ + 106, + 364, + 496, + 377 + ], + "score": 1.0, + "content": "Instead, using the authors script, we sample a much bigger out-of-sample version of YAGO 3-10.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14, + "bbox_fs": [ + 105, + 320, + 507, + 377 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 380, + 505, + 436 + ], + "lines": [ + { + "bbox": [ + 105, + 380, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 505, + 393 + ], + "score": 1.0, + "content": "As a baseline, we compare to oDistMult (Albooyeh et al., 2020) which aggregates embeddings", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 391, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 505, + 405 + ], + "score": 1.0, + "content": "of all seen neighboring nodes around the unseen one (akin to 1-layer message passing with mean", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 402, + 506, + 417 + ], + "spans": [ + { + "bbox": [ + 104, + 402, + 506, + 417 + ], + "score": 1.0, + "content": "aggregator). We adopt the same evaluation protocol - given an unseen node with its connecting", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 414, + 505, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 505, + 427 + ], + "score": 1.0, + "content": "edges, we mask one of the edges and predict its tail or head using the rest of the edges, repeating this", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 425, + 428, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 339, + 438 + ], + "score": 1.0, + "content": "procedure for each edge. We report filtered MRR and Hits", + "type": "text" + }, + { + "bbox": [ + 340, + 425, + 360, + 435 + ], + "score": 0.73, + "content": "@ 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 425, + 428, + 438 + ], + "score": 1.0, + "content": "as main metrics.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19, + "bbox_fs": [ + 104, + 380, + 506, + 438 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 442, + 505, + 496 + ], + "lines": [ + { + "bbox": [ + 106, + 441, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 505, + 453 + ], + "score": 1.0, + "content": "NodePiece enables traditional transductive-only models to perform inductive inference as both seen", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 453, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 106, + 453, + 505, + 465 + ], + "score": 1.0, + "content": "and unseen nodes are tokenized using the same vocabulary. For a smaller oFB15k-237 the NodePiece", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 463, + 504, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 168, + 475 + ], + "score": 1.0, + "content": "vocabulary has", + "type": "text" + }, + { + "bbox": [ + 169, + 464, + 192, + 474 + ], + "score": 0.34, + "content": "1 \\mathrm { k } / 2 0", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 463, + 475, + 475 + ], + "score": 1.0, + "content": "configuration with 15 relations, while in a bigger oYAGO 3-10 we use", + "type": "text" + }, + { + "bbox": [ + 475, + 464, + 504, + 474 + ], + "score": 0.48, + "content": "1 0 \\mathrm { k } / 2 0", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 474, + 507, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 507, + 487 + ], + "score": 1.0, + "content": "with 5 relations. For this task, we apply a transformer encoder instead of MLP. For a fair comparison,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 486, + 292, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 292, + 497 + ], + "score": 1.0, + "content": "we use DistMult as a scoring function as well.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 441, + 507, + 497 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 502, + 505, + 612 + ], + "lines": [ + { + "bbox": [ + 106, + 502, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 106, + 502, + 453, + 514 + ], + "score": 1.0, + "content": "Discussion. The results in Table 7 show that a simple NodePiece-based model retains", + "type": "text" + }, + { + "bbox": [ + 453, + 502, + 478, + 513 + ], + "score": 0.88, + "content": "{ \\sim } 9 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 502, + 505, + 514 + ], + "score": 1.0, + "content": "of the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 513, + 505, + 526 + ], + "spans": [ + { + "bbox": [ + 106, + 513, + 505, + 526 + ], + "score": 1.0, + "content": "baseline performance on oFB15k-237, but achieved faster and computationally inexpensive compared", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 524, + 506, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 506, + 537 + ], + "score": 1.0, + "content": "to oDistMult. Moreover, while oDistMult is tailored specifically for the out-of-sample task, we", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 534, + 506, + 548 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 506, + 548 + ], + "score": 1.0, + "content": "did not do any task-specific modifications to the NodePiece-enabled model as it is inductive by", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 546, + 506, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 506, + 559 + ], + "score": 1.0, + "content": "design. Furthermore, oDistMult is not able to scale to a bigger oYAGO 3-10 on a 256 GB RAM", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 558, + 506, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 506, + 570 + ], + "score": 1.0, + "content": "machine due to the out of memory crash. Conversely, a NodePiece-equipped model has the same", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 568, + 505, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 505, + 580 + ], + "score": 1.0, + "content": "computational requirements as in other tasks and converges rather quickly (40 epochs). Performed", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 578, + 505, + 593 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 505, + 593 + ], + "score": 1.0, + "content": "ablations underline the importance of having both anchors and relational context for tokenizing", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 590, + 506, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 506, + 603 + ], + "score": 1.0, + "content": "unseen entities. We elaborate more on possible inference strategies for transductive and inductive", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 600, + 200, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 200, + 613 + ], + "score": 1.0, + "content": "tasks in Appendix A.6.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 502, + 506, + 613 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 629, + 195, + 642 + ], + "lines": [ + { + "bbox": [ + 104, + 626, + 198, + 646 + ], + "spans": [ + { + "bbox": [ + 104, + 626, + 198, + 646 + ], + "score": 1.0, + "content": "5 CONCLUSION", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 654, + 505, + 731 + ], + "lines": [ + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "In this paper, we have introduced NodePiece, a compositional approach for representing nodes in", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "score": 1.0, + "content": "multi-relational graphs with a fixed-size vocabulary. Similar to subword units, NodePiece allows to", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 677, + 504, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 504, + 689 + ], + "score": 1.0, + "content": "tokenize every node as a combination of anchors and relations where the number of anchors can be", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 145, + 699 + ], + "score": 0.86, + "content": "1 0 { - } 1 0 0 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "smaller than the total number of nodes. We show that in some tasks, node embeddings are", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 698, + 507, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 507, + 712 + ], + "score": 1.0, + "content": "not even necessary for getting an acceptable accuracy thanks to a rich set of relation types. Moreover,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "NodePiece is inductive by design and is able to tokenize unseen entities and perform downstream", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 721, + 318, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 318, + 733 + ], + "score": 1.0, + "content": "prediction tasks in the same fashion as on seen ones.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 41, + "bbox_fs": [ + 105, + 655, + 507, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 127 + ], + "lines": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "Reproducibility Statement. The source code is openly available on GitHub. All hyperparameters", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 506, + 105 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 506, + 105 + ], + "score": 1.0, + "content": "and implementation details are presented in Appendix A. Information on the used datasets is presented", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 506, + 117 + ], + "score": 1.0, + "content": "in Table 8 and we provide more details on dataset construction for node classification and out-of-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 507, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 507, + 128 + ], + "score": 1.0, + "content": "sample link prediction tasks in Appendix F. The proof for Proposition 1 is given in the Appendix H.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 104, + 132, + 504, + 155 + ], + "lines": [ + { + "bbox": [ + 105, + 131, + 506, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 131, + 506, + 145 + ], + "score": 1.0, + "content": "Ethics Statement. As NodePiece is a general graph representation learning method, we do not", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 142, + 393, + 155 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 393, + 155 + ], + "score": 1.0, + "content": "foresee immediate ethical consequences pertaining to the method itself.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 107, + 160, + 505, + 215 + ], + "lines": [ + { + "bbox": [ + 106, + 160, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 106, + 160, + 505, + 172 + ], + "score": 1.0, + "content": "Acknowledgements. The authors would like to thank Koustuv Sinha, Gaurav Maheshwari, and", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 171, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 505, + 183 + ], + "score": 1.0, + "content": "Priyansh Trivedi for insightful and valuable discussions at earlier stages of this work. We also thank", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 182, + 504, + 194 + ], + "spans": [ + { + "bbox": [ + 106, + 182, + 504, + 194 + ], + "score": 1.0, + "content": "anonymous reviewers for the helpful comments. This work is partially supported by the Canada", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 192, + 505, + 206 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 505, + 206 + ], + "score": 1.0, + "content": "CIFAR AI Chair Program and Samsung AI grant (held at Mila). 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The source code is openly available on GitHub. All hyperparameters", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 506, + 105 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 506, + 105 + ], + "score": 1.0, + "content": "and implementation details are presented in Appendix A. Information on the used datasets is presented", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 506, + 117 + ], + "score": 1.0, + "content": "in Table 8 and we provide more details on dataset construction for node classification and out-of-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 507, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 507, + 128 + ], + "score": 1.0, + "content": "sample link prediction tasks in Appendix F. The proof for Proposition 1 is given in the Appendix H.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5, + "bbox_fs": [ + 105, + 83, + 507, + 128 + ] + }, + { + "type": "text", + "bbox": [ + 104, + 132, + 504, + 155 + ], + "lines": [ + { + "bbox": [ + 105, + 131, + 506, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 131, + 506, + 145 + ], + "score": 1.0, + "content": "Ethics Statement. 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In 36th IEEE International Conference on Data Engineering, ICDE", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 115, + 338, + 400, + 351 + ], + "spans": [ + { + "bbox": [ + 115, + 338, + 400, + 351 + ], + "score": 1.0, + "content": "2020, Dallas, TX, USA, April 20-24, 2020, pp. 433–444. IEEE, 2020b.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18, + "bbox_fs": [ + 106, + 316, + 506, + 351 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 357, + 504, + 391 + ], + "lines": [ + { + "bbox": [ + 106, + 357, + 506, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 506, + 370 + ], + "score": 1.0, + "content": "Yongqi Zhang, Quanming Yao, Wenyuan Dai, and Lei Chen. Autosf: Searching scoring functions for", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 115, + 367, + 506, + 382 + ], + "spans": [ + { + "bbox": [ + 115, + 367, + 506, + 382 + ], + "score": 1.0, + "content": "knowledge graph embedding. 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Distile: Distiling", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 115, + 409, + 506, + 423 + ], + "spans": [ + { + "bbox": [ + 115, + 409, + 506, + 423 + ], + "score": 1.0, + "content": "knowledge graph embeddings for faster and cheaper reasoning. arXiv preprint arXiv:2009.05912,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 115, + 419, + 141, + 432 + ], + "spans": [ + { + "bbox": [ + 115, + 419, + 141, + 432 + ], + "score": 1.0, + "content": "2020.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 397, + 506, + 432 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 439, + 504, + 473 + ], + "lines": [ + { + "bbox": [ + 106, + 439, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 106, + 439, + 505, + 451 + ], + "score": 1.0, + "content": "Zhaocheng Zhu, Zuobai Zhang, Louis-Pascal Xhonneux, and Jian Tang. Neural bellman-ford", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 115, + 450, + 506, + 462 + ], + "spans": [ + { + "bbox": [ + 115, + 450, + 506, + 462 + ], + "score": 1.0, + "content": "networks: A general graph neural network framework for link prediction. In Neural Information", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 115, + 461, + 264, + 473 + ], + "spans": [ + { + "bbox": [ + 115, + 461, + 264, + 473 + ], + "score": 1.0, + "content": "Processing Systems, NeurIPS, 2021.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27, + "bbox_fs": [ + 106, + 439, + 506, + 473 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 342, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 343, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 343, + 96 + ], + "score": 1.0, + "content": "A IMPLEMENTATION & HYPERPARAMETERS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table", + "bbox": [ + 108, + 149, + 504, + 239 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 117, + 505, + 140 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 117, + 506, + 130 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 506, + 130 + ], + "score": 1.0, + "content": "Table 8: Dataset statistics. LP - link prediction, RP - relation prediction, NC - node classification,", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 128, + 504, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 504, + 141 + ], + "score": 1.0, + "content": "OOS - out-of-sample. In OOS-LP, Nodes also shows the amount of unseen nodes in validation/test.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5 + }, + { + "type": "table_body", + "bbox": [ + 108, + 149, + 504, + 239 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 149, + 504, + 239 + ], + "spans": [ + { + "bbox": [ + 108, + 149, + 504, + 239 + ], + "score": 0.98, + "html": "
DatasetTaskNodesRelationsEdgesTrainValidationTest
FB15k-237(Toutanova & Chen,2015)LP,RP14,505237310.079272,11517,52620.438
WN18RR (Dettmers et al.,2018)LP,RP40,5591192.58386.83528242924
CoDEx-Large (Safavi & Koutra,2020)LP77,95169612,437551,19330,62230,622
YAGO 3-10 (Mahdisoltani et al., 2015)LP,RP123,143371,089,0001,079,04049784982
OGB WikiKG 2 (Hu et al.,2020)LP2,500,60453517,137,18116,109,182429,456598,543
WD50KNC46,164526222.5634600 (N)4600 (N)4600 (N)
oFB15k-237 (Albooyeh et al.,2020)OOS-LP11k/1395/1395234292,173193,49044,60154,082
oYAGO 3-10OOS-LP117k/2960/2959371,086,416988,12447,11251,180
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DatasetRelationsTrainValidationTest
EntityQueryFactsEntityQueryFactEntityQueryFacts
FB15k-237v11832.0004,2454,2451,5002061,9931,5002051,993
v22033,0009,7399,7392,0004694,1452.0004784,145
v32184,00017,98617,9863.0008667,4063,0008657,406
Vv42225,00027,20327,2033,5001,41611,7143,5001,42411,714
WN18RRv192,7465,4105,4109221851,6189221881,618
v2106,95415,26215,2622,9234114,0112.9234414,011
v31112.07825,90125,9015,0845386,3275.0846056.327
V493,8617,9407,9407,2081,39412,3347,2081,42912.334
NELL-995v1143,1034,6874,687225101833225100833
v2882,5648,2198,2194,9374594,5864,9374764,586
v31424,64716,39316,3934,9218118.0484,9218098.048
v4772,0927,5467,5463,2947167,0733,2947317,073
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LP - link prediction, RP - relation prediction, NC - node classification,", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 128, + 504, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 504, + 141 + ], + "score": 1.0, + "content": "OOS - out-of-sample. In OOS-LP, Nodes also shows the amount of unseen nodes in validation/test.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5 + }, + { + "type": "table_body", + "bbox": [ + 108, + 149, + 504, + 239 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 149, + 504, + 239 + ], + "spans": [ + { + "bbox": [ + 108, + 149, + 504, + 239 + ], + "score": 0.98, + "html": "
DatasetTaskNodesRelationsEdgesTrainValidationTest
FB15k-237(Toutanova & Chen,2015)LP,RP14,505237310.079272,11517,52620.438
WN18RR (Dettmers et al.,2018)LP,RP40,5591192.58386.83528242924
CoDEx-Large (Safavi & Koutra,2020)LP77,95169612,437551,19330,62230,622
YAGO 3-10 (Mahdisoltani et al., 2015)LP,RP123,143371,089,0001,079,04049784982
OGB WikiKG 2 (Hu et al.,2020)LP2,500,60453517,137,18116,109,182429,456598,543
WD50KNC46,164526222.5634600 (N)4600 (N)4600 (N)
oFB15k-237 (Albooyeh et al.,2020)OOS-LP11k/1395/1395234292,173193,49044,60154,082
oYAGO 3-10OOS-LP117k/2960/2959371,086,416988,12447,11251,180
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DatasetRelationsTrainValidationTest
EntityQueryFactsEntityQueryFactEntityQueryFacts
FB15k-237v11832.0004,2454,2451,5002061,9931,5002051,993
v22033,0009,7399,7392,0004694,1452.0004784,145
v32184,00017,98617,9863.0008667,4063,0008657,406
Vv42225,00027,20327,2033,5001,41611,7143,5001,42411,714
WN18RRv192,7465,4105,4109221851,6189221881,618
v2106,95415,26215,2622,9234114,0112.9234414,011
v31112.07825,90125,9015,0845386,3275.0846056.327
V493,8617,9407,9407,2081,39412,3347,2081,42912.334
NELL-995v1143,1034,6874,687225101833225100833
v2882,5648,2198,2194,9374594,5864,9374764,586
v31424,64716,39316,3934,9218118.0484,9218098.048
v4772,0927,5467,5463,2947167,0733,2947317,073
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The OGB WikiKG 2 experiments were", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 540, + 506, + 553 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 506, + 553 + ], + "score": 1.0, + "content": "executed on a single Tesla V100 16 GB VRAM and 64 GB RAM. 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ParameterFB15k-237WN18RRCoDEx-LYAGO 3-10OGB WikiKG 2
# Anchors, |A|100050070001000020000
# Anchors per node,k2050202020
Relational context, m1546512
Vocabulary dim, d200200200200200
Batch size512512256512512
Learning rate0.00050.00050.00050.000250.0001
Epochs400600120600300k (steps)
Encoder typeMLPMLPMLPMLPMLP
Encoder dim400400400400400
Encoder layers22222
Encoder dropout0.10.10.10.10.1
Loss functionBCENSSALBCENSSALNSSAL
Margin-15-5050
# Negative samples=20110128
Label smoothing0.410.311
Training time, hours75.5262311
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ParameterFB15k-237WN18RRCoDEx-LYAGO 3-10
Embedding dim, d200010001000 /501000 /40
Batch size1024512512/5121024 /512
Loss functionNSSALNSSALNSSALNSSAL
Margin9625/924/15
# Negative samples2561024100 /100400/100
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As RotatE is a scoring function in the complex space, the", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 114, + 506, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 506, + 127 + ], + "score": 1.0, + "content": "reported embedding dimensions are a sum of real and imaginary dimensions, e.g., 1000d means that", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 125, + 275, + 137 + ], + "spans": [ + { + "bbox": [ + 106, + 125, + 275, + 137 + ], + "score": 1.0, + "content": "both real and imaginary vectors are 500d.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2, + "bbox_fs": [ + 105, + 102, + 506, + 137 + ] + }, + { + "type": "table", + "bbox": [ + 110, + 167, + 501, + 373 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 139, + 147, + 471, + 159 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 139, + 146, + 471, + 161 + ], + "spans": [ + { + "bbox": [ + 139, + 146, + 471, + 161 + ], + "score": 1.0, + "content": "Table 10: NodePiece hyperparameters for transductive link prediction experiments", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "table_body", + "bbox": [ + 110, + 167, + 501, + 373 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 110, + 167, + 501, + 373 + ], + "spans": [ + { + "bbox": [ + 110, + 167, + 501, + 373 + ], + "score": 0.985, + "html": "
ParameterFB15k-237WN18RRCoDEx-LYAGO 3-10OGB WikiKG 2
# Anchors, |A|100050070001000020000
# Anchors per node,k2050202020
Relational context, m1546512
Vocabulary dim, d200200200200200
Batch size512512256512512
Learning rate0.00050.00050.00050.000250.0001
Epochs400600120600300k (steps)
Encoder typeMLPMLPMLPMLPMLP
Encoder dim400400400400400
Encoder layers22222
Encoder dropout0.10.10.10.10.1
Loss functionBCENSSALBCENSSALNSSAL
Margin-15-5050
# Negative samples=20110128
Label smoothing0.410.311
Training time, hours75.5262311
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ParameterFB15k-237WN18RRCoDEx-LYAGO 3-10
Embedding dim, d200010001000 /501000 /40
Batch size1024512512/5121024 /512
Loss functionNSSALNSSALNSSALNSSAL
Margin9625/924/15
# Negative samples2561024100 /100400/100
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We mostly reduce the number of epochs and negative samples as models", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 574, + 220, + 586 + ], + "spans": [ + { + "bbox": [ + 106, + 574, + 220, + 586 + ], + "score": 1.0, + "content": "converge faster on this task.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 551, + 506, + 586 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 599, + 234, + 610 + ], + "lines": [ + { + "bbox": [ + 106, + 598, + 235, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 598, + 235, + 612 + ], + "score": 1.0, + "content": "A.4 NODE CLASSIFICATION", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 619, + 505, + 653 + ], + "lines": [ + { + "bbox": [ + 104, + 618, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 104, + 618, + 505, + 634 + ], + "score": 1.0, + "content": "In this experiment (Table 13), NodePiece is used at the initial step to bootstrap a node embeddings", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 631, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 505, + 644 + ], + "score": 1.0, + "content": "matrix which is then sent to the CompGCN graph encoder. In contrast, CompGCN and MLP baselines", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 642, + 371, + 654 + ], + "spans": [ + { + "bbox": [ + 106, + 642, + 371, + 654 + ], + "score": 1.0, + "content": "use directly a trained node embedding matrix as their initial input.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20, + "bbox_fs": [ + 104, + 618, + 505, + 654 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 667, + 286, + 678 + ], + "lines": [ + { + "bbox": [ + 106, + 667, + 288, + 680 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 288, + 680 + ], + "score": 1.0, + "content": "A.5 OUT-OF-SAMPLE LINK PREDICTION", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "The set of NodePiece hyperparameters (Table 14) is similar to the set of the transductive experiments", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "except the scoring function (DistMult), encoder function (Transformer), and number of epochs as the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "model converges faster. 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Training times of the baseline oDistMult were not", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 518, + 201, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 201, + 529 + ], + "score": 1.0, + "content": "reported by its authors.", + "type": "text", + "cross_page": true + } + ], + "index": 10 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 688, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 108, + 111, + 503, + 217 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 80, + 504, + 102 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 80, + 505, + 93 + ], + "spans": [ + { + "bbox": [ + 106, + 80, + 505, + 93 + ], + "score": 1.0, + "content": "Table 12: Hyperparameters for relation prediction experiments. The content is largely identical to", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 90, + 287, + 103 + ], + "spans": [ + { + "bbox": [ + 105, + 90, + 287, + 103 + ], + "score": 1.0, + "content": "Table 10, only changed parameters are listed", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table_body", + "bbox": [ + 108, + 111, + 503, + 217 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 111, + 503, + 217 + ], + "spans": [ + { + "bbox": [ + 108, + 111, + 503, + 217 + ], + "score": 0.982, + "html": "
NodePiece+RotatERotatE
ParameterFB15k-237WN18RRYAGO 3-10FB15k-237WN18RRYAGO 3-10
Batch size512512512512512512
Epochs201507150150150
Loss functionNSSALNSSALNSSALNSSALNSSALNSSAL
Margin151225935
# Negative samples202020202020
Training time,min253025281057
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ParameterNodePiece + CompGCNCompGCNMLP
# Anchors, |Al50=
# Anchors per node, k10=
Relational context,m5=1
Vocabulary dim, d100100100
Batch size512512512
Learning rate0.0010.0010.001
Epochs400040004000
NodePiece encoderMLP--
NodePiece encoder dim2001
NodePiece encoder layers2
NodePiece encoder dropout0.1==
GNN (MLP) layers333
GNN (MLP) dropout0.50.50.5
Loss functionBCEBCEBCE
Label smoothing0.10.10.1
Training time, hours14226
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Training times of the baseline oDistMult were not", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 518, + 201, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 201, + 529 + ], + "score": 1.0, + "content": "reported by its authors.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 107, + 560, + 410, + 572 + ], + "lines": [ + { + "bbox": [ + 106, + 560, + 411, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 560, + 411, + 573 + ], + "score": 1.0, + "content": "A.6 DEPLOYMENT IN REAL-WORLD DYNAMIC KNOWLEDGE GRAPHS", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 589, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 588, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 505, + 601 + ], + "score": 1.0, + "content": "NodePiece, on account of its compositional representation, can be applied to dynamic real-world", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 600, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 505, + 613 + ], + "score": 1.0, + "content": "knowledge graphs where nodes are added and removed over time. That is, training on a graph snapshot", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 610, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 624 + ], + "score": 1.0, + "content": "we can obtain the embeddings of new nodes without re-computing and updating representations", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 622, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 505, + 635 + ], + "score": 1.0, + "content": "of every other node in the graph. This is valuable in settings where there are latency requirements", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 633, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 506, + 645 + ], + "score": 1.0, + "content": "such as many online services. 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In order to", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "reduce latency the entire embedding matrix can be materialized and cached ahead of time, updating", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "embeddings as new nodes and edges are added. The parameter efficiency and compositionality of", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "NodePeice means that for large real-world graphs NodePiece subsumes what would before have been", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "a complex system of a large-scale embedding framework like Pytorch-BigGraph (Lerer et al., 2019),", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 720, + 474, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 474, + 734 + ], + "score": 1.0, + "content": "an OOV embedding method (e.g,. ERAvg) and a shallow embedding method (e.g., RotatE).", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 18 + } + ], + "page_idx": 16, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 310, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 13 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 108, + 111, + 503, + 217 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 80, + 504, + 102 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 80, + 505, + 93 + ], + "spans": [ + { + "bbox": [ + 106, + 80, + 505, + 93 + ], + "score": 1.0, + "content": "Table 12: Hyperparameters for relation prediction experiments. The content is largely identical to", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 90, + 287, + 103 + ], + "spans": [ + { + "bbox": [ + 105, + 90, + 287, + 103 + ], + "score": 1.0, + "content": "Table 10, only changed parameters are listed", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table_body", + "bbox": [ + 108, + 111, + 503, + 217 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 111, + 503, + 217 + ], + "spans": [ + { + "bbox": [ + 108, + 111, + 503, + 217 + ], + "score": 0.982, + "html": "
NodePiece+RotatERotatE
ParameterFB15k-237WN18RRYAGO 3-10FB15k-237WN18RRYAGO 3-10
Batch size512512512512512512
Epochs201507150150150
Loss functionNSSALNSSALNSSALNSSALNSSALNSSAL
Margin151225935
# Negative samples202020202020
Training time,min253025281057
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ParameterNodePiece + CompGCNCompGCNMLP
# Anchors, |Al50=
# Anchors per node, k10=
Relational context,m5=1
Vocabulary dim, d100100100
Batch size512512512
Learning rate0.0010.0010.001
Epochs400040004000
NodePiece encoderMLP--
NodePiece encoder dim2001
NodePiece encoder layers2
NodePiece encoder dropout0.1==
GNN (MLP) layers333
GNN (MLP) dropout0.50.50.5
Loss functionBCEBCEBCE
Label smoothing0.10.10.1
Training time, hours14226
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That is, training on a graph snapshot", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 610, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 624 + ], + "score": 1.0, + "content": "we can obtain the embeddings of new nodes without re-computing and updating representations", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 622, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 505, + 635 + ], + "score": 1.0, + "content": "of every other node in the graph. This is valuable in settings where there are latency requirements", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 633, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 506, + 645 + ], + "score": 1.0, + "content": "such as many online services. 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In order to", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "reduce latency the entire embedding matrix can be materialized and cached ahead of time, updating", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "embeddings as new nodes and edges are added. The parameter efficiency and compositionality of", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "NodePeice means that for large real-world graphs NodePiece subsumes what would before have been", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "a complex system of a large-scale embedding framework like Pytorch-BigGraph (Lerer et al., 2019),", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 720, + 474, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 474, + 734 + ], + "score": 1.0, + "content": "an OOV embedding method (e.g,. ERAvg) and a shallow embedding method (e.g., RotatE).", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 18, + "bbox_fs": [ + 104, + 588, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 153, + 110, + 457, + 311 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 80, + 504, + 103 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 78, + 505, + 94 + ], + "spans": [ + { + "bbox": [ + 105, + 78, + 505, + 94 + ], + "score": 1.0, + "content": "Table 14: Hyperparameters for out-of-sample link prediction experiments. The content is largely", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 91, + 334, + 104 + ], + "spans": [ + { + "bbox": [ + 105, + 91, + 334, + 104 + ], + "score": 1.0, + "content": "identical to Table 10, only changed parameters are listed", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table_body", + "bbox": [ + 153, + 110, + 457, + 311 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 153, + 110, + 457, + 311 + ], + "spans": [ + { + "bbox": [ + 153, + 110, + 457, + 311 + ], + "score": 0.983, + "html": "
ParameterNodePiece+DistMultoDistMult
oFB15k-237oYAGO 3-10
# Anchors, |A|100010000
# Anchors per node,k2020=
Relational context, m155=
Vocabulary dim,d200200200
Batch size2562561000
Learning rate0.00050.00050.01
Epochs40401000
NodePiece encoderTransformerTransformer-
NodePiece encoder dim512512■
NodePiece encoder layers22
NodePiece encoder dropout0.10.1
Loss functionSoftplusSoftplusSoftplus
# Negative samples551
Training time,hours28
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While it is difficult to prove, we also hypothesize that", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 384, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 106, + 384, + 505, + 397 + ], + "score": 1.0, + "content": "the parameters required by NodePeice to maintain the same level of performance (as the graph", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 394, + 506, + 407 + ], + "spans": [ + { + "bbox": [ + 106, + 394, + 506, + 407 + ], + "score": 1.0, + "content": "scales) increase sublinearly according to the size of the graph. The intuition for this is twofold. 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Second,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 441, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 505, + 453 + ], + "score": 1.0, + "content": "increasing the size of the graph will only require sublinear increase in the number of anchors in", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 451, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 505, + 464 + ], + "score": 1.0, + "content": "order to maintain the same average node-anchor distance. 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Recall that there exist several ways to select the total set of anchors", + "type": "text" + }, + { + "bbox": [ + 458, + 545, + 466, + 554 + ], + "score": 0.78, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 467, + 545, + 505, + 556 + ], + "score": 1.0, + "content": "as stated", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 555, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 364, + 568 + ], + "score": 1.0, + "content": "in Section 3.1, i.e., random or centrality-measure based. Then,", + "type": "text" + }, + { + "bbox": [ + 365, + 556, + 372, + 565 + ], + "score": 0.78, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 555, + 505, + 568 + ], + "score": 1.0, + "content": "anchors per node can be chosen", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 567, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 567, + 143, + 578 + ], + "score": 1.0, + "content": "either as", + "type": "text" + }, + { + "bbox": [ + 144, + 567, + 151, + 576 + ], + "score": 0.79, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 567, + 304, + 578 + ], + "score": 1.0, + "content": "nearest (default NodePiece mode) or", + "type": "text" + }, + { + "bbox": [ + 304, + 567, + 311, + 576 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 567, + 505, + 578 + ], + "score": 1.0, + "content": "random anchors. Figure 3 depicts the effect of", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 578, + 504, + 589 + ], + "spans": [ + { + "bbox": [ + 106, + 578, + 504, + 589 + ], + "score": 1.0, + "content": "those strategies on the distribution of anchor distances (number of hops between a target node and its", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 588, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 505, + 600 + ], + "score": 1.0, + "content": "anchors). We use the configurations used in the main experiments, i.e., 1000 anchors and 20 anchors", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 599, + 437, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 437, + 611 + ], + "score": 1.0, + "content": "per node for FB15k-237, and 500 anchors with 50 anchors per node on WN18RR.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 615, + 504, + 682 + ], + "lines": [ + { + "bbox": [ + 105, + 615, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 305, + 628 + ], + "score": 1.0, + "content": "First, we observe that PPR, degree, and mixed (", + "type": "text" + }, + { + "bbox": [ + 305, + 616, + 325, + 627 + ], + "score": 0.84, + "content": "40 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 615, + 350, + 628 + ], + "score": 1.0, + "content": "PPR,", + "type": "text" + }, + { + "bbox": [ + 350, + 616, + 370, + 627 + ], + "score": 0.86, + "content": "40 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 615, + 405, + 628 + ], + "score": 1.0, + "content": "degree,", + "type": "text" + }, + { + "bbox": [ + 405, + 616, + 425, + 627 + ], + "score": 0.86, + "content": "20 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 615, + 506, + 628 + ], + "score": 1.0, + "content": "random) strategies", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 626, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 506, + 640 + ], + "score": 1.0, + "content": "generally skew the distribution towards smaller anchor distances compared to random strategies. This", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 638, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 506, + 651 + ], + "score": 1.0, + "content": "fact supports the hypothesis that deterministic strategies improve the chances to find an anchor in a", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 648, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 132, + 662 + ], + "score": 1.0, + "content": "closer", + "type": "text" + }, + { + "bbox": [ + 133, + 649, + 137, + 659 + ], + "score": 0.5, + "content": "l", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 648, + 427, + 662 + ], + "score": 1.0, + "content": "-hop neighborhood of a target node. Second, varying the way of selecting", + "type": "text" + }, + { + "bbox": [ + 427, + 649, + 434, + 659 + ], + "score": 0.75, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 648, + 506, + 662 + ], + "score": 1.0, + "content": "anchors per node", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 660, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 505, + 672 + ], + "score": 1.0, + "content": "between nearest (left column) and random (right column), we also observe the skew of a distribution", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 671, + 189, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 189, + 683 + ], + "score": 1.0, + "content": "of anchor distances.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 39.5 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "Next, we fix the anchor selection strategy to the mix, fix the number of anchors per node (50 for", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 389, + 711 + ], + "score": 1.0, + "content": "WN18RR and 20 for FB15k-237), and vary a total number of anchors", + "type": "text" + }, + { + "bbox": [ + 389, + 699, + 398, + 709 + ], + "score": 0.76, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 698, + 506, + 711 + ], + "score": 1.0, + "content": "(50 to 1000 for WN18RR", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 376, + 723 + ], + "score": 1.0, + "content": "and 20 to 1000 for FB15k-237) along with the method of sampling", + "type": "text" + }, + { + "bbox": [ + 376, + 710, + 383, + 720 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "anchors per node, i.e., nearest", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 721, + 506, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 465, + 732 + ], + "score": 1.0, + "content": "and random. Figure 4 shows that increasing the total number of anchors together with", + "type": "text" + }, + { + "bbox": [ + 465, + 721, + 472, + 730 + ], + "score": 0.79, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 721, + 506, + 732 + ], + "score": 1.0, + "content": "nearest", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 44.5 + } + ], + "page_idx": 17, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 310, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 763 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 763 + ], + "score": 1.0, + "content": "18", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 153, + 110, + 457, + 311 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 80, + 504, + 103 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 78, + 505, + 94 + ], + "spans": [ + { + "bbox": [ + 105, + 78, + 505, + 94 + ], + "score": 1.0, + "content": "Table 14: Hyperparameters for out-of-sample link prediction experiments. The content is largely", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 91, + 334, + 104 + ], + "spans": [ + { + "bbox": [ + 105, + 91, + 334, + 104 + ], + "score": 1.0, + "content": "identical to Table 10, only changed parameters are listed", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table_body", + "bbox": [ + 153, + 110, + 457, + 311 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 153, + 110, + 457, + 311 + ], + "spans": [ + { + "bbox": [ + 153, + 110, + 457, + 311 + ], + "score": 0.983, + "html": "
ParameterNodePiece+DistMultoDistMult
oFB15k-237oYAGO 3-10
# Anchors, |A|100010000
# Anchors per node,k2020=
Relational context, m155=
Vocabulary dim,d200200200
Batch size2562561000
Learning rate0.00050.00050.01
Epochs40401000
NodePiece encoderTransformerTransformer-
NodePiece encoder dim512512■
NodePiece encoder layers22
NodePiece encoder dropout0.10.1
Loss functionSoftplusSoftplusSoftplus
# Negative samples551
Training time,hours28
", + "type": "table", + "image_path": "9c7a2bc4cb2aeacc29c9bdc69d70ca3599d377ea0fa0a3c9a8f3b403f092be84.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 153, + 110, + 457, + 123.4 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 153, + 123.4, + 457, + 136.8 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 153, + 136.8, + 457, + 150.20000000000002 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 153, + 150.20000000000002, + 457, + 163.60000000000002 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 153, + 163.60000000000002, + 457, + 177.00000000000003 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 153, + 177.00000000000003, + 457, + 190.40000000000003 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 153, + 190.40000000000003, + 457, + 203.80000000000004 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 153, + 203.80000000000004, + 457, + 217.20000000000005 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 153, + 217.20000000000005, + 457, + 230.60000000000005 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 153, + 230.60000000000005, + 457, + 244.00000000000006 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 153, + 244.00000000000006, + 457, + 257.40000000000003 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 153, + 257.40000000000003, + 457, + 270.8 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 153, + 270.8, + 457, + 284.2 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 153, + 284.2, + 457, + 297.59999999999997 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 153, + 297.59999999999997, + 457, + 310.99999999999994 + ], + "spans": [], + "index": 16 + } + ] + } + ], + "index": 4.75 + }, + { + "type": "title", + "bbox": [ + 107, + 334, + 303, + 347 + ], + "lines": [ + { + "bbox": [ + 105, + 333, + 305, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 305, + 349 + ], + "score": 1.0, + "content": "B LIMITATIONS AND FUTURE WORK", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 362, + 505, + 485 + ], + "lines": [ + { + "bbox": [ + 106, + 362, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 362, + 505, + 374 + ], + "score": 1.0, + "content": "Our experimental results demonstrate the promise of using NodePeice to significantly reduce the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 372, + 506, + 386 + ], + "spans": [ + { + "bbox": [ + 104, + 372, + 506, + 386 + ], + "score": 1.0, + "content": "parameter complexity of node embeddings. While it is difficult to prove, we also hypothesize that", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 384, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 106, + 384, + 505, + 397 + ], + "score": 1.0, + "content": "the parameters required by NodePeice to maintain the same level of performance (as the graph", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 394, + 506, + 407 + ], + "spans": [ + { + "bbox": [ + 106, + 394, + 506, + 407 + ], + "score": 1.0, + "content": "scales) increase sublinearly according to the size of the graph. The intuition for this is twofold. First,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 405, + 506, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 317, + 419 + ], + "score": 1.0, + "content": "the number of unique anchor combinations of size", + "type": "text" + }, + { + "bbox": [ + 317, + 406, + 325, + 416 + ], + "score": 0.77, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 405, + 506, + 419 + ], + "score": 1.0, + "content": "that can be encoded increases according to", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 108, + 415, + 507, + 434 + ], + "spans": [ + { + "bbox": [ + 108, + 416, + 127, + 432 + ], + "score": 0.9, + "content": "\\textstyle { \\binom { | A | } { k } }", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 415, + 150, + 434 + ], + "score": 1.0, + "content": "(i.e.", + "type": "text" + }, + { + "bbox": [ + 150, + 417, + 188, + 430 + ], + "score": 0.93, + "content": "\\mathcal { O } ( | A | ^ { k } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 415, + 507, + 434 + ], + "score": 1.0, + "content": "if randomly sampled — if the sampling is done via nearest neighbor anchor", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 430, + 506, + 442 + ], + "spans": [ + { + "bbox": [ + 106, + 430, + 506, + 442 + ], + "score": 1.0, + "content": "selection then the number of unique permutations is expected to increase polynomially. Second,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 441, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 505, + 453 + ], + "score": 1.0, + "content": "increasing the size of the graph will only require sublinear increase in the number of anchors in", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 451, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 505, + 464 + ], + "score": 1.0, + "content": "order to maintain the same average node-anchor distance. Although proving causality is difficult, we", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 462, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 462, + 505, + 475 + ], + "score": 1.0, + "content": "believe that maintaining hashing uniqueness and node-anchor distances stable will be sufficient to", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 473, + 242, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 242, + 487 + ], + "score": 1.0, + "content": "maintain equivalent performance.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 23, + "bbox_fs": [ + 104, + 362, + 507, + 487 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 505, + 302, + 518 + ], + "lines": [ + { + "bbox": [ + 105, + 504, + 304, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 304, + 520 + ], + "score": 1.0, + "content": "C ANCHOR SELECTION STRATEGIES", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 533, + 505, + 610 + ], + "lines": [ + { + "bbox": [ + 105, + 533, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 354, + 545 + ], + "score": 1.0, + "content": "Here, we provide more details as to anchor configurations (", + "type": "text" + }, + { + "bbox": [ + 354, + 534, + 361, + 543 + ], + "score": 0.73, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 533, + 439, + 545 + ], + "score": 1.0, + "content": "nearest from total", + "type": "text" + }, + { + "bbox": [ + 439, + 534, + 448, + 543 + ], + "score": 0.74, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 533, + 505, + 545 + ], + "score": 1.0, + "content": "anchors) and", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 545, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 457, + 556 + ], + "score": 1.0, + "content": "anchor distances. Recall that there exist several ways to select the total set of anchors", + "type": "text" + }, + { + "bbox": [ + 458, + 545, + 466, + 554 + ], + "score": 0.78, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 467, + 545, + 505, + 556 + ], + "score": 1.0, + "content": "as stated", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 555, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 364, + 568 + ], + "score": 1.0, + "content": "in Section 3.1, i.e., random or centrality-measure based. Then,", + "type": "text" + }, + { + "bbox": [ + 365, + 556, + 372, + 565 + ], + "score": 0.78, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 555, + 505, + 568 + ], + "score": 1.0, + "content": "anchors per node can be chosen", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 567, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 567, + 143, + 578 + ], + "score": 1.0, + "content": "either as", + "type": "text" + }, + { + "bbox": [ + 144, + 567, + 151, + 576 + ], + "score": 0.79, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 567, + 304, + 578 + ], + "score": 1.0, + "content": "nearest (default NodePiece mode) or", + "type": "text" + }, + { + "bbox": [ + 304, + 567, + 311, + 576 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 567, + 505, + 578 + ], + "score": 1.0, + "content": "random anchors. Figure 3 depicts the effect of", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 578, + 504, + 589 + ], + "spans": [ + { + "bbox": [ + 106, + 578, + 504, + 589 + ], + "score": 1.0, + "content": "those strategies on the distribution of anchor distances (number of hops between a target node and its", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 588, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 505, + 600 + ], + "score": 1.0, + "content": "anchors). 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This", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 638, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 506, + 651 + ], + "score": 1.0, + "content": "fact supports the hypothesis that deterministic strategies improve the chances to find an anchor in a", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 648, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 132, + 662 + ], + "score": 1.0, + "content": "closer", + "type": "text" + }, + { + "bbox": [ + 133, + 649, + 137, + 659 + ], + "score": 0.5, + "content": "l", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 648, + 427, + 662 + ], + "score": 1.0, + "content": "-hop neighborhood of a target node. 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Figure 4 shows that increasing the total number of anchors together with", + "type": "text" + }, + { + "bbox": [ + 465, + 721, + 472, + 730 + ], + "score": 0.79, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 721, + 506, + 732 + ], + "score": 1.0, + "content": "nearest", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 681, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 106, + 681, + 505, + 694 + ], + "score": 1.0, + "content": "anchors again skews the distribution of anchor distances towards smaller values and, hence, to higher", + "type": "text", + "cross_page": true + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 692, + 402, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 692, + 402, + 705 + ], + "score": 1.0, + "content": "probabilities of finding anchors in a closer neighborhood of a target node.", + "type": "text", + "cross_page": true + } + ], + "index": 16 + } + ], + "index": 44.5, + "bbox_fs": [ + 105, + 687, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 107, + 86, + 507, + 303 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 107, + 86, + 507, + 303 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 86, + 507, + 303 + ], + "spans": [ + { + "bbox": [ + 107, + 86, + 507, + 303 + ], + "score": 0.973, + "type": "image", + "image_path": "057e081baea2ec49fd398141c91fb1aaf275530faf7f731a530bcf6c764d7cbc.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 107, + 86, + 507, + 158.33333333333331 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 107, + 158.33333333333331, + 507, + 230.66666666666663 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 107, + 230.66666666666663, + 507, + 302.99999999999994 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 312, + 506, + 367 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 312, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 312, + 506, + 324 + ], + "score": 1.0, + "content": "Figure 3: Distribution of anchor distances under various anchor selection strategies. 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On both graphs, increase in", + "type": "text" + }, + { + "bbox": [ + 460, + 636, + 468, + 645 + ], + "score": 0.74, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 469, + 635, + 506, + 647 + ], + "score": 1.0, + "content": "with the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 646, + 335, + 659 + ], + "spans": [ + { + "bbox": [ + 104, + 646, + 335, + 659 + ], + "score": 1.0, + "content": "nearest anchors always leads to shorter anchor distances.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12.5 + } + ], + "index": 10.75 + }, + { + "type": "text", + "bbox": [ + 106, + 681, + 504, + 704 + ], + "lines": [], + "index": 15.5, + "bbox_fs": [ + 105, + 681, + 505, + 705 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 359, + 722 + ], + "score": 1.0, + "content": "We would recommend using centrality-based strategies to select", + "type": "text" + }, + { + "bbox": [ + 359, + 710, + 367, + 720 + ], + "score": 0.73, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 709, + 388, + 722 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 389, + 710, + 395, + 720 + ], + "score": 0.75, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "nearest anchors per node if", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 720, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 734 + ], + "score": 1.0, + "content": "anchor distances and probability of finding anchors in a closer neighborhood are of higher importance.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 709, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 160 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 506, + 95 + ], + "score": 1.0, + "content": "Finally, we fix the anchor selection strategy as mix, obtain nearest anchors per node, and under", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 312, + 106 + ], + "score": 1.0, + "content": "this setup study average anchor distances varying", + "type": "text" + }, + { + "bbox": [ + 313, + 94, + 320, + 104 + ], + "score": 0.67, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "- the number of anchors per node in various", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 227, + 117 + ], + "score": 1.0, + "content": "combinations of total anchors", + "type": "text" + }, + { + "bbox": [ + 227, + 105, + 235, + 114 + ], + "score": 0.51, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 104, + 505, + 117 + ], + "score": 1.0, + "content": ". The results presented on Figure 5 suggest that sparser graphs (like", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 363, + 128 + ], + "score": 1.0, + "content": "WN18RR) benefit more from increasing the number of anchors", + "type": "text" + }, + { + "bbox": [ + 363, + 116, + 372, + 125 + ], + "score": 0.71, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 114, + 505, + 128 + ], + "score": 1.0, + "content": ", i.e., the delta between distances", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "score": 1.0, + "content": "is much larger than that of dense FB15k-237. The difference in distances on Figure 5 might also", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 343, + 150 + ], + "score": 1.0, + "content": "explain the performance on Figure 8, i.e., generally, smaller", + "type": "text" + }, + { + "bbox": [ + 343, + 137, + 362, + 149 + ], + "score": 0.91, + "content": "A / k", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 137, + 506, + 150 + ], + "score": 1.0, + "content": "configurations like 25/5 are inferior", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 149, + 350, + 161 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 350, + 161 + ], + "score": 1.0, + "content": "on sparser graphs but perform competitively on denser ones.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3 + }, + { + "type": "image", + "bbox": [ + 115, + 197, + 489, + 343 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 115, + 197, + 489, + 343 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 115, + 197, + 489, + 343 + ], + "spans": [ + { + "bbox": [ + 115, + 197, + 489, + 343 + ], + "score": 0.972, + "type": "image", + "image_path": "5c83762ad70aceab22b28829e29e8a4217f2f51bb030fca7fc3d9b6ffed58232.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 115, + 197, + 489, + 245.66666666666666 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 115, + 245.66666666666666, + 489, + 294.3333333333333 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 115, + 294.3333333333333, + 489, + 343.0 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 353, + 505, + 397 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 352, + 506, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 449, + 365 + ], + "score": 1.0, + "content": "Figure 5: Average node-anchor distances when varying the total number of anchors", + "type": "text" + }, + { + "bbox": [ + 450, + 353, + 458, + 363 + ], + "score": 0.69, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 352, + 506, + 365 + ], + "score": 1.0, + "content": "from 25 to", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 363, + 505, + 377 + ], + "spans": [ + { + "bbox": [ + 104, + 363, + 144, + 377 + ], + "score": 1.0, + "content": "1000 and", + "type": "text" + }, + { + "bbox": [ + 145, + 365, + 152, + 374 + ], + "score": 0.78, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 363, + 505, + 377 + ], + "score": 1.0, + "content": "nearest anchors per node from 5 to 50. Note that on a sparser WN18RR the gap between", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 375, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 505, + 387 + ], + "score": 1.0, + "content": "min and max values is much wider than of denser FB15k-237. Signs of saturation suggest that further", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 387, + 228, + 398 + ], + "spans": [ + { + "bbox": [ + 106, + 387, + 150, + 398 + ], + "score": 1.0, + "content": "increasing", + "type": "text" + }, + { + "bbox": [ + 150, + 387, + 159, + 396 + ], + "score": 0.7, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 387, + 228, + 398 + ], + "score": 1.0, + "content": "is not beneficial.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11.5 + } + ], + "index": 9.75 + }, + { + "type": "title", + "bbox": [ + 108, + 422, + 283, + 434 + ], + "lines": [ + { + "bbox": [ + 105, + 420, + 285, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 285, + 437 + ], + "score": 1.0, + "content": "D EMBEDDING VISUALIZATIONS", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 447, + 504, + 480 + ], + "lines": [ + { + "bbox": [ + 105, + 447, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 505, + 459 + ], + "score": 1.0, + "content": "To further study learned representations of anchors and capabilities of the encoder, we build tSNE", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 458, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 506, + 470 + ], + "score": 1.0, + "content": "and UMAP projections from subsamples of FB15k-237 and WN18RR based on trained models from", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 469, + 430, + 481 + ], + "spans": [ + { + "bbox": [ + 106, + 469, + 430, + 481 + ], + "score": 1.0, + "content": "the transductive link prediction experiments (hyperparameters listed in Table A).", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 485, + 505, + 551 + ], + "lines": [ + { + "bbox": [ + 105, + 484, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 506, + 498 + ], + "score": 1.0, + "content": "For FB15k-237, we randomly sample 1000 entities (out of total 15K) and find their top-100 most", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 497, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 505, + 510 + ], + "score": 1.0, + "content": "common anchors. The anchor embeddings are extracted from the learned tensor while 1000 entity", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 507, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 505, + 520 + ], + "score": 1.0, + "content": "embeddings are obtained through the NodePiece encoder. Similarly for WN18RR, we sample 4000", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 518, + 506, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 506, + 531 + ], + "score": 1.0, + "content": "entities (out of total 40K) keeping their top-100 most common anchors. As we use the RotatE decoder", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 527, + 507, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 507, + 544 + ], + "score": 1.0, + "content": "that assumes entities and anchors are modeled in a complex space, we visualize their real parts (e.g.,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 541, + 239, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 541, + 239, + 552 + ], + "score": 1.0, + "content": "first 100 dimensions out of 200).", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 107, + 557, + 505, + 657 + ], + "lines": [ + { + "bbox": [ + 105, + 557, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 332, + 569 + ], + "score": 1.0, + "content": "Recalling that link prediction performance of NodePiece", + "type": "text" + }, + { + "bbox": [ + 333, + 559, + 341, + 567 + ], + "score": 0.75, + "content": "^ +", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 557, + 400, + 569 + ], + "score": 1.0, + "content": "RotatE retains", + "type": "text" + }, + { + "bbox": [ + 400, + 558, + 433, + 568 + ], + "score": 0.87, + "content": "80 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 557, + 506, + 569 + ], + "score": 1.0, + "content": "of the state of the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 568, + 506, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 506, + 581 + ], + "score": 1.0, + "content": "art models performance, the results on Fig. 6 and Fig. 7 demonstrate that (1) NodePiece encoder is", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 580, + 506, + 592 + ], + "spans": [ + { + "bbox": [ + 106, + 580, + 506, + 592 + ], + "score": 1.0, + "content": "able to reconstruct clusters of similar entities; (2) anchors are well-scattered among communities.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 591, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 106, + 591, + 322, + 603 + ], + "score": 1.0, + "content": "Albeit entity embeddings are built as a composition of", + "type": "text" + }, + { + "bbox": [ + 323, + 591, + 330, + 600 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 591, + 505, + 603 + ], + "score": 1.0, + "content": "anchors, it can be seen that all communities", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 601, + 506, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 506, + 614 + ], + "score": 1.0, + "content": "have \"specialized\" nearby anchors. On a higher level, common anchors tend to be well-scattered in", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 610, + 507, + 627 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 507, + 627 + ], + "score": 1.0, + "content": "the space. Less common anchors, as seen on FB15k-237 and Fig. 6, tend to group together. However,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 623, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 106, + 623, + 506, + 635 + ], + "score": 1.0, + "content": "thanks to the non-linear nature of the NodePiece encoder, resulting entity embeddings still form", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 635, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 635, + 506, + 646 + ], + "score": 1.0, + "content": "different clusters and communities not concentrated around one point. We believe this is the effect of", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 646, + 400, + 658 + ], + "spans": [ + { + "bbox": [ + 106, + 646, + 400, + 658 + ], + "score": 1.0, + "content": "a compositional encoder and plan to investigate this phenomenon further.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 28 + }, + { + "type": "title", + "bbox": [ + 107, + 673, + 391, + 686 + ], + "lines": [ + { + "bbox": [ + 106, + 673, + 392, + 687 + ], + "spans": [ + { + "bbox": [ + 106, + 673, + 392, + 687 + ], + "score": 1.0, + "content": "E TRANSDUCTIVE LINK PREDICTION RESULTS: MRR", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 272, + 712 + ], + "score": 1.0, + "content": "In addition to Figure 2 that presents Hits", + "type": "text" + }, + { + "bbox": [ + 272, + 699, + 291, + 709 + ], + "score": 0.6, + "content": "@ 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 698, + 506, + 712 + ], + "score": 1.0, + "content": ", we report variations of mean reciprocal rank (MRR)", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 232, + 722 + ], + "score": 1.0, + "content": "depending on combinations of", + "type": "text" + }, + { + "bbox": [ + 233, + 710, + 241, + 720 + ], + "score": 0.78, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 710, + 259, + 722 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 260, + 710, + 267, + 720 + ], + "score": 0.84, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "on Figure 8 from the same set of experiments. On sparser", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 182, + 733 + ], + "score": 1.0, + "content": "WN18RR, smaller", + "type": "text" + }, + { + "bbox": [ + 183, + 720, + 202, + 732 + ], + "score": 0.91, + "content": "A / k", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "combinations like 25/5 or 50/10 struggle with more precise predictions like", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35 + } + ], + "page_idx": 19, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 15 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 160 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 506, + 95 + ], + "score": 1.0, + "content": "Finally, we fix the anchor selection strategy as mix, obtain nearest anchors per node, and under", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 312, + 106 + ], + "score": 1.0, + "content": "this setup study average anchor distances varying", + "type": "text" + }, + { + "bbox": [ + 313, + 94, + 320, + 104 + ], + "score": 0.67, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "- the number of anchors per node in various", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 227, + 117 + ], + "score": 1.0, + "content": "combinations of total anchors", + "type": "text" + }, + { + "bbox": [ + 227, + 105, + 235, + 114 + ], + "score": 0.51, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 104, + 505, + 117 + ], + "score": 1.0, + "content": ". The results presented on Figure 5 suggest that sparser graphs (like", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 363, + 128 + ], + "score": 1.0, + "content": "WN18RR) benefit more from increasing the number of anchors", + "type": "text" + }, + { + "bbox": [ + 363, + 116, + 372, + 125 + ], + "score": 0.71, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 114, + 505, + 128 + ], + "score": 1.0, + "content": ", i.e., the delta between distances", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "score": 1.0, + "content": "is much larger than that of dense FB15k-237. The difference in distances on Figure 5 might also", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 343, + 150 + ], + "score": 1.0, + "content": "explain the performance on Figure 8, i.e., generally, smaller", + "type": "text" + }, + { + "bbox": [ + 343, + 137, + 362, + 149 + ], + "score": 0.91, + "content": "A / k", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 137, + 506, + 150 + ], + "score": 1.0, + "content": "configurations like 25/5 are inferior", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 149, + 350, + 161 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 350, + 161 + ], + "score": 1.0, + "content": "on sparser graphs but perform competitively on denser ones.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3, + "bbox_fs": [ + 105, + 81, + 506, + 161 + ] + }, + { + "type": "image", + "bbox": [ + 115, + 197, + 489, + 343 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 115, + 197, + 489, + 343 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 115, + 197, + 489, + 343 + ], + "spans": [ + { + "bbox": [ + 115, + 197, + 489, + 343 + ], + "score": 0.972, + "type": "image", + "image_path": "5c83762ad70aceab22b28829e29e8a4217f2f51bb030fca7fc3d9b6ffed58232.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 115, + 197, + 489, + 245.66666666666666 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 115, + 245.66666666666666, + 489, + 294.3333333333333 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 115, + 294.3333333333333, + 489, + 343.0 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 353, + 505, + 397 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 352, + 506, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 449, + 365 + ], + "score": 1.0, + "content": "Figure 5: Average node-anchor distances when varying the total number of anchors", + "type": "text" + }, + { + "bbox": [ + 450, + 353, + 458, + 363 + ], + "score": 0.69, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 352, + 506, + 365 + ], + "score": 1.0, + "content": "from 25 to", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 363, + 505, + 377 + ], + "spans": [ + { + "bbox": [ + 104, + 363, + 144, + 377 + ], + "score": 1.0, + "content": "1000 and", + "type": "text" + }, + { + "bbox": [ + 145, + 365, + 152, + 374 + ], + "score": 0.78, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 363, + 505, + 377 + ], + "score": 1.0, + "content": "nearest anchors per node from 5 to 50. Note that on a sparser WN18RR the gap between", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 375, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 505, + 387 + ], + "score": 1.0, + "content": "min and max values is much wider than of denser FB15k-237. Signs of saturation suggest that further", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 387, + 228, + 398 + ], + "spans": [ + { + "bbox": [ + 106, + 387, + 150, + 398 + ], + "score": 1.0, + "content": "increasing", + "type": "text" + }, + { + "bbox": [ + 150, + 387, + 159, + 396 + ], + "score": 0.7, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 387, + 228, + 398 + ], + "score": 1.0, + "content": "is not beneficial.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11.5 + } + ], + "index": 9.75 + }, + { + "type": "title", + "bbox": [ + 108, + 422, + 283, + 434 + ], + "lines": [ + { + "bbox": [ + 105, + 420, + 285, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 285, + 437 + ], + "score": 1.0, + "content": "D EMBEDDING VISUALIZATIONS", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 447, + 504, + 480 + ], + "lines": [ + { + "bbox": [ + 105, + 447, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 505, + 459 + ], + "score": 1.0, + "content": "To further study learned representations of anchors and capabilities of the encoder, we build tSNE", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 458, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 506, + 470 + ], + "score": 1.0, + "content": "and UMAP projections from subsamples of FB15k-237 and WN18RR based on trained models from", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 469, + 430, + 481 + ], + "spans": [ + { + "bbox": [ + 106, + 469, + 430, + 481 + ], + "score": 1.0, + "content": "the transductive link prediction experiments (hyperparameters listed in Table A).", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 447, + 506, + 481 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 485, + 505, + 551 + ], + "lines": [ + { + "bbox": [ + 105, + 484, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 506, + 498 + ], + "score": 1.0, + "content": "For FB15k-237, we randomly sample 1000 entities (out of total 15K) and find their top-100 most", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 497, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 505, + 510 + ], + "score": 1.0, + "content": "common anchors. The anchor embeddings are extracted from the learned tensor while 1000 entity", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 507, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 505, + 520 + ], + "score": 1.0, + "content": "embeddings are obtained through the NodePiece encoder. Similarly for WN18RR, we sample 4000", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 518, + 506, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 506, + 531 + ], + "score": 1.0, + "content": "entities (out of total 40K) keeping their top-100 most common anchors. As we use the RotatE decoder", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 527, + 507, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 507, + 544 + ], + "score": 1.0, + "content": "that assumes entities and anchors are modeled in a complex space, we visualize their real parts (e.g.,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 541, + 239, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 541, + 239, + 552 + ], + "score": 1.0, + "content": "first 100 dimensions out of 200).", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 484, + 507, + 552 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 557, + 505, + 657 + ], + "lines": [ + { + "bbox": [ + 105, + 557, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 332, + 569 + ], + "score": 1.0, + "content": "Recalling that link prediction performance of NodePiece", + "type": "text" + }, + { + "bbox": [ + 333, + 559, + 341, + 567 + ], + "score": 0.75, + "content": "^ +", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 557, + 400, + 569 + ], + "score": 1.0, + "content": "RotatE retains", + "type": "text" + }, + { + "bbox": [ + 400, + 558, + 433, + 568 + ], + "score": 0.87, + "content": "80 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 557, + 506, + 569 + ], + "score": 1.0, + "content": "of the state of the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 568, + 506, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 506, + 581 + ], + "score": 1.0, + "content": "art models performance, the results on Fig. 6 and Fig. 7 demonstrate that (1) NodePiece encoder is", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 580, + 506, + 592 + ], + "spans": [ + { + "bbox": [ + 106, + 580, + 506, + 592 + ], + "score": 1.0, + "content": "able to reconstruct clusters of similar entities; (2) anchors are well-scattered among communities.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 591, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 106, + 591, + 322, + 603 + ], + "score": 1.0, + "content": "Albeit entity embeddings are built as a composition of", + "type": "text" + }, + { + "bbox": [ + 323, + 591, + 330, + 600 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 591, + 505, + 603 + ], + "score": 1.0, + "content": "anchors, it can be seen that all communities", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 601, + 506, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 506, + 614 + ], + "score": 1.0, + "content": "have \"specialized\" nearby anchors. On a higher level, common anchors tend to be well-scattered in", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 610, + 507, + 627 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 507, + 627 + ], + "score": 1.0, + "content": "the space. Less common anchors, as seen on FB15k-237 and Fig. 6, tend to group together. However,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 623, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 106, + 623, + 506, + 635 + ], + "score": 1.0, + "content": "thanks to the non-linear nature of the NodePiece encoder, resulting entity embeddings still form", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 635, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 635, + 506, + 646 + ], + "score": 1.0, + "content": "different clusters and communities not concentrated around one point. 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On sparser", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 182, + 733 + ], + "score": 1.0, + "content": "WN18RR, smaller", + "type": "text" + }, + { + "bbox": [ + 183, + 720, + 202, + 732 + ], + "score": 0.91, + "content": "A / k", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "combinations like 25/5 or 50/10 struggle with more precise predictions like", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35, + "bbox_fs": [ + 105, + 698, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 108, + 83, + 502, + 260 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 108, + 83, + 502, + 260 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 83, + 502, + 260 + ], + "spans": [ + { + "bbox": [ + 108, + 83, + 502, + 260 + ], + "score": 0.969, + "type": "image", + "image_path": "1af61e1e10b58335056878a25e13f8e5248e69fd3b55b7b56b177dfd7f48b627.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 108, + 83, + 502, + 142.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 108, + 142.0, + 502, + 201.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 108, + 201.0, + 502, + 260.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 105, + 271, + 504, + 294 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 271, + 505, + 284 + ], + "spans": [ + { + "bbox": [ + 106, + 271, + 505, + 284 + ], + "score": 1.0, + "content": "Figure 6: tSNE (left) and UMAP (right) projections of 1000 encoded entities sampled randomly from", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 282, + 320, + 294 + ], + "spans": [ + { + "bbox": [ + 106, + 282, + 120, + 294 + ], + "score": 1.0, + "content": "FB", + "type": "text" + }, + { + "bbox": [ + 120, + 282, + 135, + 293 + ], + "score": 0.26, + "content": "1 5 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 282, + 320, + 294 + ], + "score": 1.0, + "content": "-237 and their top 100 most common anchors.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "image", + "bbox": [ + 108, + 308, + 501, + 488 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 108, + 308, + 501, + 488 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 108, + 308, + 501, + 488 + ], + "spans": [ + { + "bbox": [ + 108, + 308, + 501, + 488 + ], + "score": 0.969, + "type": "image", + "image_path": "8957842fac60ab0a93c114967577de387a5575380933e8686ecfe472945834f4.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 108, + 308, + 501, + 368.0 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 108, + 368.0, + 501, + 428.0 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 108, + 428.0, + 501, + 488.0 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 498, + 504, + 520 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 105, + 497, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 505, + 511 + ], + "score": 1.0, + "content": "Figure 7: tSNE (left) and UMAP (right) projections of 4000 encoded entities sampled randomly from", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 508, + 313, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 508, + 313, + 521 + ], + "score": 1.0, + "content": "WN18RR and their top 100 most common anchors.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5 + } + ], + "index": 7.25 + }, + { + "type": "text", + "bbox": [ + 107, + 543, + 505, + 577 + ], + "lines": [ + { + "bbox": [ + 105, + 543, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 124, + 556 + ], + "score": 1.0, + "content": "Hits", + "type": "text" + }, + { + "bbox": [ + 124, + 545, + 138, + 554 + ], + "score": 0.7, + "content": "@ 1", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 543, + 505, + 556 + ], + "score": 1.0, + "content": "which is captured by low values of MRR. 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First, we remove all triples", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 667, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 505, + 678 + ], + "score": 1.0, + "content": "containing Wikidata properties P31 (instance of ) and P279 (subclass of ) as they already contain", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "class information. We then remove nodes that became disconnected after removing those edges.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "Third, using SPARQL queries, for each remaining node in a graph, we extract a 3-hop class hierarchy", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "of Wikidata classes and their superclasses. We only keep class labels that occur at least 50 times in", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 244, + 722 + ], + "score": 1.0, + "content": "the training set. Then, we sample", + "type": "text" + }, + { + "bbox": [ + 244, + 710, + 263, + 720 + ], + "score": 0.87, + "content": "10 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 709, + 423, + 722 + ], + "score": 1.0, + "content": "of nodes with labels for validation and", + "type": "text" + }, + { + "bbox": [ + 423, + 710, + 442, + 720 + ], + "score": 0.87, + "content": "10 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "for test, and of", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 150, + 733 + ], + "score": 1.0, + "content": "remaining", + "type": "text" + }, + { + "bbox": [ + 150, + 721, + 170, + 731 + ], + "score": 0.84, + "content": "80 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 170, + 721, + 471, + 733 + ], + "score": 1.0, + "content": "we sample a set of nodes for the semi-supervised setup, i.e., we keep only", + "type": "text" + }, + { + "bbox": [ + 472, + 721, + 487, + 731 + ], + "score": 0.85, + "content": "5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 643, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 109, + 80, + 501, + 231 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 109, + 80, + 501, + 231 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 80, + 501, + 231 + ], + "spans": [ + { + "bbox": [ + 109, + 80, + 501, + 231 + ], + "score": 0.972, + "type": "image", + "image_path": "5a8194a73210c9ecf515ecad0695a36c6cc94619e6c07cb6c8a88372a7ca6b56.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 109, + 80, + 501, + 130.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 109, + 130.33333333333334, + 501, + 180.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 109, + 180.66666666666669, + 501, + 231.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 241, + 506, + 286 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 241, + 506, + 253 + ], + "spans": [ + { + "bbox": [ + 106, + 241, + 267, + 253 + ], + "score": 1.0, + "content": "Figure 8: Combinations of total anchors", + "type": "text" + }, + { + "bbox": [ + 267, + 242, + 276, + 252 + ], + "score": 0.66, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 241, + 506, + 253 + ], + "score": 1.0, + "content": "and anchors per node. Denser FB15k-237 saturates faster", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 252, + 506, + 264 + ], + "spans": [ + { + "bbox": [ + 106, + 252, + 150, + 264 + ], + "score": 1.0, + "content": "on smaller", + "type": "text" + }, + { + "bbox": [ + 150, + 253, + 159, + 263 + ], + "score": 0.68, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 252, + 506, + 264 + ], + "score": 1.0, + "content": "while sparse WN18RR saturates at around 500 anchors. MRR metric captures all ranks.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 263, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 505, + 277 + ], + "score": 1.0, + "content": "Note that performance gap on FB15k-237 is very small indicating that saturation has occurred already", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 275, + 242, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 242, + 286 + ], + "score": 1.0, + "content": "with small anchor configurations.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "text", + "bbox": [ + 107, + 306, + 504, + 328 + ], + "lines": [ + { + "bbox": [ + 106, + 304, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 106, + 306, + 126, + 317 + ], + "score": 0.84, + "content": "10 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 304, + 505, + 320 + ], + "score": 1.0, + "content": "of those nodes. The resulting graph has 46k nodes, 526 distinct relation types, and 465 class", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 316, + 136, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 136, + 329 + ], + "score": 1.0, + "content": "labels.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5 + }, + { + "type": "title", + "bbox": [ + 106, + 342, + 349, + 353 + ], + "lines": [ + { + "bbox": [ + 105, + 342, + 349, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 349, + 354 + ], + "score": 1.0, + "content": "F.2 OUF-OF-SAMPLE LINK PREDICTION: OYAGO 3-10", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 107, + 362, + 505, + 429 + ], + "lines": [ + { + "bbox": [ + 106, + 362, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 362, + 505, + 375 + ], + "score": 1.0, + "content": "For sampling the out-of-sample version of a bigger YAGO 3-10 we largely follow the same original", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 373, + 506, + 386 + ], + "spans": [ + { + "bbox": [ + 104, + 373, + 506, + 386 + ], + "score": 1.0, + "content": "procedure described in Section 4 of (Albooyeh et al., 2020). We first merge the train, validation and", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 383, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 506, + 398 + ], + "score": 1.0, + "content": "test triples from the original dataset for transductive link prediction. Then, from all entities appearing", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 396, + 504, + 407 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 275, + 407 + ], + "score": 1.0, + "content": "in at least two triples, we randomly sample", + "type": "text" + }, + { + "bbox": [ + 275, + 396, + 290, + 406 + ], + "score": 0.86, + "content": "5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 396, + 504, + 407 + ], + "score": 1.0, + "content": "of nodes to be the out-of-sample entities for validation", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 406, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 406, + 123, + 419 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 123, + 407, + 137, + 417 + ], + "score": 0.85, + "content": "5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 406, + 505, + 419 + ], + "score": 1.0, + "content": "for test. All triples containing the out-of-sample entities on subject or object positions are put", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 417, + 485, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 485, + 430 + ], + "score": 1.0, + "content": "into validation or test, respectively, as edges that connect an unseen entity with the seen graph.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 12.5 + }, + { + "type": "title", + "bbox": [ + 107, + 444, + 354, + 457 + ], + "lines": [ + { + "bbox": [ + 106, + 443, + 354, + 459 + ], + "spans": [ + { + "bbox": [ + 106, + 443, + 354, + 459 + ], + "score": 1.0, + "content": "G NODE CLASSIFICATION: TRAINING CURVES", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 469, + 505, + 524 + ], + "lines": [ + { + "bbox": [ + 106, + 470, + 505, + 482 + ], + "spans": [ + { + "bbox": [ + 106, + 470, + 505, + 482 + ], + "score": 1.0, + "content": "Figure 9 depicts train and validation values of Hard Accuracy and PRC-AUC metrics for all the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 480, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 262, + 491 + ], + "score": 1.0, + "content": "compared models on WD50K NC with", + "type": "text" + }, + { + "bbox": [ + 262, + 480, + 277, + 491 + ], + "score": 0.86, + "content": "5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 480, + 505, + 491 + ], + "score": 1.0, + "content": "of labeled nodes. The NodePiece model has only 50 total", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 491, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 505, + 504 + ], + "score": 1.0, + "content": "anchors with 10 nearest anchors per node, and 5 unique relation types in the relational context. The", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 502, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 234, + 515 + ], + "score": 1.0, + "content": "performance on the dataset with", + "type": "text" + }, + { + "bbox": [ + 235, + 502, + 254, + 513 + ], + "score": 0.86, + "content": "10 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 502, + 490, + 515 + ], + "score": 1.0, + "content": "of nodes is almost the same, so we report the charts only on", + "type": "text" + }, + { + "bbox": [ + 490, + 502, + 505, + 513 + ], + "score": 0.84, + "content": "5 \\%", + "type": "inline_equation" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 513, + 500, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 500, + 526 + ], + "score": 1.0, + "content": "dataset. By the generalization gap we understand the delta between training and validation values.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 530, + 505, + 585 + ], + "lines": [ + { + "bbox": [ + 105, + 528, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 505, + 544 + ], + "score": 1.0, + "content": "The MLP baseline quickly overfits but fails to generalize on the validation. The generalization gap", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 541, + 505, + 553 + ], + "spans": [ + { + "bbox": [ + 106, + 541, + 505, + 553 + ], + "score": 1.0, + "content": "of CompGCN is smaller compared to MLP but is still significant, i.e., validation performance is", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 550, + 507, + 566 + ], + "spans": [ + { + "bbox": [ + 106, + 552, + 131, + 563 + ], + "score": 0.87, + "content": "2 { - } 3 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 550, + 507, + 566 + ], + "score": 1.0, + "content": "smaller than train. Finally, the NodePiece-enabled model has the smallest generalization gaps,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 562, + 506, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 506, + 577 + ], + "score": 1.0, + "content": "especially along the Hard Accuracy metric where the validation performance is very close to that of", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 574, + 363, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 363, + 587 + ], + "score": 1.0, + "content": "train. Similarly, the gap on PRC-AUC is smaller than 10 points.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 591, + 505, + 657 + ], + "lines": [ + { + "bbox": [ + 106, + 590, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 106, + 590, + 505, + 603 + ], + "score": 1.0, + "content": "As shown in the ablation study in Table 6, it appears that explicit node embeddings do not contribute", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 601, + 505, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 505, + 615 + ], + "score": 1.0, + "content": "to the classification performance. Hence, the baseline models tend to be overparameterized where", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 612, + 506, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 506, + 626 + ], + "score": 1.0, + "content": "learnable node embeddings add noise, while the NodePiece model has only a few anchors (or no an-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 623, + 505, + 637 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 505, + 637 + ], + "score": 1.0, + "content": "chors at all when using only the relational context), much fewer parameters, and therefore generalizes", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 635, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 635, + 505, + 646 + ], + "score": 1.0, + "content": "better. This hypothesis also explains the observation that the node classification performance does", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 646, + 336, + 658 + ], + "spans": [ + { + "bbox": [ + 106, + 646, + 225, + 658 + ], + "score": 1.0, + "content": "not improve when increasing", + "type": "text" + }, + { + "bbox": [ + 225, + 646, + 244, + 658 + ], + "score": 0.92, + "content": "A / k", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 646, + 336, + 658 + ], + "score": 1.0, + "content": "anchor configurations.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 29.5 + }, + { + "type": "title", + "bbox": [ + 107, + 673, + 169, + 685 + ], + "lines": [ + { + "bbox": [ + 105, + 671, + 171, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 171, + 689 + ], + "score": 1.0, + "content": "H PROOFS", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 108, + 695, + 504, + 731 + ], + "lines": [ + { + "bbox": [ + 104, + 694, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 104, + 694, + 307, + 712 + ], + "score": 1.0, + "content": "Proposition 2. The nearest-anchor encoder with", + "type": "text" + }, + { + "bbox": [ + 307, + 697, + 326, + 711 + ], + "score": 0.91, + "content": "\\binom { | A | } { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 694, + 381, + 712 + ], + "score": 1.0, + "content": "anchors and", + "type": "text" + }, + { + "bbox": [ + 381, + 699, + 396, + 710 + ], + "score": 0.92, + "content": "| m |", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 694, + 506, + 712 + ], + "score": 1.0, + "content": "subsampled relations, can", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 708, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 171, + 723 + ], + "score": 1.0, + "content": "be considered a", + "type": "text" + }, + { + "bbox": [ + 172, + 711, + 179, + 720 + ], + "score": 0.6, + "content": "\\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 708, + 274, + 723 + ], + "score": 1.0, + "content": "-SGD approximation of", + "type": "text" + }, + { + "bbox": [ + 275, + 711, + 314, + 722 + ], + "score": 0.91, + "content": "( k + | m | )", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 708, + 506, + 723 + ], + "score": 1.0, + "content": "-ary Janossy pooling with a canonical ordering", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 720, + 240, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 240, + 733 + ], + "score": 1.0, + "content": "induced by the anchor distances.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35 + } + ], + "page_idx": 21, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 105, + 25, + 294, + 39 + ], + "spans": [ + { + "bbox": [ + 105, + 25, + 294, + 39 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 765 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 765 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 15 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 109, + 80, + 501, + 231 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 109, + 80, + 501, + 231 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 80, + 501, + 231 + ], + "spans": [ + { + "bbox": [ + 109, + 80, + 501, + 231 + ], + "score": 0.972, + "type": "image", + "image_path": "5a8194a73210c9ecf515ecad0695a36c6cc94619e6c07cb6c8a88372a7ca6b56.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 109, + 80, + 501, + 130.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 109, + 130.33333333333334, + 501, + 180.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 109, + 180.66666666666669, + 501, + 231.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 241, + 506, + 286 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 241, + 506, + 253 + ], + "spans": [ + { + "bbox": [ + 106, + 241, + 267, + 253 + ], + "score": 1.0, + "content": "Figure 8: Combinations of total anchors", + "type": "text" + }, + { + "bbox": [ + 267, + 242, + 276, + 252 + ], + "score": 0.66, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 241, + 506, + 253 + ], + "score": 1.0, + "content": "and anchors per node. Denser FB15k-237 saturates faster", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 252, + 506, + 264 + ], + "spans": [ + { + "bbox": [ + 106, + 252, + 150, + 264 + ], + "score": 1.0, + "content": "on smaller", + "type": "text" + }, + { + "bbox": [ + 150, + 253, + 159, + 263 + ], + "score": 0.68, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 252, + 506, + 264 + ], + "score": 1.0, + "content": "while sparse WN18RR saturates at around 500 anchors. MRR metric captures all ranks.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 263, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 505, + 277 + ], + "score": 1.0, + "content": "Note that performance gap on FB15k-237 is very small indicating that saturation has occurred already", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 275, + 242, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 242, + 286 + ], + "score": 1.0, + "content": "with small anchor configurations.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "text", + "bbox": [ + 107, + 306, + 504, + 328 + ], + "lines": [ + { + "bbox": [ + 106, + 304, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 106, + 306, + 126, + 317 + ], + "score": 0.84, + "content": "10 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 304, + 505, + 320 + ], + "score": 1.0, + "content": "of those nodes. The resulting graph has 46k nodes, 526 distinct relation types, and 465 class", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 316, + 136, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 136, + 329 + ], + "score": 1.0, + "content": "labels.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5, + "bbox_fs": [ + 105, + 304, + 505, + 329 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 342, + 349, + 353 + ], + "lines": [ + { + "bbox": [ + 105, + 342, + 349, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 349, + 354 + ], + "score": 1.0, + "content": "F.2 OUF-OF-SAMPLE LINK PREDICTION: OYAGO 3-10", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 107, + 362, + 505, + 429 + ], + "lines": [ + { + "bbox": [ + 106, + 362, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 362, + 505, + 375 + ], + "score": 1.0, + "content": "For sampling the out-of-sample version of a bigger YAGO 3-10 we largely follow the same original", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 373, + 506, + 386 + ], + "spans": [ + { + "bbox": [ + 104, + 373, + 506, + 386 + ], + "score": 1.0, + "content": "procedure described in Section 4 of (Albooyeh et al., 2020). We first merge the train, validation and", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 383, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 506, + 398 + ], + "score": 1.0, + "content": "test triples from the original dataset for transductive link prediction. Then, from all entities appearing", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 396, + 504, + 407 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 275, + 407 + ], + "score": 1.0, + "content": "in at least two triples, we randomly sample", + "type": "text" + }, + { + "bbox": [ + 275, + 396, + 290, + 406 + ], + "score": 0.86, + "content": "5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 396, + 504, + 407 + ], + "score": 1.0, + "content": "of nodes to be the out-of-sample entities for validation", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 406, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 406, + 123, + 419 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 123, + 407, + 137, + 417 + ], + "score": 0.85, + "content": "5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 406, + 505, + 419 + ], + "score": 1.0, + "content": "for test. All triples containing the out-of-sample entities on subject or object positions are put", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 417, + 485, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 485, + 430 + ], + "score": 1.0, + "content": "into validation or test, respectively, as edges that connect an unseen entity with the seen graph.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 12.5, + "bbox_fs": [ + 104, + 362, + 506, + 430 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 444, + 354, + 457 + ], + "lines": [ + { + "bbox": [ + 106, + 443, + 354, + 459 + ], + "spans": [ + { + "bbox": [ + 106, + 443, + 354, + 459 + ], + "score": 1.0, + "content": "G NODE CLASSIFICATION: TRAINING CURVES", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 469, + 505, + 524 + ], + "lines": [ + { + "bbox": [ + 106, + 470, + 505, + 482 + ], + "spans": [ + { + "bbox": [ + 106, + 470, + 505, + 482 + ], + "score": 1.0, + "content": "Figure 9 depicts train and validation values of Hard Accuracy and PRC-AUC metrics for all the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 480, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 262, + 491 + ], + "score": 1.0, + "content": "compared models on WD50K NC with", + "type": "text" + }, + { + "bbox": [ + 262, + 480, + 277, + 491 + ], + "score": 0.86, + "content": "5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 480, + 505, + 491 + ], + "score": 1.0, + "content": "of labeled nodes. The NodePiece model has only 50 total", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 491, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 505, + 504 + ], + "score": 1.0, + "content": "anchors with 10 nearest anchors per node, and 5 unique relation types in the relational context. The", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 502, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 234, + 515 + ], + "score": 1.0, + "content": "performance on the dataset with", + "type": "text" + }, + { + "bbox": [ + 235, + 502, + 254, + 513 + ], + "score": 0.86, + "content": "10 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 502, + 490, + 515 + ], + "score": 1.0, + "content": "of nodes is almost the same, so we report the charts only on", + "type": "text" + }, + { + "bbox": [ + 490, + 502, + 505, + 513 + ], + "score": 0.84, + "content": "5 \\%", + "type": "inline_equation" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 513, + 500, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 500, + 526 + ], + "score": 1.0, + "content": "dataset. By the generalization gap we understand the delta between training and validation values.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 470, + 505, + 526 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 530, + 505, + 585 + ], + "lines": [ + { + "bbox": [ + 105, + 528, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 505, + 544 + ], + "score": 1.0, + "content": "The MLP baseline quickly overfits but fails to generalize on the validation. The generalization gap", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 541, + 505, + 553 + ], + "spans": [ + { + "bbox": [ + 106, + 541, + 505, + 553 + ], + "score": 1.0, + "content": "of CompGCN is smaller compared to MLP but is still significant, i.e., validation performance is", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 550, + 507, + 566 + ], + "spans": [ + { + "bbox": [ + 106, + 552, + 131, + 563 + ], + "score": 0.87, + "content": "2 { - } 3 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 550, + 507, + 566 + ], + "score": 1.0, + "content": "smaller than train. Finally, the NodePiece-enabled model has the smallest generalization gaps,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 562, + 506, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 506, + 577 + ], + "score": 1.0, + "content": "especially along the Hard Accuracy metric where the validation performance is very close to that of", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 574, + 363, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 363, + 587 + ], + "score": 1.0, + "content": "train. Similarly, the gap on PRC-AUC is smaller than 10 points.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 528, + 507, + 587 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 591, + 505, + 657 + ], + "lines": [ + { + "bbox": [ + 106, + 590, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 106, + 590, + 505, + 603 + ], + "score": 1.0, + "content": "As shown in the ablation study in Table 6, it appears that explicit node embeddings do not contribute", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 601, + 505, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 505, + 615 + ], + "score": 1.0, + "content": "to the classification performance. Hence, the baseline models tend to be overparameterized where", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 612, + 506, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 506, + 626 + ], + "score": 1.0, + "content": "learnable node embeddings add noise, while the NodePiece model has only a few anchors (or no an-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 623, + 505, + 637 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 505, + 637 + ], + "score": 1.0, + "content": "chors at all when using only the relational context), much fewer parameters, and therefore generalizes", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 635, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 635, + 505, + 646 + ], + "score": 1.0, + "content": "better. This hypothesis also explains the observation that the node classification performance does", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 646, + 336, + 658 + ], + "spans": [ + { + "bbox": [ + 106, + 646, + 225, + 658 + ], + "score": 1.0, + "content": "not improve when increasing", + "type": "text" + }, + { + "bbox": [ + 225, + 646, + 244, + 658 + ], + "score": 0.92, + "content": "A / k", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 646, + 336, + 658 + ], + "score": 1.0, + "content": "anchor configurations.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 590, + 506, + 658 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 673, + 169, + 685 + ], + "lines": [ + { + "bbox": [ + 105, + 671, + 171, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 171, + 689 + ], + "score": 1.0, + "content": "H PROOFS", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 108, + 695, + 504, + 731 + ], + "lines": [ + { + "bbox": [ + 104, + 694, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 104, + 694, + 307, + 712 + ], + "score": 1.0, + "content": "Proposition 2. The nearest-anchor encoder with", + "type": "text" + }, + { + "bbox": [ + 307, + 697, + 326, + 711 + ], + "score": 0.91, + "content": "\\binom { | A | } { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 694, + 381, + 712 + ], + "score": 1.0, + "content": "anchors and", + "type": "text" + }, + { + "bbox": [ + 381, + 699, + 396, + 710 + ], + "score": 0.92, + "content": "| m |", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 694, + 506, + 712 + ], + "score": 1.0, + "content": "subsampled relations, can", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 708, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 171, + 723 + ], + "score": 1.0, + "content": "be considered a", + "type": "text" + }, + { + "bbox": [ + 172, + 711, + 179, + 720 + ], + "score": 0.6, + "content": "\\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 708, + 274, + 723 + ], + "score": 1.0, + "content": "-SGD approximation of", + "type": "text" + }, + { + "bbox": [ + 275, + 711, + 314, + 722 + ], + "score": 0.91, + "content": "( k + | m | )", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 708, + 506, + 723 + ], + "score": 1.0, + "content": "-ary Janossy pooling with a canonical ordering", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 720, + 240, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 240, + 733 + ], + "score": 1.0, + "content": "induced by the anchor distances.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35, + "bbox_fs": [ + 104, + 694, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 106, + 78, + 505, + 237 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 106, + 78, + 505, + 237 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 78, + 505, + 237 + ], + "spans": [ + { + "bbox": [ + 106, + 78, + 505, + 237 + ], + "score": 0.969, + "type": "image", + "image_path": "0e115ed23e92c50662171a6d4f54a45b264834f88d611131451b294b96f29048.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 106, + 78, + 505, + 131.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 106, + 131.0, + 505, + 184.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 106, + 184.0, + 505, + 237.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 246, + 504, + 269 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 246, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 106, + 246, + 271, + 259 + ], + "score": 1.0, + "content": "Figure 9: Generalization gap on WD50K", + "type": "text" + }, + { + "bbox": [ + 271, + 246, + 286, + 257 + ], + "score": 0.82, + "content": "5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 246, + 505, + 259 + ], + "score": 1.0, + "content": "labeled nodes). NodePiece-based model has observably", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 258, + 325, + 270 + ], + "spans": [ + { + "bbox": [ + 106, + 258, + 325, + 270 + ], + "score": 1.0, + "content": "smaller generalization gaps compared to the baselines.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "title", + "bbox": [ + 107, + 293, + 133, + 304 + ], + "lines": [ + { + "bbox": [ + 105, + 293, + 136, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 136, + 306 + ], + "score": 1.0, + "content": "Proof.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 106, + 310, + 505, + 334 + ], + "lines": [ + { + "bbox": [ + 105, + 309, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 506, + 324 + ], + "score": 1.0, + "content": "We begin by providing the definition of Janossy pooling as it was presented in the original paper", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 320, + 195, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 195, + 335 + ], + "score": 1.0, + "content": "Murphy et al. 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This is done by randomly ordering anchors that are", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 299, + 506, + 312 + ], + "spans": [ + { + "bbox": [ + 106, + 299, + 506, + 312 + ], + "score": 1.0, + "content": "equidistant resulting in a uniform sampling of possible permutations during training and evaluation.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 309, + 505, + 323 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 278, + 323 + ], + "score": 1.0, + "content": "For more details on the formal definition of", + "type": "text" + }, + { + "bbox": [ + 279, + 311, + 285, + 320 + ], + "score": 0.76, + "content": "\\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 309, + 505, + 323 + ], + "score": 1.0, + "content": "-SGD we point the reader to the original paper (Murphy", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 321, + 159, + 333 + ], + "spans": [ + { + "bbox": [ + 106, + 321, + 159, + 333 + ], + "score": 1.0, + "content": "et al., 2019).", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9.5 + }, + { + "type": "title", + "bbox": [ + 107, + 349, + 245, + 362 + ], + "lines": [ + { + "bbox": [ + 105, + 348, + 246, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 246, + 363 + ], + "score": 1.0, + "content": "I RELATION PREDICTION", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 374, + 505, + 430 + ], + "lines": [ + { + "bbox": [ + 106, + 375, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 505, + 387 + ], + "score": 1.0, + "content": "Setup. We conduct the relation prediction experiment on the same FB15k-237, WN18RR, and YAGO", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 386, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 506, + 398 + ], + "score": 1.0, + "content": "3-10 datasets. While link prediction deals with entities, the relation prediction model has to rank a", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 396, + 506, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 388, + 411 + ], + "score": 1.0, + "content": "correct relation given a (head, ?, tail) query. We report MRR and Hits", + "type": "text" + }, + { + "bbox": [ + 388, + 397, + 408, + 407 + ], + "score": 0.76, + "content": "@ 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 396, + 506, + 411 + ], + "score": 1.0, + "content": "in the filtered setting as", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 407, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 441, + 421 + ], + "score": 1.0, + "content": "evaluation metrics. Similar to the link prediction configuration, we use NodePiece", + "type": "text" + }, + { + "bbox": [ + 441, + 408, + 457, + 418 + ], + "score": 0.78, + "content": "^ { + 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 407, + 505, + 421 + ], + "score": 1.0, + "content": "-layer MLP", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 419, + 359, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 359, + 432 + ], + "score": 1.0, + "content": "and compare against RotatE of the same total parameter count.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 106, + 435, + 505, + 524 + ], + "lines": [ + { + "bbox": [ + 105, + 435, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 506, + 448 + ], + "score": 1.0, + "content": "Discussion. The reported results (Table 15) demonstrate a competitive performance of NodePiece-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 446, + 505, + 458 + ], + "spans": [ + { + "bbox": [ + 106, + 446, + 374, + 458 + ], + "score": 1.0, + "content": "based models with reduced vocabulary sizes bringing more than", + "type": "text" + }, + { + "bbox": [ + 374, + 447, + 394, + 457 + ], + "score": 0.81, + "content": "9 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 395, + 446, + 414, + 458 + ], + "score": 1.0, + "content": "Hits", + "type": "text" + }, + { + "bbox": [ + 414, + 447, + 434, + 457 + ], + "score": 0.74, + "content": "@ 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 446, + 505, + 458 + ], + "score": 1.0, + "content": "across graphs of", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 457, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 505, + 470 + ], + "score": 1.0, + "content": "different sizes. In the case of WN18RR and YAGO 3-10, NodePiece models with fewer anchors even", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 469, + 505, + 481 + ], + "spans": [ + { + "bbox": [ + 106, + 469, + 505, + 481 + ], + "score": 1.0, + "content": "slightly improve the accuracy upon the shallow embedding baseline. The ablation study suggests", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 480, + 504, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 480, + 504, + 491 + ], + "score": 1.0, + "content": "that on dense graphs with a reasonable amount of unique relations having explicit learnable node", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 491, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 409, + 502 + ], + "score": 1.0, + "content": "embeddings might not be needed at all for this task. That is, we see that on", + "type": "text" + }, + { + "bbox": [ + 410, + 491, + 457, + 501 + ], + "score": 0.31, + "content": "\\mathrm { F B } 1 5 \\mathrm { k } { - } 2 3 7", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 491, + 505, + 502 + ], + "score": 1.0, + "content": "and YAGO", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 501, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 505, + 513 + ], + "score": 1.0, + "content": "3-10 the NodePiece hashes comprised only of the relational context deliver the same performance", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 512, + 447, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 447, + 525 + ], + "score": 1.0, + "content": "without any performance drop confirming the findings from the previous experiment.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 22.5 + }, + { + "type": "title", + "bbox": [ + 108, + 541, + 280, + 553 + ], + "lines": [ + { + "bbox": [ + 104, + 539, + 282, + 555 + ], + "spans": [ + { + "bbox": [ + 104, + 539, + 282, + 555 + ], + "score": 1.0, + "content": "J INDUCTIVE LINK PREDICTION", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 565, + 505, + 621 + ], + "lines": [ + { + "bbox": [ + 106, + 566, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 566, + 505, + 578 + ], + "score": 1.0, + "content": "Setup. Nodes in the inference graphs do not have any associated feature vectors which makes this", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 576, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 505, + 591 + ], + "score": 1.0, + "content": "benchmark very relevant for graph representation learning. Importantly, the set of relation types", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 587, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 505, + 601 + ], + "score": 1.0, + "content": "in the inference graphs is a subset of those seen in the training set. Since the relation embedding", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 597, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 505, + 613 + ], + "score": 1.0, + "content": "matrix can be learned on the training graph, we therefore have a uniform method for constructing", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 610, + 504, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 504, + 623 + ], + "score": 1.0, + "content": "node representations both on seen and unseen graphs. As an encoder we try MLP and Transformer.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 626, + 505, + 671 + ], + "lines": [ + { + "bbox": [ + 105, + 626, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 506, + 640 + ], + "score": 1.0, + "content": "Evaluation Protocol. Following the original work (Teru et al., 2020), we employ a filtered setting", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 400, + 650 + ], + "score": 1.0, + "content": "and rank each triple against 50 random negative triples reporting the Hits", + "type": "text" + }, + { + "bbox": [ + 400, + 639, + 419, + 648 + ], + "score": 0.33, + "content": "@ 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "metric. This setup is", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 649, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 440, + 661 + ], + "score": 1.0, + "content": "motivated by computational complexity of GraIL at inference time, while NodePiece", + "type": "text" + }, + { + "bbox": [ + 441, + 650, + 449, + 659 + ], + "score": 0.63, + "content": "^ +", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 649, + 505, + 661 + ], + "score": 1.0, + "content": "CompGCN is", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 660, + 475, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 475, + 672 + ], + "score": 1.0, + "content": "as fast in the inductive inference as in the transductive regime reported in other experiments.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34.5 + }, + { + "type": "text", + "bbox": [ + 107, + 676, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "Discussion. 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This is done by randomly ordering anchors that are", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 299, + 506, + 312 + ], + "spans": [ + { + "bbox": [ + 106, + 299, + 506, + 312 + ], + "score": 1.0, + "content": "equidistant resulting in a uniform sampling of possible permutations during training and evaluation.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 309, + 505, + 323 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 278, + 323 + ], + "score": 1.0, + "content": "For more details on the formal definition of", + "type": "text" + }, + { + "bbox": [ + 279, + 311, + 285, + 320 + ], + "score": 0.76, + "content": "\\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 309, + 505, + 323 + ], + "score": 1.0, + "content": "-SGD we point the reader to the original paper (Murphy", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 321, + 159, + 333 + ], + "spans": [ + { + "bbox": [ + 106, + 321, + 159, + 333 + ], + "score": 1.0, + "content": "et al., 2019).", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9.5, + "bbox_fs": [ + 105, + 266, + 506, + 333 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 349, + 245, + 362 + ], + "lines": [ + { + "bbox": [ + 105, + 348, + 246, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 246, + 363 + ], + "score": 1.0, + "content": "I RELATION PREDICTION", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 374, + 505, + 430 + ], + "lines": [ + { + "bbox": [ + 106, + 375, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 505, + 387 + ], + "score": 1.0, + "content": "Setup. We conduct the relation prediction experiment on the same FB15k-237, WN18RR, and YAGO", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 386, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 506, + 398 + ], + "score": 1.0, + "content": "3-10 datasets. While link prediction deals with entities, the relation prediction model has to rank a", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 396, + 506, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 388, + 411 + ], + "score": 1.0, + "content": "correct relation given a (head, ?, tail) query. We report MRR and Hits", + "type": "text" + }, + { + "bbox": [ + 388, + 397, + 408, + 407 + ], + "score": 0.76, + "content": "@ 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 396, + 506, + 411 + ], + "score": 1.0, + "content": "in the filtered setting as", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 407, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 441, + 421 + ], + "score": 1.0, + "content": "evaluation metrics. Similar to the link prediction configuration, we use NodePiece", + "type": "text" + }, + { + "bbox": [ + 441, + 408, + 457, + 418 + ], + "score": 0.78, + "content": "^ { + 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 407, + 505, + 421 + ], + "score": 1.0, + "content": "-layer MLP", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 419, + 359, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 359, + 432 + ], + "score": 1.0, + "content": "and compare against RotatE of the same total parameter count.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 375, + 506, + 432 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 435, + 505, + 524 + ], + "lines": [ + { + "bbox": [ + 105, + 435, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 506, + 448 + ], + "score": 1.0, + "content": "Discussion. The reported results (Table 15) demonstrate a competitive performance of NodePiece-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 446, + 505, + 458 + ], + "spans": [ + { + "bbox": [ + 106, + 446, + 374, + 458 + ], + "score": 1.0, + "content": "based models with reduced vocabulary sizes bringing more than", + "type": "text" + }, + { + "bbox": [ + 374, + 447, + 394, + 457 + ], + "score": 0.81, + "content": "9 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 395, + 446, + 414, + 458 + ], + "score": 1.0, + "content": "Hits", + "type": "text" + }, + { + "bbox": [ + 414, + 447, + 434, + 457 + ], + "score": 0.74, + "content": "@ 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 446, + 505, + 458 + ], + "score": 1.0, + "content": "across graphs of", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 457, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 505, + 470 + ], + "score": 1.0, + "content": "different sizes. In the case of WN18RR and YAGO 3-10, NodePiece models with fewer anchors even", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 469, + 505, + 481 + ], + "spans": [ + { + "bbox": [ + 106, + 469, + 505, + 481 + ], + "score": 1.0, + "content": "slightly improve the accuracy upon the shallow embedding baseline. The ablation study suggests", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 480, + 504, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 480, + 504, + 491 + ], + "score": 1.0, + "content": "that on dense graphs with a reasonable amount of unique relations having explicit learnable node", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 491, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 409, + 502 + ], + "score": 1.0, + "content": "embeddings might not be needed at all for this task. That is, we see that on", + "type": "text" + }, + { + "bbox": [ + 410, + 491, + 457, + 501 + ], + "score": 0.31, + "content": "\\mathrm { F B } 1 5 \\mathrm { k } { - } 2 3 7", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 491, + 505, + 502 + ], + "score": 1.0, + "content": "and YAGO", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 501, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 505, + 513 + ], + "score": 1.0, + "content": "3-10 the NodePiece hashes comprised only of the relational context deliver the same performance", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 512, + 447, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 447, + 525 + ], + "score": 1.0, + "content": "without any performance drop confirming the findings from the previous experiment.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 435, + 506, + 525 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 541, + 280, + 553 + ], + "lines": [ + { + "bbox": [ + 104, + 539, + 282, + 555 + ], + "spans": [ + { + "bbox": [ + 104, + 539, + 282, + 555 + ], + "score": 1.0, + "content": "J INDUCTIVE LINK PREDICTION", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 565, + 505, + 621 + ], + "lines": [ + { + "bbox": [ + 106, + 566, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 566, + 505, + 578 + ], + "score": 1.0, + "content": "Setup. Nodes in the inference graphs do not have any associated feature vectors which makes this", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 576, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 505, + 591 + ], + "score": 1.0, + "content": "benchmark very relevant for graph representation learning. Importantly, the set of relation types", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 587, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 505, + 601 + ], + "score": 1.0, + "content": "in the inference graphs is a subset of those seen in the training set. Since the relation embedding", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 597, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 505, + 613 + ], + "score": 1.0, + "content": "matrix can be learned on the training graph, we therefore have a uniform method for constructing", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 610, + 504, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 504, + 623 + ], + "score": 1.0, + "content": "node representations both on seen and unseen graphs. As an encoder we try MLP and Transformer.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 566, + 505, + 623 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 626, + 505, + 671 + ], + "lines": [ + { + "bbox": [ + 105, + 626, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 506, + 640 + ], + "score": 1.0, + "content": "Evaluation Protocol. 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This setup is", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 649, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 440, + 661 + ], + "score": 1.0, + "content": "motivated by computational complexity of GraIL at inference time, while NodePiece", + "type": "text" + }, + { + "bbox": [ + 441, + 650, + 449, + 659 + ], + "score": 0.63, + "content": "^ +", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 649, + 505, + 661 + ], + "score": 1.0, + "content": "CompGCN is", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 660, + 475, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 475, + 672 + ], + "score": 1.0, + "content": "as fast in the inductive inference as in the transductive regime reported in other experiments.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34.5, + "bbox_fs": [ + 105, + 626, + 506, + 672 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 676, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "Discussion. 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ParameterFB15k-237WN18RRNELL-995
# Anchors, |A|
# Anchors per node,k=
Relational context, m1241414134161416
Vocabulary dim, d100100100
Batch size512512512
Learning rate0.00010.00010.0001
Num negatives323232
Epochs2500 V1|2000 rest590|2000|210丨20002000
NodePiece encoderMLPMLPMLPIMLPIMLP|Trf
NodePiece encoder dim200200200
NodePiece encoder layers222
NodePiece encoder dropout0.10.10.1
CompGCNlayers3316161103141313
CompGCN attentionyesyesyes
CompGCN dropout0.10.10.210.110.110.1
Loss functionNSSALNSSALNSSAL
Margin25115115125151151512015|20|30120
Training time, hours2141161191118|41106151818
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ParameterFB15k-237WN18RRNELL-995
# Anchors, |A|
# Anchors per node,k=
Relational context, m1241414134161416
Vocabulary dim, d100100100
Batch size512512512
Learning rate0.00010.00010.0001
Num negatives323232
Epochs2500 V1|2000 rest590|2000|210丨20002000
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NodePiece encoder dim200200200
NodePiece encoder layers222
NodePiece encoder dropout0.10.10.1
CompGCNlayers3316161103141313
CompGCN attentionyesyesyes
CompGCN dropout0.10.10.210.110.110.1
Loss functionNSSALNSSALNSSAL
Margin25115115125151151512015|20|30120
Training time, hours2141161191118|41106151818
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ClassMethodFB15k-237WN18RRNELL-995
V1V2V3V4v1V2V3V4V1V2V3V4
PathNeural LP †0.5290.5890.5290.5590.7440.6890.4620.6710.4080.7870.8270.806
DRUM+0.5290.5870.5290.5590.7440.6890.4620.6710.1940.7860.8270.806
RuleN†0.4980.7780.8770.8560.8090.7820.5340.7160.5350.8180.7730.614
GNNGraIL +0.6420.8180.8280.8930.8250.7870.5840.7340.5950.9330.9140.732
NBFNet0.8340.9490.9510.9600.9480.9050.8930.890-、--
NP + CompGCN0.8730.9390.9440.9490.8300.8860.7850.8070.8900.9010.9360.893
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WD50K (5%labeled)WD50K(10%labeled)
|VI#P (M)ROC-AUCPRC-AUCHard AccROC-AUCPRC-AUCHard Acc
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oFB15k-2370YAGO 3-10 (117k)
|VI#P (M)MRRH@10%|VI#P (M)MRRH@10%
oDistMult-ERAvg11k + 0.5k2.40.256t0.420t100117k + 7423.40OM0OM-
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DatasetRelationsTrainValidationTest
EntityQueryFactsEntityQueryFactEntityQueryFacts
FB15k-237v11832.0004,2454,2451,5002061,9931,5002051,993
v22033,0009,7399,7392,0004694,1452.0004784,145
v32184,00017,98617,9863.0008667,4063,0008657,406
Vv42225,00027,20327,2033,5001,41611,7143,5001,42411,714
WN18RRv192,7465,4105,4109221851,6189221881,618
v2106,95415,26215,2622,9234114,0112.9234414,011
v31112.07825,90125,9015,0845386,3275.0846056.327
V493,8617,9407,9407,2081,39412,3347,2081,42912.334
NELL-995v1143,1034,6874,687225101833225100833
v2882,5648,2198,2194,9374594,5864,9374764,586
v31424,64716,39316,3934,9218118.0484,9218098.048
v4772,0927,5467,5463,2947167,0733,2947317,073
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DatasetTaskNodesRelationsEdgesTrainValidationTest
FB15k-237(Toutanova & Chen,2015)LP,RP14,505237310.079272,11517,52620.438
WN18RR (Dettmers et al.,2018)LP,RP40,5591192.58386.83528242924
CoDEx-Large (Safavi & Koutra,2020)LP77,95169612,437551,19330,62230,622
YAGO 3-10 (Mahdisoltani et al., 2015)LP,RP123,143371,089,0001,079,04049784982
OGB WikiKG 2 (Hu et al.,2020)LP2,500,60453517,137,18116,109,182429,456598,543
WD50KNC46,164526222.5634600 (N)4600 (N)4600 (N)
oFB15k-237 (Albooyeh et al.,2020)OOS-LP11k/1395/1395234292,173193,49044,60154,082
oYAGO 3-10OOS-LP117k/2960/2959371,086,416988,12447,11251,180
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ParameterFB15k-237WN18RRCoDEx-LYAGO 3-10OGB WikiKG 2
# Anchors, |A|100050070001000020000
# Anchors per node,k2050202020
Relational context, m1546512
Vocabulary dim, d200200200200200
Batch size512512256512512
Learning rate0.00050.00050.00050.000250.0001
Epochs400600120600300k (steps)
Encoder typeMLPMLPMLPMLPMLP
Encoder dim400400400400400
Encoder layers22222
Encoder dropout0.10.10.10.10.1
Loss functionBCENSSALBCENSSALNSSAL
Margin-15-5050
# Negative samples=20110128
Label smoothing0.410.311
Training time, hours75.5262311
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ParameterFB15k-237WN18RRCoDEx-LYAGO 3-10
Embedding dim, d200010001000 /501000 /40
Batch size1024512512/5121024 /512
Loss functionNSSALNSSALNSSALNSSAL
Margin9625/924/15
# Negative samples2561024100 /100400/100
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ParameterNodePiece + CompGCNCompGCNMLP
# Anchors, |Al50=
# Anchors per node, k10=
Relational context,m5=1
Vocabulary dim, d100100100
Batch size512512512
Learning rate0.0010.0010.001
Epochs400040004000
NodePiece encoderMLP--
NodePiece encoder dim2001
NodePiece encoder layers2
NodePiece encoder dropout0.1==
GNN (MLP) layers333
GNN (MLP) dropout0.50.50.5
Loss functionBCEBCEBCE
Label smoothing0.10.10.1
Training time, hours14226
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NodePiece+RotatERotatE
ParameterFB15k-237WN18RRYAGO 3-10FB15k-237WN18RRYAGO 3-10
Batch size512512512512512512
Epochs201507150150150
Loss functionNSSALNSSALNSSALNSSALNSSALNSSAL
Margin151225935
# Negative samples202020202020
Training time,min253025281057
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ParameterNodePiece+DistMultoDistMult
oFB15k-237oYAGO 3-10
# Anchors, |A|100010000
# Anchors per node,k2020=
Relational context, m155=
Vocabulary dim,d200200200
Batch size2562561000
Learning rate0.00050.00050.01
Epochs40401000
NodePiece encoderTransformerTransformer-
NodePiece encoder dim512512■
NodePiece encoder layers22
NodePiece encoder dropout0.10.1
Loss functionSoftplusSoftplusSoftplus
# Negative samples551
Training time,hours28
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ParameterFB15k-237WN18RRNELL-995
# Anchors, |A|
# Anchors per node,k=
Relational context, m1241414134161416
Vocabulary dim, d100100100
Batch size512512512
Learning rate0.00010.00010.0001
Num negatives323232
Epochs2500 V1|2000 rest590|2000|210丨20002000
NodePiece encoderMLPMLPMLPIMLPIMLP|Trf
NodePiece encoder dim200200200
NodePiece encoder layers222
NodePiece encoder dropout0.10.10.1
CompGCNlayers3316161103141313
CompGCN attentionyesyesyes
CompGCN dropout0.10.10.210.110.110.1
Loss functionNSSALNSSALNSSAL
Margin25115115125151151512015|20|30120
Training time, hours2141161191118|41106151818
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